How to Think Like a Computer Scientist: Learning

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How to Think Like a Computer
Scientist: Learning with Python 3
Documentation
Release 3rd Edition
Peter Wentworth, Jeffrey Elkner,
Allen B. Downey and Chris Meyers
Apr 26, 2017
Contents
1
The way of the program
3
2
Variables, expressions and statements
13
3
Program Flow
29
4
Functions
77
5
Data Types
109
6
Numpy
159
7
Files
167
8
Modules
175
9
More datatypes
189
10 Recursion
193
11 Classes and Objects
209
12 Exceptions
255
13 Fitting
261
14 PyGame
267
15 Copyright Notice
291
16 Contributions
293
A Modules
297
B More datatypes
311
i
C Recursion
315
D Classes and Objects
331
E Exceptions
377
F Fitting
383
G PyGame
389
H Plotting data with matplotlib
413
ii
How to Think Like a Computer Scientist: Learning with Python 3
Documentation, Release 3rd Edition
3rd Edition (Using Python 3.x)
by Jeffrey Elkner, Peter Wentworth, Allen B. Downey, and Chris Meyers
illustrated by Dario Mitchell
• Copyright Notice
• Contributor List
• Chapter 1 The way of the program
• Chapter 2 Variables, expressions, and statements
• Chapter 3 Program Flow
• Chapter 4 Functions
• Chapter 5 Datatypes
• Chapter 6 Numpy
• Chapter 7 File I/O
• Appendix A Writing Your Own Modules
• Appendix B More datatypes
• Appendix C Recursion
• Appendix D Object Oriented Programming
• Appendix E Exceptions
• Appendix F Fitting and Scientific Data Handling
• Appendix G PyGame
• Appendix H Plotting with matplotlib
• GNU Free Document License
Contents
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Contents
CHAPTER 1
The way of the program
The goal of this book is to teach you to think like a computer scientist. This way of thinking
combines some of the best features of mathematics, engineering, and natural science. Like
mathematicians, computer scientists use formal languages to denote ideas (specifically computations). Like engineers, they design things, assembling components into systems and evaluating tradeoffs among alternatives. Like scientists, they observe the behavior of complex systems,
form hypotheses, and test predictions.
The single most important skill for a computer scientist is problem solving. Problem solving
means the ability to formulate problems, think creatively about solutions, and express a solution
clearly and accurately. As it turns out, the process of learning to program is an excellent
opportunity to practice problem-solving skills. That’s why this chapter is called, The way of
the program.
On one level, you will be learning to program, a useful skill by itself. On another level, you
will use programming as a means to an end. As we go along, that end will become clearer.
The Python programming language
The programming language you will be learning is Python. Python is an example of a highlevel language; other high-level languages you might have heard of are C++, PHP, Pascal, C#,
and Java.
As you might infer from the name high-level language, there are also low-level languages,
sometimes referred to as machine languages or assembly languages. Loosely speaking, computers can only execute programs written in low-level languages. Thus, programs written in a
high-level language have to be translated into something more suitable before they can run.
Almost all programs are written in high-level languages because of their advantages. It is much
easier to program in a high-level language so programs take less time to write, they are shorter
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and easier to read, and they are more likely to be correct. Second, high-level languages are
portable, meaning that they can run on different kinds of computers with few or no modifications.
The engine that translates and runs Python is called the Python Interpreter: There are two
ways to use it: immediate mode and script mode. In immediate mode, you type Python expressions into the Python Interpreter window, and the interpreter immediately shows the result:
The >>> is called the Python prompt. The interpreter uses the prompt to indicate that it is
ready for instructions. We typed 2 + 2, and the interpreter evaluated our expression, and
replied 4, and on the next line it gave a new prompt, indicating that it is ready for more input.
Alternatively, you can write a program in a file and use the interpreter to execute the contents
of the file. Such a file is called a script. Scripts have the advantage that they can be saved to
disk, printed, and so on.
Working directly in the interpreter is convenient for testing short bits of code because you get
immediate feedback. Think of it as scratch paper used to help you work out problems. Anything
longer than a few lines should be put into a script.
What is a program?
A program is a sequence of instructions that specifies how to perform a computation. The
computation might be something mathematical, such as solving a system of equations or finding
the roots of a polynomial, but it can also be a symbolic computation, such as searching and
replacing text in a document or (strangely enough) compiling a program.
The details look different in different languages, but a few basic instructions appear in just
about every language:
input Get data from the keyboard, a file, or some other device such as a sensor.
output Display data on the screen or send data to a file or other device such as a motor.
math Perform basic mathematical operations like addition and multiplication.
conditional execution Check for certain conditions and execute the appropriate sequence of
statements.
repetition Perform some action repeatedly, usually with some variation.
Believe it or not, that’s pretty much all there is to it. Every program you’ve ever used, no matter
how complicated, is made up of instructions that look more or less like these. Thus, we can
describe programming as the process of breaking a large, complex task into smaller and smaller
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subtasks until the subtasks are simple enough to be performed with sequences of these basic
instructions.
That may be a little vague, but we will come back to this topic later when we talk about algorithms.
What is debugging?
Programming is a complex process, and because it is done by human beings, it often leads to
errors. Programming errors are called bugs and the process of tracking them down and correcting them is called debugging. Use of the term bug to describe small engineering difficulties
dates back to at least 1889, when Thomas Edison had a bug with his phonograph.
Three kinds of errors can occur in a program: syntax errors, runtime errors, and semantic errors.
It is useful to distinguish between them in order to track them down more quickly.
Syntax errors
Python can only execute a program if the program is syntactically correct; otherwise, the process fails and returns an error message. Syntax refers to the structure of a program and the
rules about that structure. For example, in English, a sentence must begin with a capital letter
and end with a period. this sentence contains a syntax error. So does this one
For most readers, a few syntax errors are not a significant problem, which is why we can read
the poetry of E. E. Cummings without problems. Python is not so forgiving. If there is a single
syntax error anywhere in your program, Python will display an error message and quit, and you
will not be able to run your program. During the first few weeks of your programming career,
you will probably spend a lot of time tracking down syntax errors. As you gain experience,
though, you will make fewer errors and find them faster.
Runtime errors
The second type of error is a runtime error, so called because the error does not appear until
you run the program. These errors are also called exceptions because they usually indicate that
something exceptional (and bad) has happened.
Runtime errors are rare in the simple programs you will see in the first few chapters, so it might
be a while before you encounter one.
Semantic errors
The third type of error is the semantic error. If there is a semantic error in your program, it
will run successfully, in the sense that the computer will not generate any error messages, but
1.3. What is debugging?
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it will not do the right thing. It will do something else. Specifically, it will do what you told it
to do.
The problem is that the program you wrote is not the program you wanted to write. The
meaning of the program (its semantics) is wrong. Identifying semantic errors can be tricky
because it requires you to work backward by looking at the output of the program and trying to
figure out what it is doing.
Experimental debugging
One of the most important skills you will acquire is debugging. Although it can be frustrating,
debugging is one of the most intellectually rich, challenging, and interesting parts of programming.
In some ways, debugging is like detective work. You are confronted with clues, and you have
to infer the processes and events that led to the results you see.
Debugging is also like an experimental science. Once you have an idea what is going wrong,
you modify your program and try again. If your hypothesis was correct, then you can predict the
result of the modification, and you take a step closer to a working program. If your hypothesis
was wrong, you have to come up with a new one. As Sherlock Holmes pointed out, When you
have eliminated the impossible, whatever remains, however improbable, must be the truth. (A.
Conan Doyle, The Sign of Four)
For some people, programming and debugging are the same thing. That is, programming is
the process of gradually debugging a program until it does what you want. The idea is that
you should start with a program that does something and make small modifications, debugging
them as you go, so that you always have a working program.
For example, Linux is an operating system kernel that contains millions of lines of code, but
it started out as a simple program Linus Torvalds used to explore the Intel 80386 chip. According to Larry Greenfield, one of Linus’s earlier projects was a program that would switch
between displaying AAAA and BBBB. This later evolved to Linux (The Linux Users’ Guide
Beta Version 1).
Later chapters will make more suggestions about debugging and other programming practices.
Formal and natural languages
Natural languages are the languages that people speak, such as English, Spanish, and French.
They were not designed by people (although people try to impose some order on them); they
evolved naturally.
Formal languages are languages that are designed by people for specific applications. For
example, the notation that mathematicians use is a formal language that is particularly good
at denoting relationships among numbers and symbols. Chemists use a formal language to
represent the chemical structure of molecules. And most importantly:
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Programming languages are formal languages that have been designed to express
computations.
Formal languages tend to have strict rules about syntax. For example, 3+3=6 is a syntactically
correct mathematical statement, but 3=+6$ is not. H2 O is a syntactically correct chemical
name, but 2 Zz is not.
Syntax rules come in two flavors, pertaining to tokens and structure. Tokens are the basic elements of the language, such as words, numbers, parentheses, commas, and so on. In Python, a
statement like print("Happy New Year for ",2013) has 6 tokens: a function name,
an open parenthesis (round bracket), a string, a comma, a number, and a close parenthesis.
It is possible to make errors in the way one constructs tokens. One of the problems with 3=+6$
is that $ is not a legal token in mathematics (at least as far as we know). Similarly, 2 Zz is not a
legal token in chemistry notation because there is no element with the abbreviation Zz.
The second type of syntax rule pertains to the structure of a statement— that is, the way the
tokens are arranged. The statement 3=+6$ is structurally illegal because you can’t place a plus
sign immediately after an equal sign. Similarly, molecular formulas have to have subscripts
after the element name, not before. And in our Python example, if we omitted the comma,
or if we changed the two parentheses around to say print)"Happy New Year for ",
2013( our statement would still have six legal and valid tokens, but the structure is illegal.
When you read a sentence in English or a statement in a formal language, you have to figure
out what the structure of the sentence is (although in a natural language you do this subconsciously). This process is called parsing.
For example, when you hear the sentence, “The other shoe fell”, you understand that the other
shoe is the subject and fell is the verb. Once you have parsed a sentence, you can figure out
what it means, or the semantics of the sentence. Assuming that you know what a shoe is and
what it means to fall, you will understand the general implication of this sentence.
Although formal and natural languages have many features in common — tokens, structure,
syntax, and semantics — there are many differences:
ambiguity Natural languages are full of ambiguity, which people deal with by using contextual clues and other information. Formal languages are designed to be nearly or completely unambiguous, which means that any statement has exactly one meaning, regardless of context.
redundancy In order to make up for ambiguity and reduce misunderstandings, natural languages employ lots of redundancy. As a result, they are often verbose. Formal languages
are less redundant and more concise.
literalness Formal languages mean exactly what they say. On the other hand, natural languages are full of idiom and metaphor. If someone says, “The other shoe fell”, there is
probably no shoe and nothing falling. You’ll need to find the original joke to understand
the idiomatic meaning of the other shoe falling. Yahoo! Answers thinks it knows!
People who grow up speaking a natural language—everyone—often have a hard time adjusting
to formal languages. In some ways, the difference between formal and natural language is like
the difference between poetry and prose, but more so:
poetry Words are used for their sounds as well as for their meaning, and the whole poem
1.8. Formal and natural languages
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together creates an effect or emotional response. Ambiguity is not only common but
often deliberate.
prose The literal meaning of words is more important, and the structure contributes more
meaning. Prose is more amenable to analysis than poetry but still often ambiguous.
program The meaning of a computer program is unambiguous and literal, and can be understood entirely by analysis of the tokens and structure.
Here are some suggestions for reading programs (and other formal languages). First, remember
that formal languages are much more dense than natural languages, so it takes longer to read
them. Also, the structure is very important, so it is usually not a good idea to read from top to
bottom, left to right. Instead, learn to parse the program in your head, identifying the tokens
and interpreting the structure. Finally, the details matter. Little things like spelling errors and
bad punctuation, which you can get away with in natural languages, can make a big difference
in a formal language.
The first program
Traditionally, the first program written in a new language is called Hello, World! because all it
does is display the words, Hello, World! In Python, the script looks like this: (For scripts, we’ll
show line numbers to the left of the Python statements.)
1
print("Hello, World!")
This is an example of using the print function, which doesn’t actually print anything on paper.
It displays a value on the screen. In this case, the result shown is
1
Hello, World!
The quotation marks in the program mark the beginning and end of the value; they don’t appear
in the result.
Some people judge the quality of a programming language by the simplicity of the Hello,
World! program. By this standard, Python does about as well as possible.
Comments
As programs get bigger and more complicated, they get more difficult to read. Formal languages are dense, and it is often difficult to look at a piece of code and figure out what it is
doing, or why.
For this reason, it is a good idea to add notes to your programs to explain in natural language
what the program is doing.
A comment in a computer program is text that is intended only for the human reader — it is
completely ignored by the interpreter.
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In Python, the # token starts a comment. The rest of the line is ignored. Here is a new version
of Hello, World!.
1
2
3
4
5
#--------------------------------------------------# This demo program shows off how elegant Python is!
# Written by Joe Soap, December 2010.
# Anyone may freely copy or modify this program.
#---------------------------------------------------
6
7
print("Hello, World!")
# Isn't this easy!
You’ll also notice that we’ve left a blank line in the program. Blank lines are also ignored by
the interpreter, but comments and blank lines can make your programs much easier for humans
to parse. Use them liberally!
Glossary
algorithm A set of specific steps for solving a category of problems.
bug An error in a program.
comment Information in a program that is meant for other programmers (or anyone reading
the source code) and has no effect on the execution of the program.
debugging The process of finding and removing any of the three kinds of programming errors.
exception Another name for a runtime error.
formal language Any one of the languages that people have designed for specific purposes,
such as representing mathematical ideas or computer programs; all programming languages are formal languages.
high-level language A programming language like Python that is designed to be easy for
humans to read and write.
immediate mode A style of using Python where we type expressions at the command prompt,
and the results are shown immediately. Contrast with script, and see the entry under
Python shell.
interpreter The engine that executes your Python scripts or expressions.
low-level language A programming language that is designed to be easy for a computer to
execute; also called machine language or assembly language.
natural language Any one of the languages that people speak that evolved naturally.
object code The output of the compiler after it translates the program.
parse To examine a program and analyze the syntactic structure.
portability A property of a program that can run on more than one kind of computer.
print function A function used in a program or script that causes the Python interpreter to
display a value on its output device.
1.11. Glossary
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problem solving The process of formulating a problem, finding a solution, and expressing the
solution.
program a sequence of instructions that specifies to a computer actions and computations to
be performed.
Python shell An interactive user interface to the Python interpreter. The user of a Python shell
types commands at the prompt (>>>), and presses the return key to send these commands
immediately to the interpreter for processing. The word shell comes from Unix. In the
PyScripter used in this RLE version of the book, the Interpreter Window is where we’d
do the immediate mode interaction.
runtime error An error that does not occur until the program has started to execute but that
prevents the program from continuing.
script A program stored in a file (usually one that will be interpreted).
semantic error An error in a program that makes it do something other than what the programmer intended.
semantics The meaning of a program.
source code A program in a high-level language before being compiled.
syntax The structure of a program.
syntax error An error in a program that makes it impossible to parse — and therefore impossible to interpret.
token One of the basic elements of the syntactic structure of a program, analogous to a word
in a natural language.
Exercises
1. Write an English (or Dutch!) sentence with understandable semantics but incorrect syntax. Write another English (or Dutch!) sentence which has correct syntax but has semantic errors.
2. Using the Python interpreter, type 1 + 2 and then hit return. Python evaluates this
expression, displays the result, and then shows another prompt. * is the multiplication
operator, and ** is the exponentiation operator. Experiment by entering different expressions and recording what is displayed by the Python interpreter.
3. Type 1 2 and then hit return. Python tries to evaluate the expression, but it can’t because
the expression is not syntactically legal. Instead, it shows the error message:
File "<interactive input>", line 1
1 2
^
SyntaxError: invalid syntax
In many cases, Python indicates where the syntax error occurred, but it is not always
right, and it doesn’t give you much information about what is wrong.
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So, for the most part, the burden is on you to learn the syntax rules.
In this case, Python is complaining because there is no operator between the numbers.
See if you can find a few more examples of things that will produce error messages when
you enter them at the Python prompt. Write down what you enter at the prompt and the
last line of the error message that Python reports back to you.
4. Type print("hello"). Python executes this, which has the effect of printing the
letters h-e-l-l-o. Notice that the quotation marks that you used to enclose the string are
not part of the output. Now type "hello" and describe your result. Make notes of when
you see the quotation marks and when you don’t.
5. Type cheese without the quotation marks. The output will look something like this:
Traceback (most recent call last):
File "<interactive input>", line 1, in ?
NameError: name 'cheese' is not defined
This is a run-time error; specifically, it is a NameError, and even more specifically, it is
an error because the name cheese is not defined. If you don’t know what that means yet,
you will soon.
6. Type 6 + 4 * 9 at the Python prompt and hit enter. Record what happens.
Now create a Python script with the following contents:
1
6 + 4 * 9
What happens when you run this script? Now change the script contents to:
1
print(6 + 4 * 9)
and run it again.
What happened this time?
Whenever an expression is typed at the Python prompt, it is evaluated and the result
is automatically shown on the line below. (Like on your calculator, if you type this
expression you’ll get the result 42.)
A script is different, however. Evaluations of expressions are not automatically displayed,
so it is necessary to use the print function to make the answer show up.
It is hardly ever necessary to use the print function in immediate mode at the command
prompt.
1.12. Exercises
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Chapter 1. The way of the program
CHAPTER 2
Variables, expressions and statements
Values and data types
A value is one of the fundamental things — like a letter or a number — that a program manipulates. The values we have seen so far are 4 (the result when we added 2 + 2), and "Hello,
World!".
These values are classified into different classes, or data types: 4 is an integer, and "Hello,
World!" is a string, so-called because it contains a string of letters. You (and the interpreter)
can identify strings because they are enclosed in quotation marks.
If you are not sure what class a value falls into, Python has a function called type which can
tell you.
>>> type("Hello, World!")
<class 'str'>
>>> type(17)
<class 'int'>
Not surprisingly, strings belong to the class str and integers belong to the class int. Less
obviously, numbers with a decimal point belong to a class called float, because these numbers
are represented in a format called floating-point. At this stage, you can treat the words class
and type interchangeably. We’ll come back to a deeper understanding of what a class is in later
chapters.
>>> type(3.2)
<class 'float'>
What about values like "17" and "3.2"? They look like numbers, but they are in quotation
marks like strings.
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>>> type("17")
<class 'str'>
>>> type("3.2")
<class 'str'>
They’re strings!
Strings in Python can be enclosed in either single quotes (') or double quotes ("), or three of
each (''' or """)
>>> type('This is a string.')
<class 'str'>
>>> type("And so is this.")
<class 'str'>
>>> type("""and this.""")
<class 'str'>
>>> type('''and even this...''')
<class 'str'>
Double quoted strings can contain single quotes inside them, as in "Bruce's beard", and
single quoted strings can have double quotes inside them, as in 'The knights who say
"Ni!"'.
Strings enclosed with three occurrences of either quote symbol are called triple quoted strings.
They can contain either single or double quotes:
>>> print('''"Oh no", she exclaimed, "Ben's bike is broken!"
˓→''')
"Oh no", she exclaimed, "Ben's bike is broken!"
>>>
Triple quoted strings can even span multiple lines:
>>> message = """This message will
... span several
... lines."""
>>> print(message)
This message will
span several
lines.
>>>
Python doesn’t care whether you use single or double quotes or the three-of-a-kind quotes to
surround your strings: once it has parsed the text of your program or command, the way it
stores the value is identical in all cases, and the surrounding quotes are not part of the value.
But when the interpreter wants to display a string, it has to decide which quotes to use to make
it look like a string.
>>> 'This is a string.'
'This is a string.'
>>> """And so is this."""
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'And so is this.'
So the Python language designers usually chose to surround their strings by single quotes.
What do think would happen if the string already contained single quotes?
When you type a large integer, you might be tempted to use commas between groups of three
digits, as in 42,000. The same goes for entering Dutch-style floating point numbers using
a comma instead of a decimal dot. This is not a legal integer in Python, but it does mean
something else, which is legal:
>>> 42000
42000
>>> 42,000
(42, 0)
Well, that’s not what we expected at all! Because of the comma, Python chose to treat this as
a pair of values. We’ll come back to learn about pairs later. But, for the moment, remember
not to put commas or spaces in your integers, no matter how big they are. Also revisit what we
said in the previous chapter: formal languages are strict, the notation is concise, and even the
smallest change might mean something quite different from what you intended.
Variables
One of the most powerful features of a programming language is the ability to manipulate
variables. A variable is a name that refers to a value.
The assignment statement gives a value to a variable:
>>> message = "What's up, Doc?"
>>> n = 17
>>> pi = 3.14159
This example makes three assignments. The first assigns the string value "What's up,
Doc?" to a variable named message. The second gives the integer 17 to n, and the third
assigns the floating-point number 3.14159 to a variable called pi.
The assignment token, =, should not be confused with equals, which uses the token ==. The
assignment statement binds a name, on the left-hand side of the operator, to a value, on the
right-hand side. Basically, an assignment is an order, and the equals operator can be read as a
question mark. This is why you will get an error if you enter:
>>> 17 = n
File "<interactive input>", line 1
SyntaxError: can't assign to literal
Tip: When reading or writing code, say to yourself “n is assigned 17” or “n gets
the value 17”. Don’t say “n equals 17”.
2.2. Variables
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A common way to represent variables on paper is to write the name with an arrow pointing
to the variable’s value. This kind of figure is called a state snapshot because it shows what
state each of the variables is in at a particular instant in time. (Think of it as the variable’s
state of mind). Some editors and programming environments do this for you, and allow you to
click through the state of the program saving you some paper. This diagram shows the result of
executing the assignment statements:
If you ask the interpreter to evaluate a variable, it will produce the value that is currently linked
to the variable:
>>> message
'What's up, Doc?'
>>> n
17
>>> pi
3.14159
We use variables in a program to “remember” things, perhaps the current score at the football
game. But variables are variable. This means they can change over time, just like the scoreboard at a football game. You can assign a value to a variable, and later assign a different value
to the same variable. (This is different from maths. In maths, if you give ‘x‘ the value 3, it
cannot change to link to a different value half-way through your calculations!)
>>> day = "Thursday"
>>> day
'Thursday'
>>> day = "Friday"
>>> day
'Friday'
>>> day = 21
>>> day
21
You’ll notice we changed the value of day three times, and on the third assignment we even
made it refer to a value that was of a different type.
A great deal of programming is about having the computer remember things, e.g. The number
of missed calls on your phone, and then arranging to update or change the variable when you
miss another call.
Variable names and keywords
Variable names can be arbitrarily long. They can contain both letters and digits, but they
have to begin with a letter or an underscore. Although it is legal to use uppercase letters, by
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convention we don’t. If you do, remember that case matters. Bruce and bruce are different
variables.
The underscore character ( _) can appear in a name. It is often used in names with multiple
words, such as my_name or price_of_tea_in_china.
There are some situations in which names beginning with an underscore have special meaning,
so a safe rule for beginners is to start all names with a letter.
If you give a variable an illegal name, you get a syntax error:
>>> 76trombones = "big parade"
SyntaxError: invalid syntax
>>> more$ = 1000000
SyntaxError: invalid syntax
>>> class = "Computer Science 101"
SyntaxError: invalid syntax
76trombones is illegal because it does not begin with a letter. more$ is illegal because it
contains an illegal character, the dollar sign. But what’s wrong with class?
It turns out that class is one of the Python keywords. Keywords define the language’s syntax
rules and structure, and they cannot be used as variable names.
Python has thirty-something keywords (and every now and again improvements to Python introduce or eliminate one or two):
and
def
finally
in
pass
yield
as
del
for
is
raise
True
assert
elif
from
lambda
return
False
break
else
global
nonlocal
try
None
class
except
if
not
while
continue
exec
import
or
with
You might want to keep this list handy. If the interpreter complains about one of your variable
names and you don’t know why, see if it is on this list.
Programmers generally choose names for their variables that are meaningful to the human
readers of the program — they help the programmer document, or remember, what the variable
is used for.
Caution: Beginners sometimes confuse “meaningful to the human readers” with “meaningful to the computer”. So they’ll wrongly think that because they’ve called some variable
average or pi, it will somehow magically calculate an average, or magically know that
the variable pi should have a value like 3.14159. No! The computer doesn’t understand
what you intend the variable to mean.
So you’ll find some instructors who deliberately don’t choose meaningful names when they
teach beginners — not because we don’t think it is a good habit, but because we’re trying
to reinforce the message that you — the programmer — must write the program code to
calculate the average, and you must write an assignment statement to give the variable pi
the value you want it to have.
2.3. Variable names and keywords
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e = 3.1415
ray = 10
size = e * ray ** 2
pi = 3.1415
radius = 10
area = pi * radius ** 2
The above two snippets do exactly the same thing, but the bottom one uses the right kind
of variable names. For the computer there is no difference at all, but for a human, using
the names and letters that are part of the conventional way of writing things make all the
difference in the world. Using e instead of pi completely confuses people, while computers
will just perform the calculation!
Statements
A statement is an instruction that the Python interpreter can execute. We have only seen the
assignment statement so far. Some other kinds of statements that we’ll see shortly are while
statements, for statements, if statements, and import statements. (There are other kinds
too!)
When you type a statement on the command line, Python executes it. Statements don’t produce
any result.
Evaluating expressions
An expression is a combination of values, variables, operators, and calls to functions. If you
type an expression at the Python prompt, the interpreter evaluates it and displays the result:
>>> 1 + 1
2
>>> len("hello")
5
In this example len is a built-in Python function that returns the number of characters in a
string. We’ve previously seen the print and the type functions, so this is our third example
of a function!
The evaluation of an expression produces a value, which is why expressions can appear on the
right hand side of assignment statements. A value all by itself is a simple expression, and so is
a variable.
>>> 17
17
>>> y = 3.14
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>>> x = len("hello")
>>> x
5
>>> y
3.14
Operators and operands
Operators are special tokens that represent computations like addition, multiplication and division. The values the operator uses are called operands.
The following are all legal Python expressions whose meaning is more or less clear:
20+32
hour-1
hour*60+minute
minute/60
5**2
(5+9)*(15-7)
The tokens +, -, and *, and the use of parenthesis for grouping, mean in Python what they
mean in mathematics. The asterisk (*) is the token for multiplication, and ** is the token for
exponentiation.
>>> 2 ** 3
8
>>> 3 ** 2
9
When a variable name appears in the place of an operand, it is replaced with its value before
the operation is performed.
Addition, subtraction, multiplication, and exponentiation all do what you expect.
Example: so let us convert 645 minutes into hours:
>>> minutes = 645
>>> hours = minutes / 60
>>> hours
10.75
Oops! In Python 3, the division operator / always yields a floating point result. What we might
have wanted to know was how many whole hours there are, and how many minutes remain.
Python gives us two different flavors of the division operator. The second, called floor division
uses the token //. Its result is always a whole number — and if it has to adjust the number it
always moves it to the left on the number line. So 6 // 4 yields 1, but -6 // 4 might surprise you!
>>> 7 / 4
1.75
>>> 7 // 4
1
>>> minutes = 645
>>> hours = minutes // 60
2.6. Operators and operands
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>>> hours
10
Take care that you choose the correct flavor of the division operator. If you’re working with expressions where you need floating point values, use the division operator that does the division
accurately.
Type converter functions
Here we’ll look at three more Python functions, int, float and str, which will (attempt
to) convert their arguments into types int, float and str respectively. We call these type
converter functions.
The int function can take a floating point number or a string, and turn it into an int. For
floating point numbers, it discards the decimal portion of the number — a process we call
truncation towards zero on the number line. Let us see this in action:
>>> int(3.14)
3
>>> int(3.9999)
˓→closest int!
3
>>> int(3.0)
3
>>> int(-3.999)
˓→closer to zero
-3
>>> int(minutes / 60)
10
>>> int("2345")
˓→int
2345
>>> int(17)
˓→already an int
17
>>> int("23 bottles")
# This doesn't round to the
# Note that the result is
# Parse a string to produce an
# It even works if arg is
This last case doesn’t look like a number — what do we expect?
Traceback (most recent call last):
File "<interactive input>", line 1, in <module>
ValueError: invalid literal for int() with base 10: '23
˓→bottles'
The type converter float can turn an integer, a float, or a syntactically legal string into a float:
>>> float(17)
17.0
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>>> float("123.45")
123.45
The type converter str turns its argument into a string:
>>> str(17)
'17'
>>> str(123.45)
'123.45'
Order of operations
When more than one operator appears in an expression, the order of evaluation depends on the
rules of precedence. Python follows the same precedence rules for its mathematical operators
that mathematics does. The acronym PEMDAS is a useful way to remember the order of
operations:
1. Parentheses have the highest precedence and can be used to force an expression to evaluate in the order you want. Since expressions in parentheses are evaluated first, 2 *
(3-1) is 4, and (1+1)**(5-2) is 8. You can also use parentheses to make an expression easier to read, as in (minute * 100) / 60, even though it doesn’t change the
result.
2. Exponentiation has the next highest precedence, so 2**1+1 is 3 and not 4, and 3*1**3
is 3 and not 27.
3. Multiplication and both Division operators have the same precedence, which is higher
than Addition and Subtraction, which also have the same precedence. So 2*3-1 yields
5 rather than 4, and 5-2*2 is 1, not 6.
4. Operators with the same precedence are evaluated from left-to-right. In algebra we say
they are left-associative. So in the expression 6-3+2, the subtraction happens first,
yielding 3. We then add 2 to get the result 5. If the operations had been evaluated from
right to left, the result would have been 6-(3+2), which is 1. (The acronym PEDMAS
could mislead you to thinking that division has higher precedence than multiplication,
and addition is done ahead of subtraction - don’t be misled. Subtraction and addition are
at the same precedence, and the left-to-right rule applies.)
Due to some historical quirk, an exception to the left-to-right left-associative rule is the
exponentiation operator **, so a useful hint is to always use parentheses to force exactly
the order you want when exponentiation is involved:
>>> 2 ** 3 ** 2
˓→done first!
512
>>> (2 ** 3) ** 2
˓→order you want!
64
2.8. Order of operations
# The right-most ** operator gets
# Use parentheses to force the
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The immediate mode command prompt of Python is great for exploring and experimenting
with expressions like this.
Operations on strings
In general, you cannot perform mathematical operations on strings, even if the strings look like
numbers. The following are illegal (assuming that message has type string):
>>>
>>>
>>>
>>>
message - 1
"Hello" / 123
message * "Hello"
"15" + 2
#
#
#
#
Error
Error
Error
Error
Interestingly, the + operator does work with strings, but for strings, the + operator represents
concatenation, not addition. Concatenation means joining the two operands by linking them
end-to-end. For example:
1
2
3
fruit = "banana"
baked_good = " nut bread"
print(fruit + baked_good)
The output of this program is banana nut bread. The space before the word nut is part
of the string, and is necessary to produce the space between the concatenated strings.
The * operator also works on strings; it performs repetition. For example, 'Fun'*3 is
'FunFunFun'. One of the operands has to be a string; the other has to be an integer.
On one hand, this interpretation of + and * makes sense by analogy with addition and multiplication. Just as 4*3 is equivalent to 4+4+4, we expect "Fun"*3 to be the same as
"Fun"+"Fun"+"Fun", and it is. On the other hand, there is a significant way in which
string concatenation and repetition are different from integer addition and multiplication. Can
you think of a property that addition and multiplication have that string concatenation and repetition do not?
Input
There is a built-in function in Python for getting input from the user:
1
name = input("Please enter your name: ")
The user of the program can enter the name and click OK, and when this happens the text that
has been entered is returned from the input function, and in this case assigned to the variable
name.
Even if you asked the user to enter their age, you would get back a string like "17". It would
be your job, as the programmer, to convert that string into a int or a float, using the int or
float converter functions we saw earlier.
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Composition
So far, we have looked at the elements of a program — variables, expressions, statements, and
function calls — in isolation, without talking about how to combine them.
One of the most useful features of programming languages is their ability to take small building
blocks and compose them into larger chunks.
For example, we know how to get the user to enter some input, we know how to convert the
string we get into a float, we know how to write a complex expression, and we know how to
print values. Let’s put these together in a small four-step program that asks the user to input a
value for the radius of a circle, and then computes the area of the circle from the formula
Area = 2
Firstly, we’ll do the four steps one at a time:
1
2
3
4
response = input("What is your radius? ")
r = float(response)
area = 3.14159 * r**2
print("The area is ", area)
Now let’s compose the first two lines into a single line of code, and compose the second two
lines into another line of code.
1
2
r = float( input("What is your radius? ") )
print("The area is ", 3.14159 * r**2)
If we really wanted to be tricky, we could write it all in one statement:
1
print("The area is ", 3.14159*float(input("What is your
˓→radius?"))**2)
Such compact code may not be most understandable for humans, but it does illustrate how we
can compose bigger chunks from our building blocks.
If you’re ever in doubt about whether to compose code or fragment it into smaller steps, try to
make it as simple as you can for the human to follow. My choice would be the first case above,
with four separate steps.
The modulus operator
The modulus operator works on integers (and integer expressions) and gives the remainder
when the first number is divided by the second. In Python, the modulus operator is a percent
sign (%). The syntax is the same as for other operators. It has the same precedence as the
multiplication operator.
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>>>
>>>
2
>>>
>>>
1
q = 7 // 3
print(q)
# This is integer division operator
r = 7 % 3
print(r)
So 7 divided by 3 is 2 with a remainder of 1.
The modulus operator turns out to be surprisingly useful. For example, you can check whether
one number is divisible by another—if x % y is zero, then x is divisible by y.
Also, you can extract the right-most digit or digits from a number. For example, x % 10
yields the right-most digit of x (in base 10). Similarly x % 100 yields the last two digits.
It is also extremely useful for doing conversions, say from seconds, to hours, minutes and
seconds. So let’s write a program to ask the user to enter some seconds, and we’ll convert them
into hours, minutes, and remaining seconds.
1
2
3
4
5
total_secs = int(input("How many seconds, in total?"))
hours = total_secs // 3600
secs_still_remaining = total_secs % 3600
minutes = secs_still_remaining // 60
secs_finally_remaining = secs_still_remaining % 60
6
7
print("Hrs=", hours, "
8
mins=", minutes,
"secs=", secs_finally_remaining)
Glossary
assignment statement A statement that assigns a value to a name (variable). To the left
of the assignment operator, =, is a name. To the right of the assignment token is an
expression which is evaluated by the Python interpreter and then assigned to the name.
The difference between the left and right hand sides of the assignment statement is often
confusing to new programmers. In the following assignment:
number = number + 1
number plays a very different role on each side of the =. On the right it is a value and
makes up part of the expression which will be evaluated by the Python interpreter before
assigning it to the name on the left.
assignment token = is Python’s assignment token. Do not confuse it with equals, which is an
operator for comparing values.
composition The ability to combine simple expressions and statements into compound statements and expressions in order to represent complex computations concisely.
concatenate To join two strings end-to-end.
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data type A set of values. The type of a value determines how it can be used in expressions.
So far, the types you have seen are integers (int), floating-point numbers (float), and
strings (str).
evaluate To simplify an expression by performing the operations in order to yield a single
value.
expression A combination of variables, operators, and values that represents a single result
value.
float A Python data type which stores floating-point numbers. Floating-point numbers are
stored internally in two parts: a base and an exponent. When printed in the standard format, they look like decimal numbers. Beware of rounding errors when you use floats,
and remember that they are only approximate values.
floor division An operator (denoted by the token //) that divides one number by another and
yields an integer, or, if the result is not already an integer, it yields the next smallest
integer.
int A Python data type that holds positive and negative whole numbers.
keyword A reserved word that is used by the compiler to parse program; you cannot use
keywords like if, def, and while as variable names.
modulus operator An operator, denoted with a percent sign ( %), that works on integers and
yields the remainder when one number is divided by another.
operand One of the values on which an operator operates.
operator A special symbol that represents a simple computation like addition, multiplication,
or string concatenation.
rules of precedence The set of rules governing the order in which expressions involving multiple operators and operands are evaluated.
state snapshot A graphical representation of a set of variables and the values to which they
refer, taken at a particular instant during the program’s execution.
statement An instruction that the Python interpreter can execute. So far we have only seen
the assignment statement, but we will soon meet the import statement and the for
statement.
str A Python data type that holds a string of characters.
value A number or string (or other things to be named later) that can be stored in a variable or
computed in an expression.
variable A name that refers to a value.
variable name A name given to a variable. Variable names in Python consist of a sequence of
letters (a..z, A..Z, and _) and digits (0..9) that begins with a letter. In best programming
practice, variable names should be chosen so that they describe their use in the program,
making the program self documenting.
2.13. Glossary
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Exercises
1. Take the sentence: All work and no play makes Jack a dull boy. Store each word in a
separate variable, then print out the sentence on one line using print.
2. Add parenthesis to the expression 6 * 1 - 2 to change its value from 4 to -6.
3. Place a comment before a line of code that previously worked, and record what happens
when you rerun the program.
4. Start the Python interpreter and enter bruce + 4 at the prompt. This will give you an
error:
NameError: name 'bruce' is not defined
Assign a value to bruce so that bruce + 4 evaluates to 10.
5. The formula for computing the final amount if one is earning compound interest is given
on Wikipedia as
Here, P is the principal amount (the amount that the interest is provided on), n the frequency that the interest is paid out (per year), and r the interest rate. The number of
years that the interest is calculated for is t. Write a program that replaces these letters
with something a bit more human-readable, and calculate the interest for some varying
amounts of money at realistic interest rates such as 1%, and -0.05%. When you have that
working, ask the user for the value of some of these variables and do the calculation.
6. Evaluate the following numerical expressions in your head, then use the Python interpreter to check your results:
(a) >>> 5 % 2
(b) >>> 9 % 5
(c) >>> 15 % 12
(d) >>> 12 % 15
(e) >>> 6 % 6
(f) >>> 0 % 7
(g) >>> 7 % 0
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What happened with the last example? Why? If you were able to correctly anticipate the
computer’s response in all but the last one, it is time to move on. If not, take time now
to make up examples of your own. Explore the modulus operator until you are confident
you understand how it works.
7. You look at the clock and it is exactly 2pm. You set an alarm to go off in 51 hours. At
what time does the alarm go off? (Hint: you could count on your fingers, but this is not
what we’re after. If you are tempted to count on your fingers, change the 51 to 5100.)
8. Write a Python program to solve the general version of the above problem. Ask the user
for the time now (in hours), and ask for the number of hours to wait. Your program
should output what the time will be on the clock when the alarm goes off.
2.14. Exercises
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CHAPTER 3
Program Flow
Hello, little turtles!
There are many modules in Python that provide very powerful features that we can use in our
own programs. Some of these can send email, or fetch web pages. The one we’ll look at in this
chapter allows us to create turtles and get them to draw shapes and patterns.
The turtles are fun, but the real purpose of the chapter is to teach ourselves a little more Python,
and to develop our theme of computational thinking, or thinking like a computer scientist. Most
of the Python covered here will be explored in more depth later.
Our first turtle program
Let’s write a couple of lines of Python program to create a new turtle and start drawing a
rectangle. (We’ll call the variable that refers to our first turtle alex, but we can choose another
name if we follow the naming rules from the previous chapter).
1
2
3
import turtle
window = turtle.Screen()
˓→turtles
alex = turtle.Turtle()
˓→alex
# Allows us to use turtles
# Creates a playground for
alex.forward(50)
˓→50 units
alex.left(90)
alex.forward(30)
˓→rectangle
# Tell alex to move forward by
# Create a turtle, assign to
4
5
6
7
# Tell alex to turn by 90 degrees
# Complete the second side of a
8
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window.mainloop()
window
# Wait for user to close
˓→
When we run this program, a new window pops up:
Here are a couple of things we’ll need to understand about this program.
The first line tells Python to load a module named turtle. That module brings us two new
types that we can use: the Turtle type, and the Screen type. The dot notation turtle.
Turtle means “The Turtle type that is defined within the turtle module”. (Remember that
Python is case sensitive, so the module name, with a lowercase t, is different from the type
Turtle.)
We then create and open what it calls a screen (we would prefer to call it a window), which
we assign to variable window. Every window contains a canvas, which is the area inside the
window on which we can draw.
In line 3 we create a turtle. The variable alex is made to refer to this turtle.
So these first three lines have set things up, we’re ready to get our turtle to draw on our canvas.
In lines 5-7, we instruct the object alex to move, and to turn. We do this by invoking, or
activating, alex‘s methods — these are the instructions that all turtles know how to respond
to.
The last line plays a part too: the window variable refers to the window shown above. When
we invoke its mainloop method, it enters a state where it waits for events (like keypresses, or
mouse movement and clicks). The program will terminate when the user closes the window.
An object can have various methods — things it can do — and it can also have attributes
— (sometimes called properties). For example, each turtle has a color attribute. The method
invocation alex.color("red") will make alex red, and drawing will be red too. (Note
the word color is spelled the American way!)
The color of the turtle, the width of its pen, the position of the turtle within the window, which
way it is facing, and so on are all part of its current state. Similarly, the window object has a
background color, and some text in the title bar, and a size and position on the screen. These
are all part of the state of the window object.
Quite a number of methods exist that allow us to modify the turtle and the window objects.
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We’ll just show a couple. In this program we’ve only commented those lines that are different
from the previous example (and we’ve used a different variable name for this turtle):
1
2
3
4
import turtle
window = turtle.Screen()
window.bgcolor("lightgreen")
˓→background color
window.title("Hello, Tess!")
# Set the window
# Set the window title
5
6
7
8
tess = turtle.Turtle()
tess.color("blue")
˓→color
tess.pensize(3)
˓→width
# Tell tess to change her
# Tell tess to set her pen
9
10
11
12
tess.forward(50)
tess.left(120)
tess.forward(50)
13
14
window.mainloop()
When we run this program, this new window pops up, and will remain on the screen until we
close it.
Extend this program ...
1. Modify this program so that before it creates the window, it prompts the user to enter
the desired background color. It should store the user’s responses in a variable, and
modify the color of the window according to the user’s wishes. (Hint: you can find a list
of permitted color names at http://www.tcl.tk/man/tcl8.4/TkCmd/colors.htm. It includes
some quite unusual ones, like “peach puff” and “HotPink”.)
2. Do similar changes to allow the user, at runtime, to set tess‘ color.
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Instances — a herd of turtles
Just like we can have many different integers in a program, we can have many turtles. Each
of them is called an instance. Each instance has its own attributes and methods — so alex
might draw with a thin black pen and be at some position, while tess might be going in her
own direction with a fat pink pen.
1
2
3
4
import turtle
window = turtle.Screen()
˓→its attributes
window.bgcolor("lightgreen")
window.title("Tess & Alex")
# Set up the window and
5
6
tess = turtle.Turtle()
attributes
tess.color("hotpink")
tess.pensize(5)
# Create tess and set some
alex = turtle.Turtle()
# Create alex
tess.forward(80)
triangle
tess.left(120)
tess.forward(80)
tess.left(120)
tess.forward(80)
tess.left(120)
# Make tess draw equilateral
tess.right(180)
tess.forward(80)
˓→origin
# Turn tess around
# Move her away from the
alex.forward(50)
alex.left(90)
alex.forward(50)
alex.left(90)
alex.forward(50)
alex.left(90)
alex.forward(50)
alex.left(90)
# Make alex draw a square
˓→
7
8
9
10
11
12
˓→
13
14
15
16
17
# Complete the triangle
18
19
20
21
22
23
24
25
26
27
28
29
30
31
window.mainloop()
Here is what happens when alex completes his rectangle, and tess completes her triangle:
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Here are some How to think like a computer scientist observations:
• There are 360 degrees in a full circle. If we add up all the turns that a turtle makes, no
matter what steps occurred between the turns, we can easily figure out if they add up to
some multiple of 360. This should convince us that alex is facing in exactly the same
direction as he was when he was first created. (Geometry conventions have 0 degrees
facing East, and that is the case here too!)
• We could have left out the last turn for alex, but that would not have been as satisfying.
If we’re asked to draw a closed shape like a square or a rectangle, it is a good idea
to complete all the turns and to leave the turtle back where it started, facing the same
direction as it started in. This makes reasoning about the program and composing chunks
of code into bigger programs easier for us humans!
• We did the same with tess: she drew her triangle, and turned through a full 360 degrees.
Then we turned her around and moved her aside. Even the blank line 18 is a hint about
how the programmer’s mental chunking is working: in big terms, tess‘ movements
were chunked as “draw the triangle” (lines 12-17) and then “move away from the origin”
(lines 19 and 20).
• One of the key uses for comments is to record our mental chunking, and big ideas.
They’re not always explicit in the code.
• And, uh-huh, two turtles may not be enough for a herd. But the important idea is that
the turtle module gives us a kind of factory that lets us create as many turtles as we need.
Each instance has its own state and behaviour.
The for loop
When we drew the square, it was quite tedious. We had to explicitly repeat the steps of moving
and turning four times. If we were drawing a hexagon, or an octogon, or a polygon with 42
sides, it would have been worse.
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So a basic building block of all programs is to be able to repeat some code, over and over again.
Python’s for loop solves this for us. Let’s say we have some friends, and we’d like to send them
each an email inviting them to our party. We don’t quite know how to send email yet, so for the
moment we’ll just print a message for each friend:
1
2
3
4
for friend in ["Joe", "Zoe", "Zuki", "Thandi", "Paris"]:
invite = "Hi " + friend + ". Please come to my party!"
print(invite)
# more code can follow here ...
When we run this, the output looks like this:
Hi
Hi
Hi
Hi
Hi
Joe. Please come to my party!
Zoe. Please come to my party!
Zuki. Please come to my party!
Thandi. Please come to my party!
Paris. Please come to my party!
• The variable friend in the for statement at line 1 is called the loop variable. We
could have chosen any other variable name instead, such as broccoli: the computer
doesn’t care.
• Lines 2 and 3 are the loop body. The loop body is always indented. The indentation
determines exactly what statements are “in the body of the loop”.
• On each iteration or pass of the loop, first a check is done to see if there are still more
items to be processed. If there are none left (this is called the terminating condition of
the loop), the loop has finished. Program execution continues at the next statement after
the loop body, (e.g. in this case the next statement below the comment in line 4).
• If there are items still to be processed, the loop variable is updated to refer to the next
item in the list. This means, in this case, that the loop body is executed here 7 times, and
each time friend will refer to a different friend.
• At the end of each execution of the body of the loop, Python returns to the for statement,
to see if there are more items to be handled, and to assign the next one to friend.
Flow of Execution of the for loop
As a program executes, the interpreter always keeps track of which statement is about to be
executed. We call this the control flow, of the flow of execution of the program. When
humans execute programs, they often use their finger to point to each statement in turn. So we
could think of control flow as “Python’s moving finger”.
Control flow until now has been strictly top to bottom, one statement at a time. The for loop
changes this.
Flowchart of a for loop
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Control flow is often easy to visualize and understand if we draw a flowchart. This shows the
exact steps and logic of how the for statement executes.
The loop simplifies our turtle program
To draw a square we’d like to do the same thing four times — move the turtle, and turn. We
previously used 8 lines to have alex draw the four sides of a square. This does exactly the
same, but using just three lines:
1
2
3
for i in [0,1,2,3]:
alex.forward(50)
alex.left(90)
Some observations:
• While “saving some lines of code” might be convenient, it is not the big deal here. What
is much more important is that we’ve found a “repeating pattern” of statements, and
reorganized our program to repeat the pattern. Finding the chunks and somehow getting
our programs arranged around those chunks is a vital skill in computational thinking.
• The values [0,1,2,3] were provided to make the loop body execute 4 times. We could
have used any four values, but these are the conventional ones to use. In fact, they are so
popular that Python gives us special built-in range objects:
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1
2
3
4
for i in range(4):
# Executes the body with i = 0, then 1, then 2,
˓→then 3
for x in range(10):
# Sets x to each of ... [0, 1, 2, 3, 4, 5, 6, 7, 8,
˓→9]
• Since we do not need or use the variable i in this case, we could replace it with _,
although this is not important for the program flow, it is good style.
• Computer scientists like to count from 0!
• range can deliver a sequence of values to the loop variable in the for loop. They start
at 0, and in these cases do not include the 4 or the 10.
• Our little trick earlier to make sure that alex did the final turn to complete 360 degrees
has paid off: if we had not done that, then we would not have been able to use a loop
for the fourth side of the square. It would have become a “special case”, different from
the other sides. When possible, we’d much prefer to make our code fit a general pattern,
rather than have to create a special case.
So to repeat something four times, a good Python programmer would do this:
1
2
3
for _ in range(4):
alex.forward(50)
alex.left(90)
By now you should be able to see how to change our previous program so that tess can also
use a for loop to draw her equilateral triangle.
But now, what would happen if we made this change?
1
2
3
4
for color in ["yellow", "red", "purple", "blue"]:
alex.color(color)
alex.forward(50)
alex.left(90)
A variable can also be assigned a value that is a list. So lists can also be used in more general
situations, not only in the for loop. The code above could be rewritten like this:
1
2
3
4
5
6
# Assign a list to a variable
colors = ["yellow", "red", "purple", "blue"]
for color in colors:
alex.color(color)
alex.forward(50)
alex.left(90)
• Notice the difference between the method alex.color, which is “part of” the instance
alex, and the variable color, which is “part of” the main body of your program.
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A few more turtle methods and tricks
Turtle methods can use negative angles or distances. So tess.forward(-100) will move
tess backwards, and tess.left(-30) turns her to the right. Additionally, because there
are 360 degrees in a circle, turning 30 to the left will get tess facing in the same direction as
turning 330 to the right! (The on-screen animation will differ, though — you will be able to tell
if tess is turning clockwise or counter-clockwise!)
This suggests that we don’t need both a left and a right turn method — we could be minimalists,
and just have one method. There is also a backward method. (If you are very nerdy, you might
enjoy saying alex.backward(-100) to move alex forward!)
Part of thinking like a scientist is to understand more of the structure and rich relationships
in our field. So revising a few basic facts about geometry and number lines, and spotting the
relationships between left, right, backward, forward, negative and positive distances or angles
values is a good start if we’re going to play with turtles.
A turtle’s pen can be picked up or put down. This allows us to move a turtle to a different place
without drawing a line. The methods are
1
2
3
alex.penup()
alex.forward(100)
˓→drawn
alex.pendown()
# This moves alex, but no line is
Every turtle can have its own shape. The ones available “out of the box” are arrow, blank,
circle, classic, square, triangle, turtle.
1
alex.shape("turtle")
We can speed up or slow down the turtle’s animation speed. (Animation controls how quickly
the turtle turns and moves forward). Speed settings can be set between 1 (slowest) to 10
(fastest). But if we set the speed to 0, it has a special meaning — turn off animation and
go as fast as possible.
1
alex.speed(10)
A turtle can “stamp” its footprint onto the canvas, and this will remain after the turtle has moved
somewhere else. Stamping works, even when the pen is up.
3.1. Hello, little turtles!
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Let’s do an example that shows off some of these new features:
1
2
3
4
5
6
import turtle
window = turtle.Screen()
window.bgcolor("lightgreen")
tess = turtle.Turtle()
tess.shape("turtle")
tess.color("blue")
7
8
9
10
11
12
13
14
tess.penup()
size = 20
for _ in range(30):
tess.stamp()
˓→canvas
size = size + 3
˓→iteration
tess.forward(size)
tess.right(24)
# This is new
# Leave an impression on the
# Increase the size on every
# Move tess along
# ... and turn her
15
16
window.mainloop()
Be careful now! How many times was the body of the loop executed? How many turtle images
do we see on the screen? All except one of the shapes we see on the screen here are footprints
created by stamp. But the program still only has one turtle instance — can you figure out
which one here is the real tess? (Hint: if you’re not sure, write a new line of code after the
for loop to change tess‘ color, or to put her pen down and draw a line, or to change her
shape, etc.)
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Conditionals
Programs get really interesting when we can test conditions and change the program behaviour
depending on the outcome of the tests. That’s what this part is about.
Boolean values and expressions
A Boolean value is either true or false. It is named after the British mathematician, George
Boole, who first formulated Boolean algebra — some rules for reasoning about and combining
these values. This is the basis of all modern computer logic.
In Python, the two Boolean values are True and False (the capitalization must be exactly as
shown), and the Python type is bool.
>>> type(True)
<class 'bool'>
>>> type(true)
Traceback (most recent call last):
File "<interactive input>", line 1, in <module>
NameError: name 'true' is not defined
A Boolean expression is an expression that evaluates to produce a result which is a Boolean
value. For example, the operator == tests if two values are equal. It produces (or yields) a
Boolean value:
>>> 5
True
>>> 5
False
>>> j
>>> j
True
== (3 + 2)
# Is five equal 5 to the result of 3 + 2?
== 6
= "hel"
+ "lo" == "hello"
In the first statement, the two operands evaluate to equal values, so the expression evaluates to
True; in the second statement, 5 is not equal to 6, so we get False.
The == operator is one of six common comparison operators which all produce a bool result;
here are all six:
x
x
x
x
x
x
== y
!= y
> y
< y
>= y
<= y
#
#
#
#
#
#
Produce True if ... x
... x is not equal to
... x is greater than
... x is less than y
... x is greater than
... x is less than or
is equal to y
y
y
or equal to y
equal to y
Although these operations are probably familiar, the Python symbols are different from the
mathematical symbols. A common error is to use a single equal sign (=) instead of a double
equal sign (==). Remember that = is an assignment operator and == is a comparison operator.
Also, there is no such thing as =< or =>.
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Like any other types we’ve seen so far, Boolean values can be assigned to variables, printed,
etc.
>>> age = 19
>>> old_enough_to_get_driving_licence = age >= 18
>>> print(old_enough_to_get_driving_licence)
True
>>> type(old_enough_to_get_driving_licence)
<class 'bool'>
Logical operators
There are three logical operators, and, or, and not, that allow us to build more complex
Boolean expressions from simpler Boolean expressions. The semantics (meaning) of these
operators is similar to their meaning in English. For example, x > 0 and x < 10 produces
True only if x is greater than 0 and at the same time, x is less than 10.
n % 2 == 0 or n % 3 == 0 is True if either of the conditions is True, that is, if the
number n is divisible by 2 or it is divisible by 3. (What do you think happens if n is divisible
by both 2 and by 3 at the same time? Will the expression yield True or False? Try it in your
Python interpreter.)
Finally, the not operator negates a Boolean value, so not (x > y) is True if (x > y) is
False, that is, if x is less than or equal to y. In other words: not True is False, and not
False is True.
The expression on the left of the or operator is evaluated first: if the result is True, Python
does not (and need not) evaluate the expression on the right — this is called short-circuit evaluation. Similarly, for the and operator, if the expression on the left yields False, Python does
not evaluate the expression on the right.
So there are no unnecessary evaluations.
Truth Tables
A truth table is a small table that allows us to list all the possible inputs, and to give the results
for the logical operators. Because the and and or operators each have two operands, there are
only four rows in a truth table that describes the semantics of and.
a
False
False
True
True
b
False
True
False
True
a and b
False
False
False
True
In a Truth Table, we sometimes use T and F as shorthand for the two Boolean values: here is
the truth table describing or:
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a
F
F
T
T
b
F
T
F
T
a or b
F
T
T
T
The third logical operator, not, only takes a single operand, so its truth table only has two
rows:
a
F
T
not a
T
F
Simplifying Boolean Expressions
A set of rules for simplifying and rearranging expressions is called an algebra. For example,
we are all familiar with school algebra rules, such as:
n * 0 == 0
Here we see a different algebra — the Boolean algebra — which provides rules for working
with Boolean values.
First, the and operator:
x and False == False
False and x == False
y and x == x and y
x and True == x
True and x == x
x and x == x
Here are some corresponding rules for the or operator:
x or False == x
False or x == x
y or x == x or y
x or True == True
True or x == True
x or x == x
Two not operators cancel each other:
not (not x) == x
Conditional execution
In order to write useful programs, we almost always need the ability to check conditions and
change the behavior of the program accordingly. Conditional statements give us this ability.
The simplest form is the if statement:
3.2. Conditionals
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1
2
3
4
5
6
7
if x % 2 == 0:
print(x, "
print("Did
˓→is prime?")
else:
print(x, "
print("Did
is even.")
you know that 2 is the only even number that
is odd.")
you know that multiplying two odd numbers " +
"always gives an
˓→odd result?")
The Boolean expression after the if statement is called the condition. If it is true, then all
the indented statements get executed. If not, then all the statements indented under the else
clause get executed.
Flowchart of an if statement with an else clause
The syntax for an if statement looks like this:
1
2
3
4
if <BOOLEAN EXPRESSION>:
<STATEMENTS_1>
˓→to True
else:
<STATEMENTS_2>
˓→to False
# Executed if condition evaluates
# Executed if condition evaluates
As with the function definition from the next chapter and other compound statements like for,
the if statement consists of a header line and a body. The header line begins with the keyword
if followed by a Boolean expression and ends with a colon (:).
The indented statements that follow are called a block. The first unindented statement marks
the end of the block.
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Each of the statements inside the first block of statements are executed in order if the Boolean
expression evaluates to True. The entire first block of statements is skipped if the Boolean
expression evaluates to False, and instead all the statements indented under the else clause
are executed.
There is no limit on the number of statements that can appear under the two clauses of an if
statement, but there has to be at least one statement in each block. Occasionally, it is useful to
have a section with no statements (usually as a place keeper, or scaffolding, for code we haven’t
written yet). In that case, we can use the pass statement, which does nothing except act as a
placeholder.
1
2
3
4
if True:
# This is always True,
pass
#
so this is always executed, but it
˓→does nothing
else:
pass
# And this is never executed
Omitting the else clause
Flowchart of an if statement with no else clause
Another form of the if statement is one in which the else clause is omitted entirely. In this
case, when the condition evaluates to True, the statements are executed, otherwise the flow of
execution continues to the statement after the if.
1
2
3
if x < 0:
print("The negative number ",
x = 42
3.2. Conditionals
x, " is not valid here.")
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4
print("I've decided to use the number 42 instead.")
5
6
print("The square root of ", x, "is", math.sqrt(x))
In this case, the print function that outputs the square root is the one after the if — not because
we left a blank line, but because of the way the code is indented. Note too that the function
call math.sqrt(x) will give an error unless we have an import math statement, usually
placed near the top of our script.
Python terminology
Python documentation sometimes uses the term suite of statements to mean what we have
called a block here. They mean the same thing, and since most other languages and computer
scientists use the word block, we’ll stick with that.
Notice too that else is not a statement. The if statement has two clauses, one of which is the
(optional) else clause.
Chained conditionals
Sometimes there are more than two possibilities and we need more than two branches. One
way to express a computation like that is a chained conditional:
1
2
3
4
5
6
if x < y:
<STATEMENTS_A>
elif x > y:
<STATEMENTS_B>
else:
<STATEMENTS_C>
# x == y
Flowchart of this chained conditional
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elif is an abbreviation of else if. Again, exactly one branch will be executed. There is no
limit of the number of elif statements but only a single (and optional) final else statement
is allowed and it must be the last branch in the statement:
1
2
3
4
5
6
7
8
if choice == "a":
function_one()
elif choice == "b":
function_two()
elif choice == "c":
function_three()
else:
print("Invalid choice.")
Each condition is checked in order. If the first is false, the next is checked, and so on. If one
of them is true, the corresponding branch executes, and the statement ends. Even if more than
one condition is true, only the first true branch executes.
Nested conditionals
One conditional can also be nested within another. (It is the same theme of composability,
again!) We could have written the previous example as follows:
Flowchart of this nested conditional
3.2. Conditionals
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1
2
3
4
5
6
7
if x < y:
<STATEMENTS_A>
else:
if x > y:
<STATEMENTS_B>
else:
<STATEMENTS_C>
The outer conditional contains two branches. The second branch contains another if statement, which has two branches of its own. Those two branches could contain conditional statements as well.
Although the indentation of the statements makes the structure apparent, nested conditionals
very quickly become very difficult to read. In general, it is a good idea to avoid them when we
can.
Logical operators often provide a way to simplify nested conditional statements. For example,
we can rewrite the following code using a single conditional:
1
2
3
if 0 < x:
# Assume x is an int here
if x < 10:
print("x is a positive single digit.")
The print function is called only if we make it past both the conditionals, so instead of the
above which uses two if statements each with a simple condition, we could make a more
complex condition using the and operator. Now we only need a single if statement:
1
2
46
if 0 < x and x < 10:
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In this case there is a third option:
1
2
if 0 < x < 10:
print("x is a positive single digit.")
Logical opposites
Each of the six relational operators has a logical opposite: for example, suppose we can get a
driving licence when our age is greater or equal to 18, we can not get the driving licence when
we are less than 18.
Notice that the opposite of >= is <.
operator
==
!=
<
<=
>
>=
logical opposite
!=
==
>=
>
<=
<
Understanding these logical opposites allows us to sometimes get rid of not operators. not
operators are often quite difficult to read in computer code, and our intentions will usually be
clearer if we can eliminate them.
For example, if we wrote this Python:
1
2
if not (age >= 18):
print("Hey, you're too young to get a driving licence!")
it would probably be clearer to use the simplification laws, and to write instead:
1
2
if age < 18:
print("Hey, you're too young to get a driving licence!")
Two powerful simplification laws (called de Morgan’s laws) that are often helpful when dealing
with complicated Boolean expressions are:
(not (x and y))
(not (x or y))
==
==
((not x) or (not y))
((not x) and (not y))
For example, suppose we can slay the dragon only if our magic lightsabre sword is charged to
90% or higher, and we have 100 or more energy units in our protective shield. We find this
fragment of Python code in the game:
1
2
3
4
if not (sword_charge >= 0.90 and shield_energy >= 100):
print("Your attack has no effect, the dragon fries you
˓→to a crisp!")
else:
print("The dragon crumples in a heap. You rescue the
˓→gorgeous princess!")
3.2. Conditionals
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de Morgan’s laws together with the logical opposites would let us rework the condition in a
(perhaps) easier to understand way like this:
1
2
3
4
if sword_charge < 0.90 or shield_energy < 100:
print("Your attack has no effect, the dragon fries you
˓→to a crisp!")
else:
print("The dragon crumples in a heap. You rescue the
˓→gorgeous princess!")
We could also get rid of the not by swapping around the then and else parts of the conditional. So here is a third version, also equivalent:
1
2
3
4
if sword_charge >= 0.90 and shield_energy >= 100:
print("The dragon crumples in a heap. You rescue the
˓→gorgeous princess!")
else:
print("Your attack has no effect, the dragon fries you
˓→to a crisp!")
To improve readability, there is this fourth version:
1
2
sword_check = sword_charge >= 0.90
shield_check = shield_energy >= 100
3
4
5
6
7
if sword_check and shield_check:
print("The dragon crumples in a heap. You rescue the
˓→gorgeous princess!")
else:
print("Your attack has no effect, the dragon fries you
˓→to a crisp!")
This version is probably the best of the four, because it very closely matches the initial English
statement. Clarity of our code (for other humans), and making it easy to see that the code does
what was expected should always be highest priority.
As our programming skills develop we’ll find we have more than one way to solve any problem.
So good programs are designed. We make choices that favour clarity, simplicity, and elegance.
The job title software architect says a lot about what we do — we are architects who engineer
our products to balance beauty, functionality, simplicity and clarity in our creations.
Tip: Once our program works, we should play around a bit trying to polish it up. Write
good comments. Think about whether the code would be clearer with different variable names.
Could we have done it more elegantly? Should we rather use a function? Can we simplify the
conditionals?
We think of our code as our creation, our work of art! We make it great.
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Iteration
Computers are often used to automate repetitive tasks. Repeating identical or similar tasks
without making errors is something that computers do well and people do poorly.
Repeated execution of a set of statements is called iteration. Because iteration is so common,
Python provides several language features to make it easier. We’ve already seen the for statement. This is the form of iteration you’ll likely be using most often. But here we’re going
to look at the while statement — another way to have your program do iteration, useful in
slightly different circumstances.
Before we do that, let’s just review a few ideas...
Assignment
As we have mentioned previously, it is legal to make more than one assignment to the same
variable. A new assignment makes an existing variable refer to a new value (and stop referring
to the old value).
1
2
3
4
airtime_remaining = 15
print(airtime_remaining)
airtime_remaining = 7
print(airtime_remaining)
The output of this program is:
15
7
because the first time airtime_remaining is printed, its value is 15, and the second time,
its value is 7.
It is especially important to distinguish between an assignment statement and a Boolean expression that tests for equality. Because Python uses the equal token (=) for assignment, it is
tempting to interpret a statement like a = b as a Boolean test. Unlike mathematics, it is not!
Remember that the Python token for the equality operator is ==.
Note too that an equality test is symmetric, but assignment is not. For example, if a == 7
then 7 == a. But in Python, the statement a = 7 is legal and 7 = a is not.
In Python, an assignment statement can make two variables equal, but because further assignments can change either of them, they don’t have to stay that way:
1
2
3
a = 5
b = a
a = 3
˓→equal
# After executing this line, a and b are now equal
# After executing this line, a and b are no longer
The third line changes the value of a but does not change the value of b, so they are no longer
equal. (In some programming languages, a different symbol is used for assignment, such as
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<- or :=, to avoid confusion. Some people also think that variable was an unfortunae word
to choose, and instead we should have called them assignables. Python chooses to follow
common terminology and token usage, also found in languages like C, C++, Java, and C#, so
we use the tokens = for assignment, == for equality, and we talk of variables.
Updating variables
When an assignment statement is executed, the right-hand side expression (i.e. the expression
that comes after the assignment token) is evaluated first. This produces a value. Then the
assignment is made, so that the variable on the left-hand side now refers to the new value.
One of the most common forms of assignment is an update, where the new value of the variable
depends on its old value. Deduct 40 cents from my airtime balance, or add one run to the
scoreboard.
1
2
n = 5
n = 3 * n + 1
Line 2 means get the current value of n, multiply it by three and add one, and assign the answer
to n, thus making n refer to the value. So after executing the two lines above, n will point/refer
to the integer 16.
If you try to get the value of a variable that has never been assigned to, you’ll get an error:
>>> w = x + 1
Traceback (most recent call last):
File "<interactive input>", line 1, in
NameError: name 'x' is not defined
Before you can update a variable, you have to initialize it to some starting value, usually with
a simple assignment:
1
2
3
runs_scored = 0
...
runs_scored = runs_scored + 1
Line 3 — updating a variable by adding 1 to it — is very common. It is called an increment
of the variable; subtracting 1 is called a decrement. Sometimes programmers also talk about
bumping a variable, which means the same as incrementing it by 1. This is commonly done
with the += operator.
1
2
3
runs_scored = 0
...
runs_scored += 1
The for loop revisited
Recall that the for loop processes each item in a list. Each item in turn is (re-)assigned to the
loop variable, and the body of the loop is executed. We saw this example before:
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1
2
3
for friend in ["Joe", "Zoe", "Zuki", "Thandi", "Paris"]:
invite = "Hi " + friend + ". Please come to my party!"
print(invite)
Running through all the items in a list is called traversing the list, or traversal.
Let us write some code now to sum up all the elements in a list of numbers. Do this by hand first,
and try to isolate exactly what steps you take. You’ll find you need to keep some “running total”
of the sum so far, either on a piece of paper, in your head, or in your calculator. Remembering
things from one step to the next is precisely why we have variables in a program: so we’ll need
some variable to remember the “running total”. It should be initialized with a value of zero, and
then we need to traverse the items in the list. For each item, we’ll want to update the running
total by adding the next number to it.
1
2
3
4
5
numbers = [5, 6, 32, 21, 9]
running_total = 0
for number in numbers:
running_total = running_total + number
print(running_total)
The while statement
Here is a fragment of code that demonstrates the use of the while statement:
2
while <CONDITION>:
<STATEMENT>
1
n = 6
1
2
3
4
5
6
7
8
current_sum = 0
i = 1
while i <= n:
current_sum += i
i += 1
print(current_sum)
You can almost read the while statement as if it were English. It means, while i is less than
or equal to n, continue executing the body of the loop. Within the body, each time, increment
i. When i passes n, return your accumulated sum. In other words: while <CONDITION> is
True, <STATEMENT> is executed. Of course, this example could be written more concisely
as sum(range(n + 1)) because the function sum already exists.
More formally, here is precise flow of execution for a while statement:
• Evaluate the condition at line 5, yielding a value which is either False or True.
• If the value is False, exit the while statement and continue execution at the next
statement (line 8 in this case).
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• If the value is True, execute each of the statements in the body (lines 6 and 7) and then
go back to the while statement at line 5.
The body consists of all of the statements indented below the while keyword.
Notice that if the loop condition is False the first time we get loop, the statements in the body
of the loop are never executed.
The body of the loop should change the value of one or more variables so that eventually the
condition becomes false and the loop terminates. Otherwise the loop will repeat forever, which
is called an infinite loop.
In the case here, we can prove that the loop terminates because we know that the value of n is
finite, and we can see that the value of i increments each time through the loop, so eventually
it will have to exceed n. In other cases, it is not so easy, even impossible in some cases, to tell
if the loop will ever terminate.
What you will notice here is that the while loop is more work for you — the programmer —
than the equivalent for loop. When using a while loop one has to manage the loop variable
yourself: give it an initial value, test for completion, and then make sure you change something
in the body so that the loop terminates. By comparison, here is an equivalent snippet that uses
for instead:
1
n = 6
2
3
4
5
6
current_sum = 0
for i in range(n+1):
current_sum += i
print(current_sum)
Notice the slightly tricky call to the range function — we had to add one onto n, because
range generates its list up to but excluding the value you give it. It would be easy to make a
programming mistake and overlook this.
So why have two kinds of loop if for looks easier? This next example shows a case where we
need the extra power that we get from the while loop.
The Collatz 3n + 1 sequence
Let’s look at a simple sequence that has fascinated and foxed mathematicians for many years.
They still cannot answer even quite simple questions about this.
The “computational rule” for creating the sequence is to start from some given n, and to generate the next term of the sequence from n, either by halving n, (whenever n is even), or else
by multiplying it by three and adding 1. The sequence terminates when n reaches 1.
This Python snippet captures that algorithm:
1
n = 1027371
2
3
4
52
while n != 1:
print(n, end=", ")
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5
6
7
8
9
if n % 2 == 0:
n = n // 2
else:
n = n * 3 + 1
print(n, end=".\n")
# n is even
# n is odd
Notice first that the print function on line 4 has an extra argument end=", ". This tells
the print function to follow the printed string with whatever the programmer chooses (in
this case, a comma followed by a space), instead of ending the line. So each time something
is printed in the loop, it is printed on the same output line, with the numbers separated by
commas. The call to print(n, end=".\n") at line 9 after the loop terminates will then
print the final value of n followed by a period and a newline character. (You’ll cover the \n
(newline character) later).
The condition for continuing with this loop is n != 1, so the loop will continue running until
it reaches its termination condition, (i.e. n == 1).
Each time through the loop, the program outputs the value of n and then checks whether it is
even or odd. If it is even, the value of n is divided by 2 using integer division. If it is odd, the
value is replaced by n * 3 + 1.
Since n sometimes increases and sometimes decreases, there is no obvious proof that n will
ever reach 1, or that the program terminates. For some particular values of n, we can prove
termination. For example, if the starting value is a power of two, then the value of n will be
even each time through the loop until it reaches 1. The previous example ends with such a
sequence, starting with 16.
See if you can find a small starting number that needs more than a hundred steps before it
terminates.
Particular values aside, the interesting question was first posed by a German mathematician
called Lothar Collatz: the Collatz conjecture (also known as the 3n + 1 conjecture), is that this
sequence terminates for all positive values of n. So far, no one has been able to prove it or
disprove it! (A conjecture is a statement that might be true, but nobody knows for sure.)
Think carefully about what would be needed for a proof or disproof of the conjecture “All
positive integers will eventually converge to 1 using the Collatz rules”. With fast computers we
have been able to test every integer up to very large values, and so far, they have all eventually
ended up at 1. But who knows? Perhaps there is some as-yet untested number which does not
reduce to 1.
You’ll notice that if you don’t stop when you reach 1, the sequence gets into its own cyclic
loop: 1, 4, 2, 1, 4, 2, 1, 4 ... So one possibility is that there might be other cycles that we just
haven’t found yet.
Wikipedia has an informative article about the Collatz conjecture. The sequence also goes
under other names (Hailstone sequence, Wonderous numbers, etc.), and you’ll find out just
how many integers have already been tested by computer, and found to converge!
Choosing between for and while
Use a for loop if you know, before you start looping, the maximum number of times that
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you’ll need to execute the body. For example, if you’re traversing a list of elements, you know
that the maximum number of loop iterations you can possibly need is “all the elements in the
list”. Or if you need to print the 12 times table, we know right away how many times the loop
will need to run.
So any problem like “iterate this weather model for 1000 cycles”, or “search this list of words”,
“find all prime numbers up to 10000” suggest that a for loop is best.
By contrast, if you are required to repeat some computation until some condition is met, and
you cannot calculate in advance when (of if) this will happen, as we did in this 3n + 1 problem,
you’ll need a while loop.
We call the first case definite iteration — we know ahead of time some definite bounds for
what is needed. The latter case is called indefinite iteration — we’re not sure how many
iterations we’ll need — we cannot even establish an upper bound!
Tracing a program
To write effective computer programs, and to build a good conceptual model of program execution, a programmer needs to develop the ability to trace the execution of a computer program.
Tracing involves becoming the computer and following the flow of execution through a sample
program run, recording the state of all variables and any output the program generates after
each instruction is executed.
To understand this process, let’s trace the call to the collatz code above with n = 3 from the
previous section. At the start of the trace, we have a variable, n, with an initial value of 3.
Since 3 is not equal to 1, the while loop body is executed. 3 is printed and 3 % 2 == 0
is evaluated. Since it evaluates to False, the else branch is executed and 3 * 3 + 1 is
evaluated and assigned to n.
To keep track of all this as you hand trace a program, make a column heading on a piece of
paper for each variable created as the program runs and another one for output. Our trace so
far would look something like this:
n
-3
10
output printed so far
--------------------3,
Since 10 != 1 evaluates to True, the loop body is again executed, and 10 is printed. 10 %
2 == 0 is true, so the if branch is executed and n becomes 5. By the end of the trace we
have:
n
-3
10
5
16
8
54
output printed so far
--------------------3,
3, 10,
3, 10, 5,
3, 10, 5, 16,
3, 10, 5, 16, 8,
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4
2
1
3, 10, 5, 16, 8, 4,
3, 10, 5, 16, 8, 4, 2,
3, 10, 5, 16, 8, 4, 2, 1.
Tracing can be a bit tedious and error prone (that’s why we get computers to do this stuff in the
first place!), but it is an essential skill for a programmer to have. From this trace we can learn
a lot about the way our code works. We can observe that as soon as n becomes a power of 2,
for example, the program will require log2 (n) executions of the loop body to complete. We can
also see that the final 1 will not be printed as output within the body of the loop, which is why
we put the special print function at the end.
Counting digits
The following snippet counts the number of decimal digits in a positive integer:
1
2
3
4
5
6
n = 3029
count = 0
while n != 0:
count = count + 1
n = n // 10
print(count)
Trace the execution to convince yourself that it works.
This snippet demonstrates an important pattern of computation called a counter. The variable
count is initialized to 0 and then incremented each time the loop body is executed. When the
loop exits, count contains the result — the total number of times the loop body was executed,
which is the same as the number of digits.
If we wanted to only count digits that are either 0 or 5, adding a conditional before incrementing
the counter will do the trick:
1
2
3
4
5
6
7
8
n = 2574301453
count = 0
while n > 0:
digit = n % 10
if digit == 0 or digit == 5:
count = count + 1
n = n // 10
print(count)
Notice, however, that if n = 0 this snippet will not print 1 as answer. Explain why. Do you
think this is a bug in the code, or a bug in the specifications, or our expectations?
Help and meta-notation
Python comes with extensive documentation for all its built-in functions, and its libraries. Different systems have different ways of accessing this help. See for example https://docs.python.
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org/3/library/stdtypes.html#typesseq-range
Notice the square brackets in the description of the arguments. These are examples of metanotation — notation that describes Python syntax, but is not part of it. The square brackets in
this documentation mean that the argument is optional — the programmer can omit it. So what
this first line of help tells us is that range must always have a stop argument, but it may have
an optional start argument (which must be followed by a comma if it is present), and it can
also have an optional step argument, preceded by a comma if it is present.
The examples from help show that range can have either 1, 2 or 3 arguments. The list can
start at any starting value, and go up or down in increments other than 1. The documentation
here also says that the arguments must be integers.
Other meta-notation you’ll frequently encounter is the use of bold and italics. The bold means
that these are tokens — keywords or symbols — typed into your Python code exactly as they
are, whereas the italic terms stand for “something of this type”. So the syntax description
for variable in list :
means you can substitute any legal variable and any legal list when you write your Python code.
This (simplified) description of the print function, shows another example of meta-notation
in which the ellipses (...) mean that you can have as many objects as you like (even zero),
separated by commas:
print( [object, ... ] )
Meta-notation gives us a concise and powerful way to describe the pattern of some syntax or
feature.
Tables
One of the things loops are good for is generating tables. Before computers were readily available, people had to calculate logarithms, sines and cosines, and other mathematical functions
by hand. To make that easier, mathematics books contained long tables listing the values of
these functions. Creating the tables was slow and boring, and they tended to be full of errors.
When computers appeared on the scene, one of the initial reactions was, “This is great! We can
use the computers to generate the tables, so there will be no errors.” That turned out to be true
(mostly) but shortsighted. Soon thereafter, computers and calculators were so pervasive that
the tables became obsolete.
Well, almost. For some operations, computers use tables of values to get an approximate answer
and then perform computations to improve the approximation. In some cases, there have been
errors in the underlying tables, most famously in the table the Intel Pentium processor chip
used to perform floating-point division.
Although a log table is not as useful as it once was, it still makes a good example of iteration.
The following program outputs a sequence of values in the left column and 2 raised to the
power of that value in the right column:
1
2
56
for x in range(13):
# Generate numbers 0 to 12
print(x, "\t", 2**x)
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The string "\t" represents a tab character. The backslash character in "\t" indicates the
beginning of an escape sequence. Escape sequences are used to represent invisible characters
like tabs and newlines. The sequence \n represents a newline.
An escape sequence can appear anywhere in a string; in this example, the tab escape sequence
is the only thing in the string. How do you think you represent a backslash in a string?
As characters and strings are displayed on the screen, an invisible marker called the cursor
keeps track of where the next character will go. After a print function, the cursor normally
goes to the beginning of the next line.
The tab character shifts the cursor to the right until it reaches one of the tab stops. Tabs are
useful for making columns of text line up, as in the output of the previous program:
0
1
2
3
4
5
6
7
8
9
10
11
12
1
2
4
8
16
32
64
128
256
512
1024
2048
4096
Because of the tab characters between the columns, the position of the second column does not
depend on the number of digits in the first column.
Two-dimensional tables
A two-dimensional table is a table where you read the value at the intersection of a row and a
column. A multiplication table is a good example. Let’s say you want to print a multiplication
table for the values from 1 to 6.
A good way to start is to write a loop that prints the multiples of 2, all on one line:
1
2
3
for i in range(1, 7):
print(2 * i, end="
print()
")
Here we’ve used the range function, but made it start its sequence at 1. As the loop executes,
the value of i changes from 1 to 6. When all the elements of the range have been assigned to
i, the loop terminates. Each time through the loop, it displays the value of 2 * i, followed
by three spaces.
Again, the extra end=" " argument in the print function suppresses the newline, and uses
three spaces instead. After the loop completes, the call to print at line 3 finishes the current
line, and starts a new line.
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The output of the program is:
2
4
6
8
10
12
So far, so good. The next step is to encapsulate and generalize. We will continue this topic in
the next chapter.
The break statement
The break statement is used to immediately leave the body of its loop. The next statement to
be executed is the first one after the body:
1
2
3
4
5
for i in [12, 16, 17, 24, 29]:
if i % 2 == 1: # If the number is odd
break
# ... immediately exit the loop
print(i)
print("done")
This prints:
12
16
done
The pre-test loop — standard loop behaviour
for and while loops do their tests at the start, before executing any part of the body. They’re
called pre-test loops, because the test happens before (pre) the body. break and return
(discussed later) are our tools for adapting this standard behaviour.
Other flavours of loops
Sometimes we’d like to have the middle-test loop with the exit test in the middle of the body,
rather than at the beginning or at the end. Or a post-test loop that puts its exit test as the
last thing in the body. Other languages have different syntax and keywords for these different
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flavours, but Python just uses a combination of while and if <CONDITION>:
to get the job done.
break
A typical example is a problem where the user has to input numbers to be summed. To indicate
that there are no more inputs, the user enters a special value, often the value -1, or the empty
string. This needs a middle-exit loop pattern: input the next number, then test whether to exit,
or else process the number:
The middle-test loop flowchart
1
2
3
4
5
6
7
total = 0
while True:
response = input("Enter the next number. (Leave blank
˓→to end)")
if response == "" or response == "-1":
break
total += int(response)
print("The total of the numbers you entered is ", total)
Convince yourself that this fits the middle-exit loop flowchart: line 3 does some useful work,
lines 4 and 5 can exit the loop, and if they don’t line 6 does more useful work before the next
iteration starts.
The while bool-expr: uses the Boolean expression to determine whether to iterate again.
True is a trivial Boolean expression, so while True: means always do the loop body
again. This is a language idiom — a convention that most programmers will recognize immediately. Since the expression on line 2 will never terminate the loop, (it is a dummy test) the
programmer must arrange to break (or return) out of the loop body elsewhere, in some other
way (i.e. in lines 4 and 5 in this sample). A clever compiler or interpreter will understand that
line 2 is a fake test that must always succeed, so it won’t even generate a test, and our flowchart
never even put the diamond-shape dummy test box at the top of the loop!
Similarly, by just moving the if condition: break to the end of the loop body we
create a pattern for a post-test loop. Post-test loops are used when you want to be sure that the
loop body always executes at least once (because the first test only happens at the end of the
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execution of the first loop body). This is useful, for example, if we want to play an interactive
game against the user — we always want to play at least one game:
1
2
3
4
5
6
while True:
play_the_game_once()
response = input("Play again? (yes or no)")
if response != "yes":
break
print("Goodbye!")
Hint: Think about where you want the exit test to happen
Once you’ve recognized that you need a loop to repeat something, think about its terminating
condition — when will I want to stop iterating? Then figure out whether you need to do the test
before starting the first (and every other) iteration, or at the end of the first (and every other)
iteration, or perhaps in the middle of each iteration. Interactive programs that require input
from the user or read from files often need to exit their loops in the middle or at the end of an
iteration, when it becomes clear that there is no more data to process, or the user doesn’t want
to play our game anymore.
An example
The following program implements a simple guessing game:
1
2
3
import random
# We cover random numbers
˓→in the
rng = random.Random()
# modules chapter, so peek
˓→ahead if you want. "rng" stands for "random number
˓→generator".
number = rng.randrange(1, 1000) # Get random number between
˓→[1 and 1000).
4
5
6
guesses = 0
message = ""
7
8
9
10
11
12
13
14
15
16
while True:
guess = int(input(message + "\nGuess my number between
˓→1 and 1000: "))
guesses += 1
if guess > number:
message += str(guess) + " is too high.\n"
elif guess < number:
message += str(guess) + " is too low.\n"
else:
break
17
18
60
input("\n\nGreat, you got it in "+str(guesses)+" guesses!
˓→\n\n")
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This program makes use of the mathematical law of trichotomy (given real numbers a and b,
exactly one of these three must be true: a > b, a < b, or a == b).
At line 18 there is a call to the input function, but we don’t do anything with the result, not
even assign it to a variable. This is legal in Python. Here it has the effect of popping up
the input dialog window and waiting for the user to respond before the program terminates.
Programmers often use the trick of doing some extra input at the end of a script, just to keep
the window open.
Also notice the use of the message variable, initially an empty string, on lines 6, 12 and 14.
Each time through the loop we extend the message being displayed: this allows us to display
the program’s feedback right at the same place as we’re asking for the next guess.
The continue statement
This is a control flow statement that causes the program to immediately skip the processing of
the rest of the body of the loop, for the current iteration. But the loop still carries on running
for its remaining iterations:
1
2
3
4
5
for i in [12, 16, 17, 24, 29, 30]:
if i % 2 == 1:
# If the number is odd
continue
# Don't process it
print(i)
print("done")
This prints:
12
16
24
30
done
Paired Data
We’ve already seen lists of names and lists of numbers in Python. We’re going to peek ahead
in the textbook a little, and show a more advanced way of representing our data. Making a pair
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of things in Python is as simple as putting them into parentheses, like this:
1
year_born = ("Paris Hilton", 1981)
We can put many pairs into a list of pairs:
1
2
celebs = [("Brad Pitt", 1963), ("Jack Nicholson", 1937),
("Justin Bieber", 1994)]
Here is a quick sample of things we can do with structured data like this. First, print all the
celebs:
1
2
print(celebs)
print(len(celebs))
[("Brad Pitt", 1963), ("Jack Nicholson", 1937), ("Justin
˓→Bieber", 1994)]
3
Notice that the celebs list has just 3 elements, each of them pairs.
Now we print the names of those celebrities born before 1980:
1
2
3
for name, year in celebs:
if year < 1980:
print(name)
Brad Pitt
Jack Nicholson
This demonstrates something we have not seen yet in the for loop: instead of using a single
loop control variable, we’ve used a pair of variable names, (name, year), instead. The
loop is executed three times — once for each pair in the list, and on each iteration both the
variables are assigned values from the pair of data that is being handled.
Nested Loops for Nested Data
Now we’ll come up with an even more adventurous list of structured data. In this case, we have
a list of students. Each student has a name which is paired up with another list of subjects that
they are enrolled for:
1
2
3
4
5
6
62
students = [
("John", ["CompSci", "Physics"]),
("Vusi", ["Maths", "CompSci", "Stats"]),
("Jess", ["CompSci", "Accounting", "Economics",
˓→"Management"]),
("Sarah", ["InfSys", "Accounting", "Economics", "CommLaw
˓→"]),
("Zuki", ["Sociology", "Economics", "Law", "Stats",
˓→"Music"])]
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Here we’ve assigned a list of five elements to the variable students. Let’s print out each
student name, and the number of subjects they are enrolled for:
1
2
3
# Print all students with a count of their courses.
for name, subjects in students:
print(name, "takes", len(subjects), "courses")
Python agreeably responds with the following output:
John takes 2 courses
Vusi takes 3 courses
Jess takes 4 courses
Sarah takes 4 courses
Zuki takes 5 courses
Now we’d like to ask how many students are taking CompSci. This needs a counter, and for
each student we need a second loop that tests each of the subjects in turn:
1
2
3
4
5
6
# Count how many students are taking CompSci
counter = 0
for name, subjects in students:
for s in subjects:
# A nested loop!
if s == "CompSci":
counter += 1
7
8
print("The number of students taking CompSci is", counter)
The number of students taking CompSci is 3
A more concise of doing this would be the following:
1
2
3
4
counter = 0
for name, subjects in students:
if "CompSci" in subjects:
counter += 1
You should set up a list of your own data that interests you — perhaps a list of your CDs, each
containing a list of song titles on the CD, or a list of movie titles, each with a list of movie stars
who acted in the movie. You could then ask questions like “Which movies starred Angelina
Jolie?”
Newton’s method for finding square roots
Loops are often used in programs that compute numerical results by starting with an approximate answer and iteratively improving it.
For example, before we had calculators or computers, people needed to calculate square roots
manually. Newton used a particularly good method (there is some evidence that this method
was known many years before). Suppose that you want to know the square root of n. If you
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start with almost any approximation, you can compute a better approximation (closer to the
actual answer) with the following formula:
1
better = (approximation + n/approximation)/2
Repeat this calculation a few times using your calculator. Can you see why each iteration brings
your estimate a little closer? One of the amazing properties of this particular algorithm is how
quickly it converges to an accurate answer — a great advantage for doing it manually.
By using a loop and repeating this formula until the better approximation gets close enough to
the previous one, we can write a function for computing the square root. (In fact, this is how
your calculator finds square roots — it may have a slightly different formula and method, but
it is also based on repeatedly improving its guesses.)
This is an example of an indefinite iteration problem: we cannot predict in advance how many
times we’ll want to improve our guess — we just want to keep getting closer and closer. Our
stopping condition for the loop will be when our old guess and our improved guess are “close
enough” to each other.
Ideally, we’d like the old and new guess to be exactly equal to each other when we stop. But
exact equality is a tricky notion in computer arithmetic when real numbers are involved. Because real numbers are not represented absolutely accurately (after all, a number like pi or the
square root of two has an infinite number of decimal places because it is irrational), we need
to formulate the stopping test for the loop by asking “is a close enough to b”? This stopping
condition can be coded like this:
1
2
3
threshold = 0.001
if abs(a-b) < threshold:
˓→accuracy
break
# Make this smaller for better
Notice that we take the absolute value of the difference between a and b!
This problem is also a good example of when a middle-exit loop is appropriate:
1
2
3
4
5
6
7
8
9
n = 8
threshold = 0.001
approximation = n/2
# Start with some or other guess at
˓→the answer
while True:
better = (approximation + n/approximation)/2
if abs(approximation - better) < threshold:
print(better)
break
approximation = better
See if you can improve the approximations by changing the stopping condition. Also, step
through the algorithm (perhaps by hand, using your calculator) to see how many iterations
were needed before it achieved this level of accuracy for sqrt(25).
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Algorithms
Newton’s method is an example of an algorithm: it is a mechanical process for solving a
category of problems (in this case, computing square roots).
Some kinds of knowledge are not algorithmic. For example, learning dates from history or your
multiplication tables involves memorization of specific solutions.
But the techniques you learned for addition with carrying, subtraction with borrowing, and long
division are all algorithms. Or if you are an avid Sudoku puzzle solver, you might have some
specific set of steps that you always follow.
One of the characteristics of algorithms is that they do not require any intelligence to carry out.
They are mechanical processes in which each step follows from the last according to a simple
set of rules. And they’re designed to solve a general class or category of problems, not just a
single problem.
Understanding that hard problems can be solved by step-by-step algorithmic processes (and
having technology to execute these algorithms for us) is one of the major breakthroughs that has
had enormous benefits. So while the execution of the algorithm may be boring and may require
no intelligence, algorithmic or computational thinking — i.e. using algorithms and automation
as the basis for approaching problems — is rapidly transforming our society. Some claim that
this shift towards algorithmic thinking and processes is going to have even more impact on our
society than the invention of the printing press. And the process of designing algorithms is
interesting, intellectually challenging, and a central part of what we call programming.
Some of the things that people do naturally, without difficulty or conscious thought, are the
hardest to express algorithmically. Understanding natural language is a good example. We all
do it, but so far no one has been able to explain how we do it, at least not in the form of a
step-by-step mechanical algorithm.
Some Tips, Tricks, and Common Errors
These are small summaries of ideas, tips, and commonly seen errors that might be helpful to
those beginning Python.
Problems with logic and flow of control
We often want to know if some condition holds for any item in a list, e.g. “does the list have
any odd numbers?” This is a common mistake:
1
numbers = [10, 5, 24, 8, 6]
2
3
4
5
6
7
# Buggy version
for number in numbers:
if number % 2 == 1:
print(True)
break
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8
9
10
else:
print(False)
break
Can we spot two problems here? As soon as we execute a break, we’ll leave the loop. So the
logic of saying “If I find an odd number I can return True” is fine. However, we cannot return
False after only looking at one item — we can only return False if we’ve been through all
the items, and none of them are odd. So line 10 should not be there, and lines 8 and 9 have to
be outside the loop. Here is a corrected version:
1
2
3
4
5
6
7
numbers = [10, 5, 24, 8, 6]
for number in numbers:
if number % 2 == 1:
print(True)
break
else:
print(False)
We’ll see This “eureka”, or “short-circuit” style of breaking from a loop as soon as we are
certain what the outcome will be again later.
Note that this uses a for ... else construct.
The else clause is executed when a loop has looped without encountering any break statements. This is ideal for our case here. Also note that the else is not, in this case, related to
the if statement that occurs inside the loop.
It is preferred over this one, which also works correctly:
1
2
3
4
5
6
7
8
9
numbers = [10, 5, 24, 8, 6]
count = 0
for number in numbers:
if number % 2 == 1:
count += 1
# Count the odd numbers
if count > 0:
print(True)
else:
print(False)
The performance disadvantage of this one is that it traverses the whole list, even if it knows the
outcome very early on.
Tip: Think about the return conditions of the loop
Do I need to look at all elements in all cases? Can I shortcut and take an early exit? Under
what conditions? When will I have to examine all the items in the list?
The code in lines 6-9 can also be tightened up. The expression count > 0 itself represents
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a Boolean value, either True or False (we can say it ‘evaluates’ to either True or False).
That True/False value can be used directly in the print statement. So we could cut out
that code and simply have the following:
1
2
3
4
5
6
7
numbers = [10, 5, 24, 8, 6]
count = 0
for number in numbers:
if number % 2 == 1:
count += 1
# Count the odd numbers
print(count > 0)
# Aha! a programmer who understands that
˓→Boolean
#
expressions are not just used in if
˓→statements!
Although this code is tighter, it is not as nice as the one that did the short-circuit return as soon
as the first odd number was found.
Even shorter:
1
2
3
4
5
6
numbers = [10, 5, 24, 8, 6]
count = 0
for number in numbers:
count += number % 2 == 1
print(count > 0)
# Aha! a programmer who understands that
˓→Boolean
#
expressions are not just used in if
˓→statements!
Tip: Generalize your use of Booleans
Programmers won’t write if is_prime(n) == True: when they could say instead if
is_prime(n): Think more generally about Boolean values, not just in the context of if or
while statements. Like arithmetic expressions, they have their own set of operators (and, or,
not) and values (True, False) and can be assigned to variables, put into lists, etc. A good resource for improving your use of Booleans is http://en.wikibooks.org/wiki/Non-Programmer%
27s_Tutorial_for_Python_3/Boolean_Expressions
Exercise time:
• How would we adapt this to print True if all the numbers are odd? Can you still use a
short-circuit style?
• How would we adapt it to print True if at least three of the numbers are odd? Shortcircuit the traversal when the third odd number is found — don’t traverse the whole list
unless we have to.
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Looping and lists
Computers are useful because they can repeat computation, accurately and fast. So loops are
going to be a central feature of almost all programs you encounter.
Tip: Don’t create unnecessary lists
Lists are useful if you need to keep data for later computation. But if you don’t need lists, it is
probably better not to generate them.
Here are two functions that both generate ten million random numbers, and return the sum of
the numbers. They both work.
1
2
import random
joe = random.Random()
3
4
5
6
7
8
9
# Version 1
# Build a list of random numbers, then sum them
numbers = []
for _ in range(10000000):
num = joe.randrange(1000) # Generate one random number
numbers.append(num)
# Save it in our list,
˓→see the next chapter
10
11
12
tot = sum(numbers)
print(tot)
13
14
15
16
17
18
19
20
# Version 2
# Sum the random numbers as we generate them
tot = 0
for _ in range(10000000):
num = joe.randrange(1000)
tot += num
print(tot)
What reasons are there for preferring the second version here? (Hint: open a tool like the
Performance Monitor on your computer, and watch the memory usage. How big can you make
the list before you get a fatal memory error in the first version?)
In a similar way, when working with files, we often have an option to read the whole file
contents into a single string, or we can read one line at a time and process each line as we read
it. Line-at-a-time is the more traditional and perhaps safer way to do things — you’ll be able
to work comfortably no matter how large the file is. (And, of course, this mode of processing
the files was essential in the old days when computer memories were much smaller.) But you
may find whole-file-at-once is sometimes more convenient!
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Glossary
attribute Some state or value that belongs to a particular object. For example, tess has a
color.
canvas A surface within a window where drawing takes place.
control flow See flow of execution.
for loop A statement in Python for convenient repetition of statements in the body of the loop.
loop body Any number of statements nested inside a loop. The nesting is indicated by the fact
that the statements are indented under the for loop statement.
loop variable A variable used as part of a for loop. It is assigned a different value on each
iteration of the loop.
instance An object of a certain type, or class. tess and alex are different instances of the
class Turtle.
method A function that is attached to an object. Invoking or activating the method causes
the object to respond in some way, e.g. forward is the method when we say tess.
forward(100).
invoke An object has methods. We use the verb invoke to mean activate the method. Invoking
a method is done by putting parentheses after the method name, with some possible
arguments. So tess.forward() is an invocation of the forward method.
module A file containing Python definitions and statements intended for use in other Python
programs. The contents of a module are made available to the other program by using
the import statement.
object A “thing” to which a variable can refer. This could be a screen window, or one of the
turtles we have created.
range A built-in function in Python for generating sequences of integers. It is especially useful
when we need to write a for loop that executes a fixed number of times.
terminating condition A condition that occurs which causes a loop to stop repeating its body.
In the for loops we saw in this chapter, the terminating condition has been when there
are no more elements to assign to the loop variable.
block A group of consecutive statements with the same indentation.
body The block of statements in a compound statement that follows the header.
Boolean algebra Some rules for rearranging and reasoning about Boolean expressions.
Boolean expression An expression that is either true or false.
Boolean value There are exactly two Boolean values: True and False. Boolean values
result when a Boolean expression is evaluated by the Python interepreter. They have type
bool.
branch One of the possible paths of the flow of execution determined by conditional execution.
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chained conditional A conditional branch with more than two possible flows of execution. In
Python chained conditionals are written with if ... elif ... else statements.
comparison operator One of the six operators that compares two values: ==, !=, >, <, >=,
and <=.
condition The Boolean expression in a conditional statement that determines which branch is
executed.
conditional statement A statement that controls the flow of execution depending on some
condition. In Python the keywords if, elif, and else are used for conditional statements.
logical operator One of the operators that combines Boolean expressions: and, or, and not.
nesting One program structure within another, such as a conditional statement inside a branch
of another conditional statement.
prompt A visual cue that tells the user that the system is ready to accept input data.
truth table A concise table of Boolean values that can describe the semantics of an operator.
type conversion An explicit function call that takes a value of one type and computes a corresponding value of another type.
algorithm A step-by-step process for solving a category of problems.
body The statements inside a loop.
bump Programmer slang. Synonym for increment.
continue statement A statement that causes the remainder of the current iteration of a loop
to be skipped. The flow of execution goes back to the top of the loop, evaluates the
condition, and if this is true the next iteration of the loop will begin.
counter A variable used to count something, usually initialized to zero and incremented in the
body of a loop.
cursor An invisible marker that keeps track of where the next character will be printed.
decrement Decrease by 1.
definite iteration A loop where we have an upper bound on the number of times the body will
be executed. Definite iteration is usually best coded as a for loop.
escape sequence An escape character, \, followed by one or more printable characters used to
designate a nonprintable character.
increment Both as a noun and as a verb, increment means to increase by 1.
infinite loop A loop in which the terminating condition is never satisfied.
indefinite iteration A loop where we just need to keep going until some condition is met. A
while statement is used for this case.
initialization (of a variable) To initialize a variable is to give it an initial value. Since in
Python variables don’t exist until they are assigned values, they are initialized when
they are created. In other programming languages this is not the case, and variables can
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be created without being initialized, in which case they have either default or garbage
values.
iteration Repeated execution of a set of programming statements.
loop The construct that allows allows us to repeatedly execute a statement or a group of statements until a terminating condition is satisfied.
loop variable A variable used as part of the terminating condition of a loop.
meta-notation Extra symbols or notation that helps describe other notation. Here we introduced square brackets, ellipses, italics, and bold as meta-notation to help describe
optional, repeatable, substitutable and fixed parts of the Python syntax.
middle-test loop A loop that executes some of the body, then tests for the exit condition, and
then may execute some more of the body. We don’t have a special Python construct for
this case, but can use while and break together.
nested loop A loop inside the body of another loop.
newline A special character that causes the cursor to move to the beginning of the next line.
post-test loop A loop that executes the body, then tests for the exit condition. We don’t have
a special Python construct for this, but can use while and break together.
pre-test loop A loop that tests before deciding whether the execute its body. for and while
are both pre-test loops.
tab A special character that causes the cursor to move to the next tab stop on the current line.
trichotomy Given any real numbers a and b, exactly one of the following relations holds: a <
b, a > b, or a == b. Thus when you can establish that two of the relations are false, you
can assume the remaining one is true.
trace To follow the flow of execution of a program by hand, recording the change of state of
the variables and any output produced.
Exercises
1. Assume the days of the week are numbered 0,1,2,3,4,5,6 from Sunday to Saturday. Write
a function which is given the day number, and it returns the day name (a string).
2. You go on a wonderful holiday (perhaps to jail, if you don’t like happy exercises) leaving
on day number 3 (a Wednesday). You return home after 137 sleeps. Write a general
version of the program which asks for the starting day number, and the length of your
stay, and it will tell you the name of day of the week you will return on.
3. Give the logical opposites of these conditions
(a) a > b
(b) a >= b
(c) a >= 18 and day == 3
(d) a >= 18 and day != 3
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4. What do these expressions evaluate to?
(a) 3 == 3
(b) 3 != 3
(c) 3 >= 4
(d) not (3 < 4)
5. Complete this truth table:
p
F
F
F
F
T
T
T
T
q
F
F
T
T
F
F
T
T
r
F
T
F
T
F
T
F
T
(not (p and q)) or r
?
?
?
?
?
?
?
?
6. Write a program which is given an exam mark, and it returns a string — the grade for
that mark — according to this scheme:
Mark
>= 75
[70-75)
[60-70)
[50-60)
[45-50)
[40-45)
< 40
Grade
First
Upper Second
Second
Third
F1 Supp
F2
F3
The square and round brackets denote closed and open intervals. A closed interval includes the number, and open interval excludes it. So 39.99999 gets grade F3, but 40 gets
grade F2. Assume
numbers = [83, 75, 74.9, 70, 69.9, 65, 60, 59.9, 55, 50,
49.9, 45, 44.9, 40, 39.9, 2, 0]
Test your code by printing the mark and the grade for all the elements in this list.
7. Write a program which, given the length of two sides of a right-angled triangle, returns
the length of the hypotenuse. (Hint: x ** 0.5 will return the square root.)
8. Write a program which, given the length of three sides of a triangle, will determine
whether the triangle is right-angled. Assume that the third argument to the function is
always the longest side. It will return True if the triangle is right-angled, or False
otherwise.
Hint: Floating point arithmetic is not always exactly accurate, so it is not safe to test
floating point numbers for equality. If a good programmer wants to know whether x is
equal or close enough to y, they would probably code it up as:
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threshold = 1e-7
if abs(x-y) < threshold:
...
# If x is approximately equal to y
9. Extend the above program so that the sides can be given to the function in any order.
10. If you’re intrigued by why floating point arithmetic is sometimes inaccurate, on a piece
of paper, divide 10 by 3 and write down the decimal result. You’ll find it does not
terminate, so you’ll need an infinitely long sheet of paper. The representation of numbers
in computer memory or on your calculator has similar problems: memory is finite, and
some digits may have to be discarded. So small inaccuracies creep in. Try this script:
1
2
3
4
import math
a = math.sqrt(2.0)
print(a, a*a)
print(a*a == 2.0)
1. Write a program that prints We like Python's turtles! 1000 times.
2. Write a program that uses a for loop to print
One of the months of the year is January
One of the months of the year is February
...
3. Suppose our turtle tess is at heading 0 — facing east. We execute the statement tess.
left(3645). What does tess do, and what is her final heading?
4. Assume you have the assignment numbers = [12, 10, 32, 3, 66, 17,
42, 99, 20]
(a) Write a loop that prints each of the numbers on a new line.
(b) Write a loop that prints each number and its square on a new line.
(c) Write a loop that adds all the numbers from the list into a variable called total. You
should set the total variable to have the value 0 before you start adding them up,
and print the value in total after the loop has completed.
(d) Print the product of all the numbers in the list. (product means all multiplied together)
5. Use for loops to make a turtle draw these regular polygons (regular means all sides the
same lengths, all angles the same):
• An equilateral triangle
• A square
• A hexagon (six sides)
• An octagon (eight sides)
6. A drunk pirate makes a random turn and then takes 100 steps forward, makes another random turn, takes another 100 steps, turns another random amount, etc. A social science
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student records the angle of each turn before the next 100 steps are taken. Her experimental data is [160, -43, 270, -97, -43, 200, -940, 17, -86]. (Positive
angles are counter-clockwise.) Use a turtle to draw the path taken by our drunk friend.
7. Enhance your program above to also tell us what the drunk pirate’s heading is after he
has finished stumbling around. (Assume he begins at heading 0).
8. If you were going to draw a regular polygon with 18 sides, what angle would you need
to turn the turtle at each corner?
9. At the interactive prompt, anticipate what each of the following lines will do, and then
record what happens. Score yourself, giving yourself one point for each one you anticipate correctly:
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
import turtle
window = turtle.Screen()
tess = turtle.Turtle()
tess.right(90)
tess.left(3600)
tess.right(-90)
tess.speed(10)
tess.left(3600)
tess.speed(0)
tess.left(3645)
tess.forward(-100)
10. Write a program to draw a shape like this:
Hints:
• Try this on a piece of paper, moving and turning your cellphone as if it was a turtle.
Watch how many complete rotations your cellphone makes before you complete the
star. Since each full rotation is 360 degrees, you can figure out the total number of
degrees that your phone was rotated through. If you divide that by 5, because there
are five points to the star, you’ll know how many degrees to turn the turtle at each
point.
• You can hide a turtle behind its invisibility cloak if you don’t want it shown. It
will still draw its lines if its pen is down. The method is invoked as tess.
hideturtle() . To make the turtle visible again, use tess.showturtle()
.
11. Write a program to draw a face of a clock that looks something like this:
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12. Create a turtle, and assign it to a variable. When you ask for its type, what do you get?
This chapter showed us how to sum a list of items, and how to count items. The counting
example also had an if statement that let us only count some selected items. we have break
to exit a loop, and continue to abandon the current iteration of the loop without ending the
loop.
Composition of list traversal, summing, counting, testing conditions and early exit is a rich
collection of building blocks that can be combined in powerful ways to create many functions
that are all slightly different.
The first six questions are typical functions you should be able to write using only these building
blocks.
1. Write a program to count how many odd numbers are in a list.
2. Sum up all the even numbers in a list.
3. Sum up all the negative numbers in a list.
4. Count how many words in a list have length 5.
5. Sum all the elements in a list up to but not including the first even number. (What if there
is no even number?)
6. Count how many words occur in a list up to and including the first occurrence of the word
“sam”. (What if “sam” does not occur?)
7. Add a print function to Newton’s sqrt algorithm that prints out better each time it is
calculated. Call your modified program with 25 as an argument and record the results.
8. Write a program that prints out the first n triangular numbers. A call to with n = 5
would produce the following output:
1
2
3
4
5
1
3
6
10
15
(hint: use a web search to find out what a triangular number is.)
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9. Write a program which prints True when n is a prime number and False otherwise.
10. Revisit the drunk pirate problem. This time, the drunk pirate makes a turn, and then
takes some steps forward, and repeats this. Our social science student now records pairs
of data: the angle of each turn, and the number of steps taken after the turn. Her experimental data is [(160, 20), (-43, 10), (270, 8), (-43, 12)]. Use a turtle to draw the path
taken by our drunk friend.
11. Many interesting shapes can be drawn by the turtle by giving a list of pairs like we did
above, where the first item of the pair is the angle to turn, and the second item is the
distance to move forward. Set up a list of pairs so that the turtle draws a house with a
cross through the centre, as show here. This should be done without going over any of
the lines / edges more than once, and without lifting your pen.
12. Recall the digit counting program. What will it print with n = 0? Modify it to print
1 for this case. Why does a call with n = -24 result in an infinite loop? (hint: -1//10
evaluates to -1) Modify num_digits so that it works correctly with any integer value.
13. Write a program that counts the number of even digits in n.
14. Write a program that computes the sum of the squares of the numbers in the list
numbers. For example a call with, numbers = [2, 3, 4] should print 4+9+16
which is 29.
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CHAPTER 4
Functions
Functions
In Python, a function is a named sequence of statements that belong together. Their primary
purpose is to help us organize programs into chunks that match how we think about the problem.
The syntax for a function definition is:
def <NAME>( <PARAMETERS> ):
<STATEMENTS>
We can make up any names we want for the functions we create, except that we can’t use a
name that is a Python keyword, and the names must follow the rules for legal identifiers.
There can be any number of statements inside the function, but they have to be indented from
the def. In the examples in this book, we will use the standard indentation of four spaces.
Function definitions are the second of several compound statements we will see, all of which
have the same pattern:
1. A header line which begins with a keyword and ends with a colon.
2. A body consisting of one or more Python statements, each indented the same amount —
the Python style guide recommends 4 spaces — from the header line.
We’ve already seen the for loop which follows this pattern.
So looking again at the function definition, the keyword in the header is def, which is followed
by the name of the function and some parameters enclosed in parentheses. The parameter list
may be empty, or it may contain any number of parameters separated from one another by commas. In either case, the parentheses are required. The parameters specifies what information, if
any, we have to provide in order to use the new function.
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Suppose we’re working with turtles, and a common operation we need is to draw squares.
“Draw a square” is an abstraction, or a mental chunk, of a number of smaller steps. So let’s
write a function to capture the pattern of this “building block”:
1
import turtle
2
3
4
5
6
7
8
9
def draw_square(animal, size):
"""
Make animal draw a square with sides of length size.
"""
for _ in range(4):
animal.forward(size)
animal.left(90)
10
11
12
window = turtle.Screen()
# Set up the window and
its attributes
window.bgcolor("lightgreen")
window.title("Alex meets a function")
˓→
13
14
15
16
17
18
alex = turtle.Turtle()
draw_square(alex, 50)
˓→the square
window.mainloop()
# Create alex
# Call the function to draw
This function is named draw_square. It has two parameters: one to tell the function which
turtle to move around, and the other to tell it the size of the square we want drawn. Make sure
you know where the body of the function ends — it depends on the indentation, and the blank
lines don’t count for this purpose!
Docstrings for documentation
If the first thing after the function header is a string, it is treated as a docstring and gets special
treatment in Python and in some programming tools.
Docstrings are the key way to document our functions in Python and the documentation part is
important. Because whoever calls our function shouldn’t have to need to know what is going
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on in the function or how it works; they just need to know what arguments our function takes,
what it does, and what the expected result is. Enough to be able to use the function without
having to look underneath. This goes back to the concept of abstraction of which we’ll talk
more about.
Docstrings are usually formed using triple-quoted strings as they allow us to easily expand the
docstring later on should we want to write more than a one-liner.
Just to differentiate from comments, a string at the start of a function (a docstring) is retrievable by Python tools at runtime. By contrast, comments are completely eliminated when the
program is parsed.
Defining the function just tells Python how to do a particular task, not to perform it. In order to
execute a function we need to make a function call. We’ve already seen how to call some builtin functions like print, range and int. Function calls contain the name of the function being
executed followed by a list of values, called arguments, which are assigned to the parameters
in the function definition. So in the second last line of the program, we call the function, and
pass alex as the turtle to be manipulated, and 50 as the size of the square we want. While the
function is executing, then, the variable size refers to the value 50, and the variable animal
refers to the same turtle instance that the variable alex refers to. We called it animal to signify
that there is no meaning to the name you give a function argument.
Once we’ve defined a function, we can call it as often as we like, and its statements will be
executed each time we call it. And we could use it to get any of our turtles to draw a square. In
the next example, we’ve changed the draw_square function a little, and we get tess to draw
15 squares, with some variations.
1
import turtle
2
3
4
5
6
7
8
def draw_multicolor_square(animal, size):
"""Make animal draw a multi-color square of given size."
˓→""
for color in ["red", "purple", "hotpink", "blue"]:
animal.color(color)
animal.forward(size)
animal.left(90)
9
10
11
window = turtle.Screen()
˓→attributes
window.bgcolor("lightgreen")
# Set up the window and its
12
13
14
tess = turtle.Turtle()
˓→attributes
tess.pensize(3)
# Create tess and set some
15
16
17
18
19
20
size = 20
# Size of the smallest square
for _ in range(15):
draw_multicolor_square(tess, size)
size += 10
# Increase the size for next
˓→time
tess.forward(10)
# Move tess along a little
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tess.right(18)
#
and give her some turn
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window.mainloop()
Functions can call other functions
Let’s assume now we want a function to draw a rectangle. We need to be able to call the
function with different arguments for width and height. And, unlike the case of the square, we
cannot repeat the same thing 4 times, because the four sides are not equal.
So we eventually come up with this rather nice code that can draw a rectangle.
1
2
3
4
5
6
7
def draw_rectangle(animal, width, height):
"""Get animal to draw a rectangle of given width and
˓→height."""
for _ in range(2):
animal.forward(width)
animal.left(90)
animal.forward(height)
animal.left(90)
Thinking like a scientist involves looking for patterns and relationships. In the code above,
we’ve done that to some extent. We did not just draw four sides. Instead, we spotted that we
could draw the rectangle as two halves, and used a loop to repeat that pattern twice.
But now we might spot that a square is a special kind of rectangle. We already have a function
that draws a rectangle, so we can use that to draw our square.
1
2
def draw_square(animal, size):
# A new version of
˓→draw_square
draw_rectangle(animal, size, size)
There are some points worth noting here:
• Functions can call other functions.
• Rewriting draw_square like this captures the relationship that we’ve spotted between
squares and rectangles.
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• A caller of this function might say draw_square(tess, 50). The parameters of
this function, animal and size, are assigned the values of the tess object, and the int
50 respectively.
• In the body of the function they are just like any other variable.
• When the call is made to draw_rectangle, the values in variables animal and
size are fetched first, then the call happens. So as we enter the top of function
draw_rectangle, its variable animal is assigned the tess object, and width and
height in that function are both given the value 50.
So far, it may not be clear why it is worth the trouble to create all of these new functions.
Actually, there are a lot of reasons, but this example demonstrates two:
1. Creating a new function gives us an opportunity to name a group of statements. Functions
can simplify a program by hiding a complex computation behind a single command. The
function (including its name) can capture our mental chunking, or abstraction, of the
problem.
2. Creating a new function can make a program smaller by eliminating repetitive code.
As we might expect, we have to create a function before we can execute it. In other words, the
function definition has to be executed before the function is called.
Flow of execution
In order to ensure that a function is defined before its first use, we have to know the order in
which statements are executed, which is called the flow of execution.
Execution always begins at the first statement of the program. Statements are executed one at
a time, in order from top to bottom.
Function definitions do not alter the flow of execution of the program, but remember that statements inside the function are not executed until the function is called. Although it is not common, we can define one function inside another. In this case, the inner definition isn’t executed
until the outer function is called.
Function calls are like a detour in the flow of execution. Instead of going to the next statement,
the flow jumps to the first line of the called function, executes all the statements there, and then
comes back to pick up where it left off.
That sounds simple enough, until we remember that one function can call another. While in the
middle of one function, the program might have to execute the statements in another function.
But while executing that new function, the program might have to execute yet another function!
Fortunately, Python is adept at keeping track of where it is, so each time a function completes,
the program picks up where it left off in the function that called it. When it gets to the end of
the program, it terminates.
What’s the moral of this sordid tale? When we read a program, don’t read from top to bottom.
Instead, follow the flow of execution.
As a simple example, let’s consider the following program:
4.3. Flow of execution
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import turtle
2
3
4
5
6
def draw_square(animal, size):
for _ in range(4):
animal.forward(size)
animal.left(90)
7
8
window = turtle.Screen()
˓→attributes
# Set up the window and its
9
10
tess = turtle.Turtle()
˓→attributes
# Create tess and set some
11
12
draw_square(tess, 50)
13
14
window.mainloop()
The Python interpreter reads this script line by line. At the first line the turtle module is
imported. We then define draw_square, which contains the instructions for a given turtle
to draw a square. However, nothing happens yet. We then go on to define a window, and
our charming turtle tess. The next line calls ``draw_square, asking tess to
draw a square with sides of length 50. Finally, window.mainloop() actually runs these
executions, and you will see tess draw a square on the screen.
Being able to trace your program is a valuable skill for a programmer.
Functions that require arguments
Most functions require arguments: the arguments provide for generalization. For example, if
we want to find the absolute value of a number, we have to indicate what the number is. Python
has a built-in function for computing the absolute value:
>>> abs(5)
5
>>> abs(-5)
5
In this example, the arguments to the abs function are 5 and -5.
Some functions take more than one argument. For example the built-in function pow takes
two arguments, the base and the exponent. Inside the function, the values that are passed get
assigned to variables called parameters.
>>> pow(2, 3)
8
>>> pow(7, 4)
2401
Another built-in function that takes more than one argument is max.
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>>> max(7, 11)
11
>>> max(4, 1, 17, 2, 12)
17
>>> max(3 * 11, 5**3, 512 - 9, 1024**0)
503
max can be passed any number of arguments, separated by commas, and will return the largest
value passed. The arguments can be either simple values or expressions. In the last example,
503 is returned, since it is larger than 33, 125, and 1.
Functions that return values
All the functions in the previous section return values. Calling each of these functions generates
a value, which we usually assign to a variable or use as part of an expression.
1
2
biggest = max(3, 7, 2, 5)
x = abs(3 - 11) + 10
So an important difference between these functions and one like draw_square is that
draw_square was not executed because we wanted it to compute a value — on the contrary,
we wrote draw_square because we wanted it to execute a sequence of steps that caused the
turtle to draw.
A function that returns a value is called a fruitful function in this book. The opposite of
a fruitful function is void function — one that is not executed for its resulting value, but is
executed because it does something useful. (Languages like Java, C#, C and C++ use the term
“void function”, other languages like Pascal call it a procedure.) Even though void functions
are not executed for their resulting value, Python always wants to return something. So if the
programmer doesn’t arrange to return a value, Python will automatically return the value None.
How do we write our own fruitful function? In the exercises at the end of chapter 2 we saw the
standard formula for compound interest, which we’ll now write as a fruitful function:
1
2
3
4
5
def final_amount(p, r, n, t):
"""
Apply the compound interest formula to p
to produce the final amount.
"""
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7
8
a = p * (1 + r/n) ** (n*t)
return a
# This is new, and makes the function
˓→fruitful.
9
10
11
12
13
# now that we have the function above, let us call it.
toInvest = float(input("How much do you want to invest?"))
fnl = final_amount(toInvest, 0.08, 12, 5)
print("At the end of the period you'll have", fnl)
• The return statement is followed an expression (a in this case). This expression will be
evaluated and returned to the caller as the “fruit” of calling this function.
• We prompted the user for the principal amount. The type of toInvest is a string, but
we need a number before we can work with it. Because it is money, and could have
decimal places, we’ve used the float type converter function to parse the string and
return a float.
• Notice how we entered the arguments for 8% interest, compounded 12 times per year,
for 5 years.
• When we run this, we get the output
At the end of the period you’ll have 14898.457083
This is a bit messy with all these decimal places, but remember that Python doesn’t
understand that we’re working with money: it just does the calculation to the best of its
ability, without rounding. Later we’ll see how to format the string that is printed in such
a way that it does get nicely rounded to two decimal places before printing.
• The
line
toInvest = float(input("How much do you want to
invest?")) also shows yet another example of composition — we can call a
function like float, and its arguments can be the results of other function calls (like
input) that we’ve called along the way.
Notice something else very important here. The name of the variable we pass as an argument
— toInvest — has nothing to do with the name of the parameter — p. It is as if p =
toInvest is executed when final_amount is called. It doesn’t matter what the value was
named in the caller, in final_amount its name is p.
These short variable names are getting quite tricky, so perhaps we’d prefer one of these versions
instead:
1
2
3
4
5
def final_amount_v2(principal_amount, nominal_percentage_
˓→rate,
num_times_per_year,
˓→years):
a = principal_amount * (1 + nominal_percentage_rate /
num_times_per_year) ** (num_times_
˓→per_year*years)
return a
6
7
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def final_amount_v3(amount, rate, compounded, years):
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9
a = amount * (1 + rate/compounded) ** (componded*years)
return a
10
11
12
13
14
15
16
def final_amount_v4(amount, rate, compounded, years):
"""
The a in final_amount_v3 was a useless asignment.
We might as well skip it.
"""
return amount * (1 + rate/compounded) **
˓→(componded*years)
They all do the same thing. Use your judgement to write code that can be best understood by
other humans! Short variable names should generally be avoided, unless when short variables
make more sense. This happens in particular with mathematical equations, where it’s perfectly
fine to use x, y, etc.
Variables and parameters are local
When we create a local variable inside a function, it only exists inside the function, and we
cannot use it outside. For example, consider again this function:
1
2
3
def final_amount(p, r, n, t):
a = p * (1 + r/n) ** (n*t)
return a
If we try to use a, outside the function, we’ll get an error:
>>> a
NameError: name 'a' is not defined
The variable a is local to final_amount, and is not visible outside the function.
Additionally, a only exists while the function is being executed — we call this its lifetime.
When the execution of the function terminates, the local variables are destroyed.
Parameters are also local, and act like local variables. For example, the lifetimes of p, r, n, t
begin when final_amount is called, and the lifetime ends when the function completes its
execution.
So it is not possible for a function to set some local variable to a value, complete its execution,
and then when it is called again next time, recover the local variable. Each call of the function
creates new local variables, and their lifetimes expire when the function returns to the caller.
Turtles Revisited
Now that we have fruitful functions, we can focus our attention on reorganizing our code so that
it fits more nicely into our mental chunks. This process of rearrangement is called refactoring
4.6. Variables and parameters are local
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the code.
Two things we’re always going to want to do when working with turtles is to create the window
for the turtle, and to create one or more turtles. We could write some functions to make these
tasks easier in future:
1
import turtle
2
3
4
5
6
7
8
9
10
11
def make_window(color, title):
"""
Set up the window with the given background color and
˓→title.
Returns the new window.
"""
window = turtle.Screen()
window.bgcolor(color)
window.title(title)
return window
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13
14
15
16
17
18
19
20
21
22
def make_turtle(color, size):
"""
Set up a turtle with the given color and pensize.
Returns the new turtle.
"""
animal = turtle.Turtle()
animal.color(color)
animal.pensize(size)
return animal
23
24
25
26
27
28
wn =
tess
alex
dave
make_window("lightgreen", "Tess and Alex dancing")
= make_turtle("hotpink", 5)
= make_turtle("black", 1)
= make_turtle("yellow", 2)
The trick about refactoring code is to anticipate which things we are likely to want to change
each time we call the function: these should become the parameters, or changeable parts, of the
functions we write.
Glossary
argument A value provided to a function when the function is called. This value is assigned
to the corresponding parameter in the function. The argument can be the result of an
expression which may involve operators, operands and calls to other fruitful functions.
body The second part of a compound statement. The body consists of a sequence of statements
all indented the same amount from the beginning of the header. The standard amount of
indentation used within the Python community is 4 spaces.
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compound statement A statement that consists of two parts:
1. header - which begins with a keyword determining the statement type, and ends
with a colon.
2. body - containing one or more statements indented the same amount from the
header.
The syntax of a compound statement looks like this:
keyword ... :
statement
statement ...
docstring A special string that is attached to a function as its __doc__ attribute. Tools like
Spyder can use docstrings to provide documentation or hints for the programmer. When
we get to modules, classes, and methods, we’ll see that docstrings can also be used there.
flow of execution The order in which statements are executed during a program run.
frame A box in a stack diagram that represents a function call. It contains the local variables
and parameters of the function.
function A named sequence of statements that performs some useful operation. Functions
may or may not take parameters and may or may not produce a result.
function call A statement that executes a function. It consists of the name of the function
followed by a list of arguments enclosed in parentheses.
function composition Using the output from one function call as the input to another.
function definition A statement that creates a new function, specifying its name, parameters,
and the statements it executes.
fruitful function A function that returns a value when it is called.
header line The first part of a compound statement. A header line begins with a keyword and
ends with a colon (:)
import statement A statement which permits functions and variables defined in another
Python module to be brought into the environment of another script. To use the features
of the turtle, we need to first import the turtle module.
lifetime Variables and objects have lifetimes — they are created at some point during program
execution, and will be destroyed at some time.
local variable A variable defined inside a function. A local variable can only be used inside
its function. Parameters of a function are also a special kind of local variable.
parameter A name used inside a function to refer to the value which was passed to it as an
argument.
refactor A fancy word to describe reorganizing our program code, usually to make it more
understandable. Typically, we have a program that is already working, then we go back
to “tidy it up”. It often involves choosing better variable names, or spotting repeated
patterns and moving that code into a function.
4.8. Glossary
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stack diagram A graphical representation of a stack of functions, their variables, and the
values to which they refer.
traceback A list of the functions that are executing, printed when a runtime error occurs. A
traceback is also commonly refered to as a stack trace, since it lists the functions in the
order in which they are stored in the runtime stack.
void function The opposite of a fruitful function: one that does not return a value. It is
executed for the work it does, rather than for the value it returns.
Exercises
1. Write a void (non-fruitful) function to draw a square. Use it in a program to draw the
image shown below. Assume each side is 20 units. (Hint: notice that the turtle has
already moved away from the ending point of the last square when the program ends.)
2. Write a program to draw this. Assume the innermost square is 20 units per side, and each
successive square is 20 units bigger, per side, than the one inside it.
3. Write a void function draw_poly(t, n, sz) which makes a turtle draw a regular
polygon. When called with draw_poly(tess, 8, 50), it will draw a shape like
this:
4. Draw this pretty pattern.
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5. The two spirals in this picture differ only by the turn angle. Draw both.
6. Write a void function draw_equitriangle(t, sz) which calls draw_poly
from the previous question to have its turtle draw a equilateral triangle.
7. Write a fruitful function sum_to(n) that returns the sum of all integer numbers up to
and including n. So sum_to(10) would be 1+2+3...+10 which would return the value
55.
8. Write a function area_of_circle(r) which returns the area of a circle of radius r.
9. Write a void function to draw a star, where the length of each side is 100 units. (Hint:
You should turn the turtle by 144 degrees at each point.)
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10. Extend your program above. Draw five stars, but between each, pick up the pen, move
forward by 350 units, turn right by 144, put the pen down, and draw the next star. You’ll
get something like this:
What would it look like if you didn’t pick up the pen?
Fruitful functions
Return values
The built-in functions we have used, such as abs, pow, int, max, and range, have produced
results. Calling each of these functions generates a value, which we usually assign to a variable
or use as part of an expression.
1
2
biggest = max(3, 7, 2, 5)
x = abs(3 - 11) + 10
We also wrote our own function to return the final amount for a compound interest calculation.
In this chapter, we are going to write more functions that return values, which we will call
fruitful functions, for want of a better name. The first example is area, which returns the area
of a circle with the given radius:
1
2
3
90
def area(radius):
b = 3.14159 * radius**2
return b
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We have seen the return statement before, but in a fruitful function the return statement
includes a return value. This statement means: evaluate the return expression, and then return it immediately as the result (the fruit) of this function. The expression provided can be
arbitrarily complicated, so we could have written this function like this:
1
2
def area(radius):
return 3.14159 * radius * radius
On the other hand, temporary variables like b above often make debugging easier.
Sometimes it is useful to have multiple return statements, one in each branch of a conditional.
We have already seen the built-in abs, now we see how to write our own:
1
2
3
4
5
def absolute_value(x):
if x < 0:
return -x
else:
return x
Another way to write the above function is to leave out the else and just follow the if
condition by the second return statement.
1
2
3
4
def absolute_value(x):
if x < 0:
return -x
return x
Think about this version and convince yourself it works the same as the first one.
Code that appears after a return statement, or any other place the flow of execution can never
reach, is called dead code, or unreachable code.
In a fruitful function, it is a good idea to ensure that every possible path through the program
hits a return statement. The following version of absolute_value fails to do this:
1
2
3
4
5
def bad_absolute_value(x):
if x < 0:
return -x
elif x > 0:
return x
This version is not correct because if x happens to be 0, neither condition is true, and the
function ends without hitting a return statement. In this case, the return value is a special
value called None:
>>> print(bad_absolute_value(0))
None
All Python functions return None whenever they do not return another value.
It is also possible to use a return statement in the middle of a for loop, in which case control
immediately returns from the function. Let us assume that we want a function which looks
4.10. Return values
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through a list of words. It should return the first 2-letter word. If there is not one, it should
return the empty string:
1
2
3
4
5
def find_first_2_letter_word(words):
for word in words:
if len(word) == 2:
return word
return ""
>>> find_first_2_letter_word(["This", "is", "a", "dead",
˓→"parrot"])
'is'
>>> find_first_2_letter_word(["I", "like", "cheese"])
''
Single-step through this code and convince yourself that in the first test case that we’ve provided, the function returns while processing the second element in the list: it does not have to
traverse the whole list.
Program development
At this point, you should be able to look at complete functions and tell what they do. Also, if
you have been doing the exercises, you have written some small functions. As you write larger
functions, you might start to have more difficulty, especially with runtime and semantic errors.
To deal with increasingly complex programs, we are going to suggest a technique called incremental development. The goal of incremental development is to avoid long debugging
sessions by adding and testing only a small amount of code at a time.
As an example, suppose we want to find the distance between two points, given by the coordinates (x1 , y1 ) and (x2 , y2 ). By the Pythagorean theorem, the distance is:
The first step is to consider what a distance function should look like in Python. In other
words, what are the inputs (parameters) and what is the output (return value)?
In this case, the two points are the inputs, which we can represent using four parameters. The
return value is the distance, which is a floating-point value.
Already we can write an outline of the function that captures our thinking so far:
1
2
def distance(x1, y1, x2, y2):
return 0.0
Obviously, this version of the function doesn’t compute distances; it always returns zero. But it
is syntactically correct, and it will run, which means that we can test it before we make it more
complicated.
To test the new function, we call it with sample values:
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>>> distance(1, 2, 4, 6)
0.0
We chose these values so that the horizontal distance equals 3 and the vertical distance equals
4; that way, the result is 5 (the hypotenuse of a 3-4-5 triangle). When testing a function, it is
useful to know the right answer.
At this point we have confirmed that the function is syntactically correct, and we can start
adding lines of code. After each incremental change, we test the function again. If an error
occurs at any point, we know where it must be — in the last line we added.
A logical first step in the computation is to find the differences x2 - x1 and y2 - y1 . We will refer
to those values using temporary variables named dx and dy.
1
2
3
4
def distance(x1, y1, x2, y2):
dx = x2 - x1
dy = y2 - y1
return 0.0
If we call the function with the arguments shown above, when the flow of execution gets to
the return statement, dx should be 3 and dy should be 4. We can check this by running the
function and printing the returned variable.
Next we compute the sum of squares of dx and dy:
1
2
3
4
5
def distance(x1, y1, x2, y2):
dx = x2 - x1
dy = y2 - y1
dsquared = dx*dx + dy*dy
return 0.0
Again, we could run the program at this stage and check the value of dsquared (which should
be 25).
Finally, using the fractional exponent 0.5 to find the square root, we compute and return the
result:
1
2
3
4
5
6
def distance(x1, y1, x2, y2):
dx = x2 - x1
dy = y2 - y1
dsquared = dx*dx + dy*dy
result = dsquared**0.5
return result
If that works correctly, you are done. Otherwise, you might want to inspect the value of
result before the return statement.
When you start out, you might add only a line or two of code at a time. As you gain more
experience, you might find yourself writing and debugging bigger conceptual chunks. Either
way, stepping through your code one line at a time and verifying that each step matches your
expectations can save you a lot of debugging time. As you improve your programming skills
you should find yourself managing bigger and bigger chunks: this is very similar to the way
4.11. Program development
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we learned to read letters, syllables, words, phrases, sentences, paragraphs, etc., or the way we
learn to chunk music — from individual notes to chords, bars, phrases, and so on.
The key aspects of the process are:
1. Start with a working skeleton program and make small incremental changes. At any
point, if there is an error, you will know exactly where it is.
2. Use temporary variables to refer to intermediate values so that you can easily inspect and
check them.
3. Once the program is working, relax, sit back, and play around with your options. (There
is interesting research that links “playfulness” to better understanding, better learning,
more enjoyment, and a more positive mindset about what you can achieve — so spend
some time fiddling around!) You might want to consolidate multiple statements into one
bigger compound expression, or rename the variables you’ve used, or see if you can make
the function shorter. A good guideline is to aim for making code as easy as possible for
others to read.
Here is another version of the function. It makes use of a square root function that is in the
math module (we’ll learn about modules shortly). Which do you prefer? Which looks “closer”
to the Pythagorean formula we started out with?
1
import math
2
3
4
def distance(x1, y1, x2, y2):
return math.sqrt( (x2-x1)**2 + (y2-y1)**2 )
>>> distance(1, 2, 4, 6)
5.0
Debugging with print
A powerful technique for debugging, is to insert extra print functions in carefully selected
places in your code. Then, by inspecting the output of the program, you can check whether the
algorithm is doing what you expect it to. Be clear about the following, however:
• You must have a clear solution to the problem, and must know what should happen before
you can debug a program. Work on solving the problem on a piece of paper (perhaps
using a flowchart to record the steps you take) before you concern yourself with writing
code. Writing a program doesn’t solve the problem — it simply automates the manual
steps you would take. So first make sure you have a pen-and-paper manual solution that
works. Programming then is about making those manual steps happen automatically.
• Do not write chatterbox functions. A chatterbox is a fruitful function that, in addition
to its primary task, also asks the user for input, or prints output, when it would be more
useful if it simply shut up and did its work quietly.
For example, we’ve seen built-in functions like range, max and abs. None of these
would be useful building blocks for other programs if they prompted the user for input,
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or printed their results while they performed their tasks.
So a good tip is to avoid calling print and input functions inside fruitful functions,
unless the primary purpose of your function is to perform input and output. The one
exception to this rule might be to temporarily sprinkle some calls to print into your
code to help debug and understand what is happening when the code runs, but these will
then be removed once you get things working.
Composition
As you should expect by now, you can call one function from within another. This ability is
called composition.
As an example, we’ll write a function that takes two points, the center of the circle and a point
on the perimeter, and computes the area of the circle.
Assume that the center point is stored in the variables xc and yc, and the perimeter point is in
xp and yp. The first step is to find the radius of the circle, which is the distance between the
two points. Fortunately, we’ve just written a function, distance, that does just that, so now
all we have to do is use it:
1
radius = distance(xc, yc, xp, yp)
The second step is to find the area of a circle with that radius and return it. Again we will use
one of our earlier functions:
1
2
result = area(radius)
return result
Wrapping that up in a function, we get:
1
2
3
4
def area_of_circle(xc, yc, xp, yp):
radius = distance(xc, yc, xp, yp)
result = area(radius)
return result
The temporary variables radius and result are useful for development, debugging, and
single-stepping through the code to inspect what is happening, but once the program is working,
we can make it more concise by composing the function calls:
1
2
def area_of_circle(xc, yc, xp, yp):
return area(distance(xc, yc, xp, yp))
Boolean functions
Functions can return Boolean values, which is often convenient for hiding complicated tests
inside functions. For example:
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2
3
4
5
6
def is_divisible(x, y):
""" Test if x is exactly divisible by y """
if x % y == 0:
return True
else:
return False
It is common to give Boolean functions names that sound like yes/no questions.
is_divisible returns either True or False to indicate whether the x is or is not divisible
by y.
We can make the function more concise by taking advantage of the fact that the condition of
the if statement is itself a Boolean expression. We can return it directly, avoiding the if
statement altogether:
1
2
def is_divisible(x, y):
return x % y == 0
This session shows the new function in action:
>>> is_divisible(6, 4)
False
>>> is_divisible(6, 3)
True
Boolean functions are often used in conditional statements:
1
2
3
4
if is_divisible(x, y):
... # Do something ...
else:
... # Do something else ...
It might be tempting to write something like:
1
if is_divisible(x, y) == True:
but the extra comparison is unnecessary.
Programming with style
Readability is very important to programmers, since in practice programs are read and modified
far more often then they are written. But, like most rules, we occasionaly break them. Most
of the code examples in this book will be consistent with the Python Enhancement Proposal 8
(PEP 8), a style guide developed by the Python community.
We’ll have more to say about style as our programs become more complex, but a few pointers
will be helpful already:
• use 4 spaces (instead of tabs) for indentation
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• limit line length to 78 characters
• when naming identifiers, use CamelCase for classes (we’ll get to those) and
lowercase_with_underscores for functions and variables
• place imports at the top of the file
• keep function definitions together below the import statements
• use docstrings to document functions
• use two blank lines to separate function definitions from each other
• keep top level statements, including function calls, together at the bottom of the program
Glossary
Boolean function A function that returns a Boolean value. The only possible values of the
bool type are False and True.
chatterbox function A function which interacts with the user (using input or print) when
it should not. Silent functions that just convert their input arguments into their output
results are usually the most useful ones.
composition (of functions) Calling one function from within the body of another, or using
the return value of one function as an argument to the call of another.
dead code Part of a program that can never be executed, often because it appears after a
return statement.
fruitful function A function that yields a return value instead of None.
incremental development A program development plan intended to simplify debugging by
adding and testing only a small amount of code at a time.
None A special Python value. One use in Python is that it is returned by functions that do not
execute a return statement with a return argument.
return value The value provided as the result of a function call.
scaffolding Code that is used during program development to assist with development and
debugging. The unit test code that we added in this chapter are examples of scaffolding.
temporary variable A variable used to store an intermediate value in a complex calculation.
Exercises
After completing each exercise, confirm that all the tests pass.
1. The four compass points can be abbreviated by single-letter strings as “N”, “E”, “S”, and
“W”. Write a function turn_clockwise that takes one of these four compass points
as its parameter, and returns the next compass point in the clockwise direction. Here are
some tests that should pass:
4.16. Glossary
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>>>turn_clockwise("N") == "E"
True
>>>turn_clockwise("W") == "N"
True
You might ask “What if the argument to the function is some other value?” For all other
cases, the function should return the value None.
2. Write a function day_name that converts an integer number 0 to 6 into the name of a
day. Assume day 0 is “Sunday”. Once again, return None if the arguments to the function
are not valid.
3. Write the inverse function day_num which is given a day name, and returns its number.
Once again, if this function is given an invalid argument, it should return None.
4. Write a function that helps answer questions like “‘Today is Wednesday. I leave on
holiday in 19 days time. What day will that be?”’ So the function must take a day name
and a delta argument — the number of days to add — and should return the resulting
day name:
day_add("Monday", 4) == "Friday"
day_add("Tuesday", 0) == "Tuesday"
day_add("Tuesday", 14) == "Tuesday"
day_add("Sunday", 100) == "Tuesday"
Hint: use the first two functions written above to help you write this one.
5. Can your day_add function already work with negative deltas? For example, -1 would
be yesterday, or -7 would be a week ago:
day_add("Sunday", -1) == "Saturday"
day_add("Sunday", -7) == "Sunday"
day_add("Tuesday", -100) == "Sunday"
If your function already works, explain why. If it does not work, make it work.
Hint: Play with some cases of using the modulus function % (introduced at the beginning of the previous chapter). Specifically, explore what happens to x % 7 when x is
negative.
6. Write a function days_in_month which takes the name of a month, and returns the
number of days in the month. Ignore leap years:
days_in_month("February") == 28
days_in_month("December") == 31
If the function is given invalid arguments, it should return None.
7. Write a function to_secs that converts hours, minutes and seconds to a total number
of seconds. Here are some tests that should pass:
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to_secs(2,
to_secs(2,
to_secs(0,
to_secs(0,
to_secs(0,
30, 10) == 9010
0, 0) == 7200
2, 0) == 120
0, 42) == 42
-10, 10) == -590
8. Extend to_secs so that it can cope with real values as inputs. It should always return
an integer number of seconds (truncated towards zero):
to_secs(2.5, 0, 10.71) == 9010
to_secs(2.433,0,0) == 8758
9. Write three functions that are the “inverses” of to_secs:
(a) hours_in returns the whole integer number of hours represented by a total number of seconds.
(b) minutes_in returns the whole integer number of left over minutes in a total
number of seconds, once the hours have been taken out.
(c) seconds_in returns the left over seconds represented by a total number of seconds.
You may assume that the total number of seconds passed to these functions is an integer.
Here are some test cases:
hours_in(9010) == 2
minutes_in(9010) == 30
seconds_in(9010) == 10
10. Which of these tests fail? Explain why.
3 % 4 == 0
3 % 4 == 3
3 / 4 == 0
3 // 4 == 0
3+4 * 2 == 14
4-2+2 == 0
len("hello, world!") == 13
11. Write a compare function that returns 1 if a > b, 0 if a == b, and -1 if a < b
compare(5, 4) == 1
compare(7, 7) == 0
compare(2, 3) == -1
compare(42, 1) == 1
12. Write a function called hypotenuse that returns the length of the hypotenuse of a right
triangle given the lengths of the two legs as parameters:
hypotenuse(3, 4) == 5.0
hypotenuse(12, 5) == 13.0
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hypotenuse(24, 7) == 25.0
hypotenuse(9, 12) == 15.0
13. Write a function slope(x1, y1, x2, y2) that returns the slope of the line through
the points (x1, y1) and (x2, y2). Be sure your implementation of slope can pass the
following tests:
slope(5,
slope(1,
slope(1,
slope(2,
3,
2,
2,
4,
4,
3,
3,
1,
2)
2)
3)
2)
==
==
==
==
1.0
0.0
0.5
2.0
Then use a call to slope in a new function named intercept(x1, y1, x2, y2)
that returns the y-intercept of the line through the points (x1, y1) and (x2, y2)
intercept(1, 6, 3, 12) == 3.0
intercept(6, 1, 1, 6) == 7.0
intercept(4, 6, 12, 8) == 5.0
14. Write a function called is_even(n) that takes an integer as an argument and returns
True if the argument is an even number and False if it is odd.
Add your own tests to the test suite.
15. Now write the function is_odd(n) that returns True when n is odd and False otherwise. Include unit tests for this function too.
Finally, modify it so that it uses a call to is_even to determine if its argument is an odd
integer, and ensure that its test still pass.
16. Write a function is_factor(f, n) that passes these tests:
is_factor(3, 12)
not is_factor(5, 12)
is_factor(7, 14)
not is_factor(7, 15)
is_factor(1, 15)
is_factor(15, 15)
not is_factor(25, 15)
17. Write is_multiple to satisfy these statements using is_factor from the previous
execise.
is_multiple(12, 3) is_multiple(12, 4) not is_multiple(12, 5) is_multiple(12, 6)
not is_multiple(12, 7)
18. Write the function f2c(t) designed to return the integer value of the nearest degree
Celsius for given temperature in Fahrenheit. (hint: you may want to make use of the
built-in function, round. Try printing round.__doc__ in a Python shell or looking
up help for the round function, and experimenting with it until you are comfortable with
how it works.)
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f2c(212) == 100
f2c(32) == 0
f2c(-40) == -40
f2c(36) == 2
f2c(37) == 3
f2c(38) == 3
f2c(39) == 4
# Boiling point of water
# Freezing point of water
# Wow, what an interesting case!
19. Now do the opposite: write the function c2f which converts Celsius to Fahrenheit:
c2f(0) == 32
c2f(100) == 212
c2f(-40) == -40
c2f(12) == 54
c2f(18) == 64
c2f(-48) == -54
Modifiers vs Pure Functions
Functions which take lists as arguments and change them during execution are called modifiers
and the changes they make are called side effects.
A pure function does not produce side effects. It communicates with the calling program only
through parameters, which it does not modify, and a return value. Let’s make a function which
doubles the items in a list:
1
2
3
4
5
6
7
8
def double_stuff(values):
""" Return a new list which contains
doubles of the elements in the list values.
"""
new_list = []
for value in values:
new_elem = 2 * value
new_list.append(new_elem)
9
return new_list
10
This version of double_stuff does not change its arguments:
>>>
>>>
>>>
[2,
>>>
[4,
things = [2, 5, 9]
more_things = double_stuff(things)
things
5, 9]
more_things
10, 18]
An early rule we saw for assignment said “first evaluate the right hand side, then assign the
resulting value to the variable”. So it is quite safe to assign the function result to the same
variable that was passed to the function:
4.17. Exercises
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>>>
>>>
>>>
[4,
things = [2, 5, 9]
things = double_stuff(things)
things
10, 18]
If however, we change the definition of double_stuff to the following:
1
2
3
4
def double_stuff(values):
""" Double the elements of values in-place. """
for index, value in enumerate(values):
values[index] = 2 * value
We get upon execution:
>>> things = [2, 5, 9]
>>> more_things = double_stuff(things)
>>> things
[4, 10, 18]
>>> more_things
None
We see that the original list was modified, while the function doesn’t return anything. This is a
good idea when building modifiers.
Which style is better?
In general, we recommend that you always use pure functions, and only use modifiers when
you are prepared to stick your head into a lion’s mouth, and have thought about the risks.
Some Tips, Tricks, and Common Errors
These are small summaries of ideas, tips, and commonly seen errors that might be helpful to
those beginning Python.
Functions
Functions help us with our mental chunking: they allow us to group together statements for a
high-level purpose, e.g. a function to sort a list of items, a function to make the turtle draw a
spiral, or a function to compute the mean and standard deviation of some measurements.
There are two kinds of functions: fruitful, or value-returning functions, which calculate and
return a value, and we use them because we’re primarily interested in the value they’ll return.
Void (non-fruitful) functions are used because they perform actions that we want done — e.g.
make a turtle draw a rectangle, or print the first thousand prime numbers. They always return
None — a special dummy value.
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Tip: None is not a string
Values like None, True and False are not strings: they are special values in Python, and are
in the list of keywords we gave in chapter 2 (Variables, expressions, and statements). Keywords
are special in the language: they are part of the syntax. So we cannot create our own variable
or function with a name True — we’ll get a syntax error. (Built-in functions are not privileged
like keywords: we can define our own variable or function called len, but we’d be silly to do
so!)
Along with the fruitful/void families of functions, there are two flavors of the return statement in Python: one that returns a useful value, and the other that returns nothing, or None.
And if we get to the end of any function and we have not explicitly executed any return
statement, Python automatically returns the value None.
Tip: Understand what the function needs to return
Perhaps nothing — some functions exists purely to perform actions rather than to calculate and
return a result. But if the function should return a value, make sure all execution paths do return
the value.
To make functions more useful, they are given parameters. So a function to make a turtle
draw a square might have two parameters — one for the turtle that needs to do the drawing,
and another for the size of the square. See the first example in Chapter 4 (Functions) — that
function can be used with any turtle, and for any size square. So it is much more general than a
function that always uses a specific turtle, say tess to draw a square of a specific size, say 30.
Tip: Use parameters to generalize functions
Understand which parts of the function will be hard-coded and unchangeable, and which parts
should become parameters so that they can be customized by the caller of the function.
Tip: Try to relate Python functions to ideas we already know
In math, we’re familiar with functions like f(x) = 3x + 5. We already understand that
when we call the function f(3) we make some association between the parameter x and the
argument 3. Try to draw parallels to argument passing in Python.
Quiz: Is the function f(z) = 3z + 5 the same as function f above?
Problems with logic and flow of control
We often want to know if some condition holds for any item in a list, e.g. “does the list have
any odd numbers?” This is a common mistake:
4.19. Problems with logic and flow of control
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2
3
4
5
6
7
def any_odd(xs): # Buggy version
""" Return True if there is an odd number in xs, a list
˓→of integers. """
for v in xs:
if v % 2 == 1:
return True
else:
return False
Can we spot two problems here? As soon as we execute a return, we’ll leave the function.
So the logic of saying “If I find an odd number I can return True” is fine. However, we
cannot return False after only looking at one item — we can only return False if we’ve
been through all the items, and none of them are odd. So line 6 should not be there, and line
7 has to be outside the loop. To find the second problem above, consider what happens if you
call this function with an argument that is an empty list. Here is a corrected version:
1
2
3
4
5
6
def any_odd(xs):
""" Return True if there is an odd number in xs, a list
˓→of integers. """
for v in xs:
if v % 2 == 1:
return True
return False
This “eureka”, or “short-circuit” style of returning from a function as soon as we are certain
what the outcome will be was first seen in Section 8.10, in the chapter on strings.
It is preferred over this one, which also works correctly:
1
2
3
4
5
6
7
8
9
10
def any_odd(xs):
""" Return True if there is an odd number in xs, a list
˓→of integers. """
count = 0
for v in xs:
if v % 2 == 1:
count += 1
# Count the odd numbers
if count > 0:
return True
else:
return False
The performance disadvantage of this one is that it traverses the whole list, even if it knows the
outcome very early on.
Tip: Think about the return conditions of the function
Do I need to look at all elements in all cases? Can I shortcut and take an early exit? Under
what conditions? When will I have to examine all the items in the list?
The code in lines 7-10 can also be tightened up. The expression count > 0 evaluates to
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a Boolean value, either True or False. The value can be used directly in the return
statement. So we could cut out that code and simply have the following:
1
2
3
4
5
6
7
8
def any_odd(xs):
""" Return True if there is an odd number in xs, a list
˓→of integers. """
count = 0
for v in xs:
if v % 2 == 1:
count += 1
# Count the odd numbers
return count > 0
# Aha! a programmer who understands
˓→that Boolean
#
expressions are not just used in
˓→if statements!
Although this code is tighter, it is not as nice as the one that did the short-circuit return as soon
as the first odd number was found.
Tip: Generalize your use of Booleans
Mature programmers won’t write if is_prime(n) == True: when they could say instead if is_prime(n): Think more generally about Boolean values, not just in the context
of if or while statements. Like arithmetic expressions, they have their own set of operators (and, or, not) and values (True, False) and can be assigned to variables, put into
lists, etc. A good resource for improving your use of Booleans is http://en.wikibooks.org/wiki/
Non-Programmer%27s_Tutorial_for_Python_3/Boolean_Expressions
Exercise time:
• How would we adapt this to make another function which returns True if all the numbers
are odd? Can you still use a short-circuit style?
• How would we adapt it to return True if at least three of the numbers are odd? Shortcircuit the traversal when the third odd number is found — don’t traverse the whole list
unless we have to.
Local variables
Functions are called, or activated, and while they’re busy they create their own stack frame
which holds local variables. A local variable is one that belongs to the current activation.
As soon as the function returns (whether from an explicit return statement or because Python
reached the last statement), the stack frame and its local variables are all destroyed. The important consequence of this is that a function cannot use its own variables to remember any kind
of state between different activations. It cannot count how many times it has been called, or
remember to switch colors between red and blue UNLESS it makes use of variables that are
global. Global variables will survive even after our function has exited, so they are the correct
way to maintain information between calls.
4.20. Local variables
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sz = 2
def h2():
""" Draw the next step of a spiral on each call. """
global sz
tess.turn(42)
tess.forward(sz)
sz += 1
1
2
3
4
5
6
7
This fragment assumes our turtle is tess. Each time we call h2() it turns, draws, and increases the global variable sz. Python always assumes that an assignment to a variable (as in
line 7) means that we want a new local variable, unless we’ve provided a global declaration
(on line 4). So leaving out the global declaration means this does not work.
Tip: Local variables do not survive when you exit the function
Use a Python visualizer like the one at http://netserv.ict.ru.ac.za/python3_viz to build a strong
understanding of function calls, stack frames, local variables, and function returns.
Tip: Assignment in a function creates a local variable
Any assignment to a variable within a function means Python will make a local variable, unless
we override with global.
String handling
There are only four really important operations on strings, and we’ll be able to do just about
anything. There are many more nice-to-have methods (we’ll call them sugar coating) that can
make life easier, but if we can work with the basic four operations smoothly, we’ll have a great
grounding.
• len(str) finds the length of a string.
• str[i] the subscript operation extracts the i’th character of the string, as a new string.
• str[i:j] the slice operation extracts a substring out of a string.
• str.find(target) returns the index where target occurs within the string, or -1 if it is not
found.
So if we need to know if “snake” occurs as a substring within s, we could write
1
2
if s.find("snake") >= 0:
if "snake" in s: ...
˓→sugar coating!
...
# Also works, nice-to-know
It would be wrong to split the string into words unless we were asked whether the word “snake”
occurred in the string.
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Suppose we’re asked to read some lines of data and find function definitions, e.g.: def
some_function_name(x, y):, and we are further asked to isolate and work with the
name of the function. (Let’s say, print it.)
1
2
3
4
5
6
s = "..."
˓→somewhere
def_pos = s.find("def ")
˓→line
if def_pos == 0:
˓→left margin
op_index = s.find("(")
˓→open parenthesis
fnname = s[4:op_index]
˓→name
print(fnname)
# Get the next line from
# Look for "def " in the
# If it occurs at the
# Find the index of the
# Slice out the function
# ... and work with it.
One can extend these ideas:
• What if the function def was indented, and didn’t start at column 0? The code would need
a bit of adjustment, and we’d probably want to be sure that all the characters in front of
the def_pos position were spaces. We would not want to do the wrong thing on data
like this: # I def initely like Python!
• We’ve assumed on line 3 that we will find an open parenthesis. It may need to be checked
that we did!
• We have also assumed that there was exactly one space between the keyword def and
the start of the function name. It will not work nicely for def f(x)
As we’ve already mentioned, there are many more “sugar-coated” methods that let us work
more easily with strings. There is an rfind method, like find, that searches from the end
of the string backwards. It is useful if we want to find the last occurrence of something. The
lower and upper methods can do case conversion. And the split method is great for
breaking a string into a list of words, or into a list of lines. We’ve also made extensive use in this
book of the format method. In fact, if we want to practice reading the Python documentation
and learning some new methods on our own, the string methods are an excellent resource.
Exercises:
• Suppose any line of text can contain at most one url that starts with “http://” and ends at
the next space in the line. Write a fragment of code to extract and print the full url if it is
present. (Hint: read the documentation for find. It takes some extra arguments, so you
can set a starting point from which it will search.)
• Suppose a string contains at most one substring “< ... >”. Write a fragment of code to
extract and print the portion of the string between the angle brackets.
Looping and lists
Computers are useful because they can repeat computation, accurately and fast. So loops are
going to be a central feature of almost all programs you encounter.
4.22. Looping and lists
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Tip: Don’t create unnecessary lists
Lists are useful if you need to keep data for later computation. But if you don’t need lists, it is
probably better not to generate them.
Here are two functions that both generate ten million random numbers, and return the sum of
the numbers. They both work.
1
2
import random
joe = random.Random()
3
4
5
6
7
8
9
def sum1():
""" Build a list of random numbers, then sum them """
xs = []
for i in range(10000000):
num = joe.randrange(1000) # Generate one random
˓→number
xs.append(num)
# Save it in our list
10
11
12
tot = sum(xs)
return tot
13
14
15
16
17
18
19
20
def sum2():
""" Sum the random numbers as we generate them """
tot = 0
for i in range(10000000):
num = joe.randrange(1000)
tot += num
return tot
21
22
23
print(sum1())
print(sum2())
What reasons are there for preferring the second version here? (Hint: open a tool like the
Performance Monitor on your computer, and watch the memory usage. How big can you make
the list before you get a fatal memory error in sum1?)
In a similar way, when working with files, we often have an option to read the whole file
contents into a single string, or we can read one line at a time and process each line as we read
it. Line-at-a-time is the more traditional and perhaps safer way to do things — you’ll be able
to work comfortably no matter how large the file is. (And, of course, this mode of processing
the files was essential in the old days when computer memories were much smaller.) But you
may find whole-file-at-once is sometimes more convenient!
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CHAPTER 5
Data Types
Strings
A compound data type
So far we have seen built-in types like int, float, bool, str and we’ve seen lists and pairs.
Strings, lists, and pairs are qualitatively different from the others because they are made up of
smaller pieces. In the case of strings, they’re made up of smaller strings each containing one
character.
Types that comprise smaller pieces are called compound data types. Depending on what we
are doing, we may want to treat a compound data type as a single thing, or we may want to
access its parts. This ambiguity is useful.
Working with strings as single things
We previously saw that each turtle instance has its own attributes and a number of methods
that can be applied to the instance. For example, we could set the turtle’s color, and we wrote
tess.turn(90).
Just like a turtle, a string is also an object. So each string instance has its own attributes and
methods.
For example:
>>> our_string = "Hello, World!"
>>> all_caps = our_string.upper()
>>> all_caps
'HELLO, WORLD!'
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upper is a method that can be invoked on any string object to create a new string, in which all
the characters are in uppercase. (The original string our_string remains unchanged.)
There are also methods such as lower, capitalize, and swapcase that do other interesting stuff.
To learn what methods are available, you can consult the Help documentation, look for string
methods, and read the documentation. Or, if you’re a bit lazier, simply type the following into
an editor like Spyder or PyScripter script:
1
2
our_string = "Hello, World!"
new_string = our_string.
When you type the period to select one of the methods of our_string, your editor might
pop up a selection window showing all the methods (there are around 70 of them — thank
goodness we’ll only use a few of those!) that could be used on your string.
When you type the name of the method, some further help about its parameter and return type,
and its docstring, will be displayed. This is a good example of a tool — PyScripter — using
the meta-information — the docstrings — provided by the module programmers.
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Working with the parts of a string
The indexing operator (Python uses square brackets to enclose the index) selects a single
character substring from a string:
>>> fruit = "banana"
>>> letter = fruit[1]
>>> print(letter)
The expression fruit[1] selects character number 1 from fruit, and creates a new string
containing just this one character. The variable m refers to the result. When we display m, we
could get a surprise:
a
Computer scientists always start counting from zero! The letter at subscript position zero of
"banana" is b. So at position [1] we have the letter a.
If we want to access the zero-eth letter of a string, we just place 0, or any expression that
evaluates to 0, inbetween the brackets:
>>> letter = fruit[0]
>>> print(letter)
b
The expression in brackets is called an index. An index specifies a member of an ordered
collection, in this case the collection of characters in the string. The index indicates which one
you want, hence the name. It can be any integer expression.
We can use enumerate to visualize the indices:
>>> fruit = "banana"
>>> list(enumerate(fruit))
[(0, 'b'), (1, 'a'), (2, 'n'), (3, 'a'), (4, 'n'), (5, 'a')]
Do not worry about enumerate at this point, we will see more of it in the chapter on lists.
Note that indexing returns a string — Python has no special type for a single character. It is
just a string of length 1.
We’ve also seen lists previously. The same indexing notation works to extract elements from a
list:
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>>> prime_numbers = [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31]
>>> prime_numbers[4]
11
>>> friends = ["Joe", "Zoe", "Brad", "Angelina", "Zuki",
˓→"Thandi", "Paris"]
>>> friends[3]
'Angelina'
Length
The len function, when applied to a string, returns the number of characters in a string:
>>> word = "banana"
>>> len(word)
6
To get the last letter of a string, you might be tempted to try something like this:
1
2
size = len(word)
last = word[size]
# ERROR!
That won’t work. It causes the runtime error IndexError: string index out of
range. The reason is that there is no character at index position 6 in "banana". Because we
start counting at zero, the six indexes are numbered 0 to 5. To get the last character, we have to
subtract 1 from the length of word:
1
2
size = len(word)
last = word[size-1]
Alternatively, we can use negative indices, which count backward from the end of the string.
The expression word[-1] yields the last letter, word[-2] yields the second to last, and so
on.
As you might have guessed, indexing with a negative index also works like this for lists.
Traversal and the for loop
A lot of computations involve processing a string one character at a time. Often they start at
the beginning, select each character in turn, do something to it, and continue until the end. This
pattern of processing is called a traversal. One way (a very bad way) to encode a traversal is
with a while statement:
1
2
3
4
5
112
ix = 0
while ix < len(fruit):
letter = fruit[ix]
print(letter)
ix += 1
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This loop traverses the string and displays each letter on a line by itself. It uses ix for the index,
which does not make it any clearer. The loop condition is ix < len(fruit), so when
ix is equal to the length of the string, the condition is false, and the body of the loop is not
executed. The last character accessed is the one with the index len(fruit)-1, which is the
last character in the string. However, this code is a lot longer than it needs to be, and not very
clear at all.
But we’ve previously seen how the for loop can easily iterate over the elements in a list and it
can do so for strings as well:
1
2
3
word="Banana"
for letter in word:
print(letter)
Each time through the loop, the next character in the string is assigned to the variable c. The
loop continues until no characters are left. Here we can see the expressive power the for loop
gives us compared to the while loop when traversing a string.
The following example shows how to use concatenation and a for loop to generate an
abecedarian series. Abecedarian refers to a series or list in which the elements appear in alphabetical order. For example, in Robert McCloskey’s book Make Way for Ducklings, the names of
the ducklings are Jack, Kack, Lack, Mack, Nack, Ouack, Pack, and Quack. This loop outputs
these names in order:
1
2
prefixes = "JKLMNOPQ"
suffix = "ack"
3
4
5
for p in prefixes:
print(p + suffix)
The output of this program is:
Jack
Kack
Lack
Mack
Nack
Oack
Pack
Qack
Of course, that’s not quite right because Ouack and Quack are misspelled. You’ll fix this as an
exercise below.
Slices
A substring of a string is obtained by taking a slice. Similarly, we can slice a list to refer to
some sublist of the items in the list:
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>>> phrase = "Pirates of the Caribbean"
>>> print(phrase[0:7])
Pirates
>>> print(phrase[11:14])
the
>>> print(phrase[13:24])
e Caribbean
>>> friends = ["Joe", "Zoe", "Brad", "Angelina", "Zuki",
˓→"Thandi", "Paris"]
>>> print(friends[2:4])
['Brad', 'Angelina']
The operator [n:m] returns the part of the string from the n’th character to the m’th character,
including the first but excluding the last. This behavior makes sense if you imagine the indices
pointing between the characters, as in the following diagram:
If you imagine this as a piece of paper, the slice operator [n:m] copies out the part of the paper
between the n and m positions. Provided m and n are both within the bounds of the string, your
result will be of length (m-n).
Three tricks are added to this: if you omit the first index (before the colon), the slice starts at
the beginning of the string (or list). If you omit the second index, the slice extends to the end
of the string (or list). Similarly, if you provide value for n that is bigger than the length of the
string (or list), the slice will take all the values up to the end. (It won’t give an “out of range”
error like the normal indexing operation does.) Thus:
>>> word = "banana"
>>> word[:3]
'ban'
>>> word[3:]
'ana'
>>> word[3:999]
'ana'
What do you think phrase[:] means? What about friends[4:]? phrase[-5:-3]?
String comparison
The comparison operators work on strings. To see if two strings are equal:
1
2
if word == "banana":
print("Yes, we have no bananas!")
Other comparison operations are useful for putting words in lexicographical order:
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1
2
3
4
5
6
if word < "banana":
print("Your word, " + word + ", comes before banana.")
elif word > "banana":
print("Your word, " + word + ", comes after banana.")
else:
print("Yes, we have no bananas!")
This is similar to the alphabetical order you would use with a dictionary, except that all the
uppercase letters come before all the lowercase letters. As a result:
Your word, Zebra, comes before banana.
A common way to address this problem is to convert strings to a standard format, such as all
lowercase, before performing the comparison. A more difficult problem is making the program
realize that zebras are not fruit.
Strings are immutable
It is tempting to use the [] operator on the left side of an assignment, with the intention of
changing a character in a string. For example:
1
2
3
greeting = "Hello, world!"
greeting[0] = 'J'
print(greeting)
# ERROR!
Instead of producing the output Jello, world!, this code produces the runtime error
TypeError: 'str' object does not support item assignment.
Strings are immutable, which means you can’t change an existing string. The best you can do
is create a new string that is a variation on the original:
1
2
3
greeting = "Hello, world!"
new_greeting = "J" + greeting[1:]
print(new_greeting)
The solution here is to concatenate a new first letter onto a slice of greeting. This operation
has no effect on the original string.
The in and not in operators
The in operator tests for membership. When both of the arguments to in are strings, in
checks whether the left argument is a substring of the right argument.
>>> "p" in "apple"
True
>>> "i" in "apple"
False
>>> "ap" in "apple"
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True
>>> "pa" in "apple"
False
Note that a string is a substring of itself, and the empty string is a substring of any other string.
(Also note that computer scientists like to think about these edge cases quite carefully!)
>>> "a" in "a"
True
>>> "apple" in "apple"
True
>>> "" in "a"
True
>>> "" in "apple"
True
The not in operator returns the logical opposite results of in:
>>> "x" not in "apple"
True
Combining the in operator with string concatenation using +, we can write a function that
removes all the vowels from a string:
1
2
3
4
5
6
7
def remove_vowels(phrase):
vowels = "aeiou"
string_sans_vowels = ""
for letter in phrase:
if letter.lower() not in vowels:
string_sans_vowels += letter
return string_sans_vowels
Important to note is the letter.lower() in line 5, without it, any uppercase vowels would
not be removed.
A find function
What does the following function do?
1
2
3
4
5
6
7
8
9
def my_find(haystack, needle):
"""
Find and return the index of needle in haystack.
Return -1 if needle does not occur in haystack.
"""
for index, letter in enumerate(haystack):
if letter == needle:
return index
return -1
Compare the output of the code above with what Python does itself with the code below:
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2
3
haystack = "Bananarama!"
print(haystack.find('a'))
print(my_find(haystack,'a'))
In a sense, find is the opposite of the indexing operator. Instead of taking an index and extracting the corresponding character, it takes a character and finds the index where that character
appears. If the character is not found, the function returns -1.
This is another example where we see a return statement inside a loop. If letter ==
needle, the function returns immediately, breaking out of the loop prematurely.
If the character doesn’t appear in the string, then the program exits the loop normally and
returns -1.
This pattern of computation is sometimes called a eureka traversal or short-circuit evaluation, because as soon as we find what we are looking for, we can cry “Eureka!”, take the
short-circuit, and stop looking.
Looping and counting
The following program counts the number of times the letter a appears in a string, and is
another example of the counter pattern introduced in Counting digits:
1
2
3
4
5
6
def count_a(text):
count = 0
for letter in text:
if letter == "a":
count += 1
return(count)
7
8
test(count_a("banana") == 3)
Optional parameters
To find the locations of the second or third occurrence of a character in a string, we can modify
the find function, adding a third parameter for the starting position in the search string:
1
2
3
4
5
def find2(haystack, needle, start):
for index,letter in enumerate(haystack[start:])
if letter == needle:
return index
return -1
6
7
8
9
test(find2("banana", "a", 2) == 3)
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The call find2("banana", "a", 2) now returns 3, the index of the first occurrence of
“a” in “banana” starting the search at index 2. What does find2("banana", "n", 3)
return? If you said, 4, there is a good chance you understand how find2 works.
Better still, we can combine find and find2 using an optional parameter:
1
2
3
4
5
def find(haystack, needle, start=0):
for index,letter in enumerate(haystack[start:]):
if letter == needle:
return index + start
return -1
When a function has an optional parameter, the caller may provide a matching argument. If
the third argument is provided to find, it gets assigned to start. But if the caller leaves the
argument out, then start is given a default value indicated by the assignment start=0 in the
function definition.
So the call find("banana", "a", 2) to this version of find behaves just like find2,
while in the call find("banana", "a"), start will be set to the default value of 0.
Adding another optional parameter to find makes it search from a starting position, up to but
not including the end position:
1
2
3
4
5
def find(haystack, needle, start=0,end=-1):
for index,letter in enumerate(haystack[start:end])
if letter == needle:
return index
return -1
The semantics of start and end in this function are precisely the same as they are in the
range function.
The built-in find method
Now that we’ve done all this work to write a powerful find function, we can reveal that strings
already have their own built-in find method. It can do everything that our code can do, and
more! Try all the examples listed above, and check the results!
The built-in find method is more general than our version. It can find substrings, not just
single characters:
>>> "banana".find("nan")
2
>>> "banana".find("na", 3)
4
Usually we’d prefer to use the methods that Python provides rather than reinvent our own
equivalents. But many of the built-in functions and methods make good teaching exercises,
and the underlying techniques you learn are your building blocks to becoming a proficient
programmer.
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The split method
One of the most useful methods on strings is the split method: it splits a single multi-word
string into a list of individual words, removing all the whitespace between them. (Whitespace
means any tabs, newlines, or spaces.) This allows us to read input as a single string, and split it
into words.
>>> phrase = "Well I never did said Alice"
>>> words = phrase.split()
>>> words
['Well', 'I', 'never', 'did', 'said', 'Alice']
Cleaning up your strings
We’ll often work with strings that contain punctuation, or tab and newline characters, especially, as we’ll see in a future chapter, when we read our text from files or from the Internet.
But if we’re writing a program, say, to count word frequencies or check the spelling of each
word, we’d prefer to strip off these unwanted characters.
We’ll show just one example of how to strip punctuation from a string. Remember that strings
are immutable, so we cannot change the string with the punctuation — we need to traverse the
original string and create a new string, omitting any punctuation:
1
punctuation = "!\"#$%&'()*+,-./:;<=>[email protected][\\]^_`{|}~"
2
3
4
5
6
7
8
def remove_punctuation(phrase):
phrase_sans_punct = ""
for letter in phrase:
if letter not in punctuation:
phrase_sans_punct += letter
return phrase_sans_punct
Setting up that first assignment is messy and error-prone. Fortunately, the Python string
module already does it for us. So we will make a slight improvement to this program — we’ll
import the string module and use its definition:
1
import string
2
3
4
5
6
7
8
def remove_punctuation(phrase):
phrase_sans_punct = ""
for letter in phrase:
if letter not in string.punctuation:
phrase_sans_punct += letter
return phrase_sans_punct
Try the examples below: “Well, I never did!”, said Alice. “Are you very, very, sure?”
Composing together this function and the split method from the previous section makes a
useful combination — we’ll clean out the punctuation, and split will clean out the newlines
and tabs while turning the string into a list of words:
5.1. Strings
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1
2
3
4
5
6
7
8
9
my_story = """
Pythons are constrictors, which means that they will
˓→'squeeze' the life
out of their prey. They coil themselves around their prey
˓→and with
each breath the creature takes the snake will squeeze a
˓→little tighter
until they stop breathing completely. Once the heart stops
˓→the prey
is swallowed whole. The entire animal is digested in the
˓→snake's
stomach except for fur or feathers. What do you think
˓→happens to the fur,
feathers, beaks, and eggshells? The 'extra stuff' gets
˓→passed out as --you guessed it --- snake POOP! """
10
11
12
words = remove_punctuation(my_story).split()
print(words)
The output:
['Pythons', 'are', 'constrictors', ... , 'it', 'snake',
˓→'POOP']
There are other useful string methods, but this book isn’t intended to be a reference manual. On
the other hand, the Python Library Reference is. Along with a wealth of other documentation,
it is available at the Python website.
The string format method
The easiest and most powerful way to format a string in Python 3 is to use the format method.
To see how this works, let’s start with a few examples:
1
2
phrase = "His name is {0}!".format("Arthur")
print(phrase)
3
4
5
6
7
8
9
name = "Alice"
age = 10
phrase = "I am {1} and I am {0} years old.".format(age,
˓→name)
print(phrase)
phrase = "I am {0} and I am {1} years old.".format(age,
˓→name)
print(phrase)
10
11
12
13
120
x = 4
y = 5
phrase = "2**10 = {0} and {1} * {2} = {3:f}".format(2**10,
˓→x, y, x * y)
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14
print(phrase)
Running the script produces:
His name is Arthur!
I am Alice and I am 10 years old.
I am 10 and I am Alice years old.
2**10 = 1024 and 4 * 5 = 20.000000
The template string contains place holders, ... {0} ... {1} ... {2} ... etc. The
format method substitutes its arguments into the place holders. The numbers in the place
holders are indexes that determine which argument gets substituted — make sure you understand line 6 above!
But there’s more! Each of the replacement fields can also contain a format specification —
it is always introduced by the : symbol (Line 13 above uses one.) This modifies how the
substitutions are made into the template, and can control things like:
• whether the field is aligned to the left <, center ^, or right >
• the width allocated to the field within the result string (a number like 10)
• the type of conversion (we’ll initially only force conversion to float, f, as we did in line 13
of the code above, or perhaps we’ll ask integer numbers to be converted to hexadecimal
using x)
• if the type conversion is a float, you can also specify how many decimal places are wanted
(typically, .2f is useful for working with currencies to two decimal places.)
Let’s do a few simple and common examples that should be enough for most needs. If you
need to do anything more esoteric, use help and read all the powerful, gory details.
1
2
3
name1 = "Paris"
name2 = "Whitney"
name3 = "Hilton"
4
5
6
7
8
9
10
print("Pi to three decimal places is {0:.3f}".format(3.
˓→1415926))
print("123456789 123456789 123456789 123456789 123456789
˓→123456789")
print("|||{0:<15}|||{1:^15}|||{2:>15}|||Born in {3}|||"
.format(name1,name2,name3,1981))
print("The decimal value {0} converts to hex value {0:x}"
.format(123456))
This script produces the output:
Pi to three decimal places is 3.142
123456789 123456789 123456789 123456789 123456789 123456789
|||Paris
|||
Whitney
|||
˓→Hilton|||Born in 1981|||
The decimal value 123456 converts to hex value 1e240
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You can have multiple placeholders indexing the same argument, or perhaps even have extra
arguments that are not referenced at all:
1
2
3
4
5
6
letter = """
Dear {0} {2}.
{0}, I have an interesting money-making proposition for
˓→you!
If you deposit $10 million into my bank account, I can
double your money ...
"""
7
8
9
print(letter.format("Paris", "Whitney", "Hilton"))
print(letter.format("Bill", "Henry", "Gates"))
This produces the following:
Dear Paris Hilton.
Paris, I have an interesting money-making proposition for
˓→you!
If you deposit $10 million into my bank account, I can
double your money ...
Dear Bill Gates.
Bill, I have an interesting money-making proposition for
˓→you!
If you deposit $10 million into my bank account I can
double your money ...
As you might expect, you’ll get an index error if your placeholders refer to arguments that you
do not provide:
>>> "hello {3}".format("Dave")
Traceback (most recent call last):
File "<interactive input>", line 1, in <module>
IndexError: tuple index out of range
The following example illustrates the real utility of string formatting. First, we’ll try to print a
table without using string formatting:
1
2
3
4
print("i\ti**2\ti**3\ti**5\ti**10\ti**20")
for i in range(1, 11):
print(i, "\t", i**2, "\t", i**3, "\t", i**5, "\t",
i**10, "\t",
˓→i**20)
This program prints out a table of various powers of the numbers from 1 to 10. (This assumes
that the tab width is 8. You might see something even worse than this if you tab width is set to
4.) In its current form it relies on the tab character ( \t) to align the columns of values, but this
breaks down when the values in the table get larger than the tab width:
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i
1
2
3
4
i**2
i**3
1
1
4
8
9
27
16
64
˓→1099511627776
5
25
125
˓→95367431640625
6
36
216
˓→3656158440062976
7
49
343
˓→79792266297612001
8
64
512
˓→1152921504606846976
9
81
729
˓→12157665459056928801
10
100
1000
˓→100000000000000000000
i**5
1
32
243
1024
i**10
1
1024
59049
1048576
i**20
1
1048576
3486784401
3125
9765625
7776
60466176
16807
282475249
32768
1073741824
59049
3486784401
100000
10000000000
One possible solution would be to change the tab width, but the first column already has more
space than it needs. The best solution would be to set the width of each column independently.
As you may have guessed by now, string formatting provides a much nicer solution. We can
also right-justify each field:
1
layout = "{0:>4}{1:>6}{2:>6}{3:>8}{4:>13}{5:>24}"
2
3
4
5
print(layout.format("i", "i**2", "i**3", "i**5", "i**10",
˓→"i**20"))
for i in range(1, 11):
print(layout.format(i, i**2, i**3, i**5, i**10, i**20))
Running this version produces the following (much more satisfying) output:
i
1
2
3
4
5
6
7
8
9
10
i**2
1
4
9
16
25
36
49
64
81
100
i**3
1
8
27
64
125
216
343
512
729
1000
i**5
1
32
243
1024
3125
7776
16807
32768
59049
100000
i**10
1
1024
59049
1048576
9765625
60466176
282475249
1073741824
3486784401
10000000000
i**20
1
1048576
3486784401
1099511627776
95367431640625
3656158440062976
79792266297612001
1152921504606846976
12157665459056928801
100000000000000000000
Summary
This chapter introduced a lot of new ideas. The following summary may prove helpful in
remembering what you learned.
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indexing ([]) Access a single character in a string using its position (starting from 0). Example: "This"[2] evaluates to "i".
length function (len) Returns the number of characters in a string.
len("happy") evaluates to 5.
Example:
for loop traversal (for) Traversing a string means accessing each character in the string, one
at a time. For example, the following for loop:
for ch in "Example":
...
executes the body of the loop 7 times with different values of ch each time.
slicing ([:]) A slice is a substring of a string. Example: 'bananas and cream'[3:6]
evaluates to ana (so does 'bananas and cream'[1:4]).
string comparison (>, <, >=, <=, ==, !=) The six common comparison operators
work with strings, evaluating according to lexicographical order. Examples: "apple"
< "banana" evaluates to True. "Zeta" < "Appricot" evaluates to False.
"Zebra" <= "aardvark" evaluates to True because all upper case letters precede
lower case letters.
in and not in operator (in, not in) The in operator tests for membership. In the case
of strings, it tests whether one string is contained inside another string. Examples:
"heck" in "I'll be checking for you." evaluates to True. "cheese"
in "I'll be checking for you." evaluates to False.
Glossary
compound data type A data type in which the values are made up of components, or elements, that are themselves values.
default value The value given to an optional parameter if no argument for it is provided in the
function call.
docstring A string constant on the first line of a function or module definition (and as we will
see later, in class and method definitions as well). Docstrings provide a convenient way
to associate documentation with code. Docstrings are also used by programming tools to
provide interactive help.
dot notation Use of the dot operator, ., to access methods and attributes of an object.
immutable data value A data value which cannot be modified. Assignments to elements or
slices (sub-parts) of immutable values cause a runtime error.
index A variable or value used to select a member of an ordered collection, such as a character
from a string, or an element from a list.
mutable data value A data value which can be modified. The types of all mutable values are
compound types. Lists and dictionaries are mutable; strings and tuples are not.
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optional parameter A parameter written in a function header with an assignment to a default
value which it will receive if no corresponding argument is given for it in the function
call.
short-circuit evaluation A style of programming that shortcuts extra work as soon as the
outcome is know with certainty. In this chapter our find function returned as soon as it
found what it was looking for; it didn’t traverse all the rest of the items in the string.
slice A part of a string (substring) specified by a range of indices. More generally, a
subsequence of any sequence type in Python can be created using the slice operator
(sequence[start:stop]).
traverse To iterate through the elements of a collection, performing a similar operation on
each.
whitespace Any of the characters that move the cursor without printing visible characters.
The constant string.whitespace contains all the white-space characters.
Exercises
1. What is the result of each of the following:
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
"Python"[1]
"Strings are sequences of characters."[5]
len("wonderful")
"Mystery"[:4]
"p" in "Pineapple"
"apple" in "Pineapple"
"pear" not in "Pineapple"
"apple" > "pineapple"
"pineapple" < "Peach"
2. Modify:
1
2
prefixes = "JKLMNOPQ"
suffix = "ack"
3
4
5
for letter in prefixes:
print(letter + suffix)
so that Ouack and Quack are spelled correctly.
3. Encapsulate
1
2
3
4
5
6
word = "banana"
count = 0
for letter in word:
if letter == "a":
count += 1
print(count)
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in a function named count_letters, and generalize it so that it accepts the string and
the letter as arguments. Make the function return the number of characters, rather than
print the answer. The caller should do the printing.
4. Now rewrite the count_letters function so that instead of traversing the string, it
repeatedly calls the find method, with the optional third parameter to locate new occurrences of the letter being counted.
5. Assign to a variable in your program a triple-quoted string that contains your favourite
paragraph of text — perhaps a poem, a speech, instructions to bake a cake, some inspirational verses, etc.
Write a function which removes all punctuation from the string, breaks the string into
a list of words, and counts the number of words in your text that contain the letter “e”.
Your program should print an analysis of the text like this:
Your text contains 243 words, of which 109 (44.8%)
˓→contain an "e".
6. Print a neat looking multiplication table like this:
1
2
3
4
5
6
7
8
9 10 11 12
:-------------------------------------------------1:
1
2
3
4
5
6
7
8
9 10 11 12
2:
2
4
6
8 10 12 14 16 18 20 22 24
3:
3
6
9 12 15 18 21 24 27 30 33 36
4:
4
8 12 16 20 24 28 32 36 40 44 48
5:
5 10 15 20 25 30 35 40 45 50 55 60
6:
6 12 18 24 30 36 42 48 54 60 66 72
7:
7 14 21 28 35 42 49 56 63 70 77 84
8:
8 16 24 32 40 48 56 64 72 80 88 96
9:
9 18 27 36 45 54 63 72 81 90 99 108
10:
10 20 30 40 50 60 70 80 90 100 110 120
11:
11 22 33 44 55 66 77 88 99 110 121 132
12:
12 24 36 48 60 72 84 96 108 120 132 144
7. Write a function that reverses its string argument, and satisfies these tests:
1
2
3
4
reverse("happy") == "yppah"
reverse("Python") == "nohtyP"
reverse("") == ""
reverse("a") == "a"
8. Write a function that mirrors its argument:
1
2
3
4
mirror("good") == "gooddoog"
mirror("Python") == "PythonnohtyP"
mirror("") == ""
mirror("a") == "aa"
9. Write a function that removes all occurrences of a given letter from a string:
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1
2
3
4
5
6
remove_letter("a",
remove_letter("a",
remove_letter("z",
remove_letter("i",
remove_letter("b",
remove_letter("b",
"apple") == "pple"
"banana") == "bnn"
"banana") == "banana"
"Mississippi") == "Msssspp"
"") = ""
"c") = "c"
10. Write a function that recognizes palindromes. (Hint: use your reverse function to
make this easy!):
1
2
3
4
5
6
7
is_palindrome("abba")
not is_palindrome("abab")
is_palindrome("tenet")
not is_palindrome("banana")
is_palindrome("straw warts")
is_palindrome("a")
# is_palindrome(""))
# Is an empty string a
˓→palindrome?
11. Write a function that counts how many times a substring occurs in a string:
1
2
3
4
5
6
count("is", "Mississippi") == 2
count("an", "banana") == 2
count("ana", "banana") == 2
count("nana", "banana") == 1
count("nanan", "banana") == 0
count("aaa", "aaaaaa") == 4
12. Write a function that removes the first occurrence of a string from another string:
1
2
3
4
remove("an", "banana") == "bana"
remove("cyc", "bicycle") == "bile"
remove("iss", "Mississippi") == "Missippi"
remove("eggs", "bicycle") == "bicycle"
13. Write a function that removes all occurrences of a string from another string:
1
2
3
4
remove_all("an", "banana") == "ba"
remove_all("cyc", "bicycle") == "bile"
remove_all("iss", "Mississippi") == "Mippi"
remove_all("eggs", "bicycle") == "bicycle"
There are only four really important operations on strings, and we’ll be able to do just about
anything. There are many more nice-to-have methods (we’ll call them sugar coating) that can
make life easier, but if we can work with the basic four operations smoothly, we’ll have a great
grounding.
• len(str) finds the length of a string.
• str[i] the subscript operation extracts the i’th character of the string, as a new string.
• str[i:j] the slice operation extracts a substring out of a string.
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• str.find(target) returns the index where target occurs within the string, or -1 if it is not
found.
So if we need to know if “snake” occurs as a substring within s, we could write
if s.find("snake") >= 0:
if "snake" in s: ...
˓→sugar coating!
1
2
...
# Also works, nice-to-know
It would be wrong to split the string into words unless we were asked whether the word “snake”
occurred in the string.
Suppose we’re asked to read some lines of data and find function definitions, e.g.: def
some_function_name(x, y):, and we are further asked to isolate and work with the
name of the function. (Let’s say, print it.)
s = "..."
˓→somewhere
def_pos = s.find("def ")
˓→line
if def_pos == 0:
˓→left margin
op_index = s.find("(")
˓→open parenthesis
fnname = s[4:op_index]
˓→name
print(fnname)
1
2
3
4
5
6
# Get the next line from
# Look for "def " in the
# If it occurs at the
# Find the index of the
# Slice out the function
# ... and work with it.
One can extend these ideas:
• What if the function def was indented, and didn’t start at column 0? The code would need
a bit of adjustment, and we’d probably want to be sure that all the characters in front of
the def_pos position were spaces. We would not want to do the wrong thing on data
like this: # I def initely like Python!
• We’ve assumed on line 3 that we will find an open parenthesis. It may need to be checked
that we did!
• We have also assumed that there was exactly one space between the keyword def and
the start of the function name. It will not work nicely for def f(x)
As we’ve already mentioned, there are many more “sugar-coated” methods that let us work
more easily with strings. There is an rfind method, like find, that searches from the end
of the string backwards. It is useful if we want to find the last occurrence of something. The
lower and upper methods can do case conversion. And the split method is great for
breaking a string into a list of words, or into a list of lines. We’ve also made extensive use in this
book of the format method. In fact, if we want to practice reading the Python documentation
and learning some new methods on our own, the string methods are an excellent resource.
Exercises:
• Suppose any line of text can contain at most one url that starts with “http://” and ends at
the next space in the line. Write a fragment of code to extract and print the full url if it is
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present. (Hint: read the documentation for find. It takes some extra arguments, so you
can set a starting point from which it will search.)
• Suppose a string contains at most one substring “< ... >”. Write a fragment of code to
extract and print the portion of the string between the angle brackets.
Tuples
Tuples are used for grouping data
We saw earlier that we could group together pairs of values by surrounding with parentheses.
Recall this example:
>>> year_born = ("Paris Hilton", 1981)
This is an example of a data structure — a mechanism for grouping and organizing data to
make it easier to use.
The pair is an example of a tuple. Generalizing this, a tuple can be used to group any number
of items into a single compound value. Syntactically, a tuple is a comma-separated sequence
of values. Although it is not necessary, it is conventional to enclose tuples in parentheses:
>>> julia = ("Julia", "Roberts", 1967, "Duplicity", 2009,
˓→"Actress", "Atlanta, Georgia")
The other thing that could be said somewhere around here, is that the parentheses are there to
disambiguate. For example, if we have a tuple nested within another tuple and the parentheses
weren’t there, how would we tell where the nested tuple begins/ends? Also: the creation of an
empty tuple is done like this: empty_tuple=()
Tuples are useful for representing what other languages often call records (or structs) — some
related information that belongs together, like your student record. There is no description of
what each of these fields means, but we can guess. A tuple lets us “chunk” together related
information and use it as a single thing.
Tuples support the same sequence operations as strings. The index operator selects an element
from a tuple.
>>> julia[2]
1967
But if we try to use item assignment to modify one of the elements of the tuple, we get an error:
>>> julia[0] = "X"
TypeError: 'tuple' object does not support item assignment
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So like strings, tuples are immutable. Once Python has created a tuple in memory, it cannot be
changed.
Of course, even if we can’t modify the elements of a tuple, we can always make the julia
variable reference a new tuple holding different information. To construct the new tuple, it is
convenient that we can slice parts of the old tuple and join up the bits to make the new tuple.
So if julia has a new recent film, we could change her variable to reference a new tuple that
used some information from the old one:
>>> julia = julia[:3] + ("Eat Pray Love", 2010) + julia[5:]
>>> julia
("Julia", "Roberts", 1967, "Eat Pray Love", 2010, "Actress",
˓→ "Atlanta, Georgia")
To create a tuple with a single element (but you’re probably not likely to do that too often),
we have to include the final comma, because without the final comma, Python treats the (5)
below as an integer in parentheses:
>>> tup = (5,)
>>> type(tup)
<class 'tuple'>
>>> x = (5)
>>> type(x)
<class 'int'>
Tuple assignment
Python has a very powerful tuple assignment feature that allows a tuple of variables on the left
of an assignment to be assigned values from a tuple on the right of the assignment. (We already
saw this used for pairs, but it generalizes.)
(name, surname, year_born, movie, year_movie, profession,
˓→birthplace) = julia
This does the equivalent of seven assignment statements, all on one easy line. One requirement
is that the number of variables on the left must match the number of elements in the tuple.
One way to think of tuple assignment is as tuple packing/unpacking.
In tuple packing, the values on the left are ‘packed’ together in a tuple:
>>> bob = ("Bob", 19, "CS")
# tuple packing
In tuple unpacking, the values in a tuple on the right are ‘unpacked’ into the variables/names
on the right:
>>> bob = ("Bob", 19, "CS")
>>> (name, age, studies) = bob
>>> name
'Bob'
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>>> age
19
>>> studies
'CS'
Once in a while, it is useful to swap the values of two variables. With conventional assignment
statements, we have to use a temporary variable. For example, to swap a and b:
1
2
3
temp = a
a = b
b = temp
Tuple assignment solves this problem neatly:
1
(a, b) = (b, a)
The left side is a tuple of variables; the right side is a tuple of values. Each value is assigned
to its respective variable. All the expressions on the right side are evaluated before any of the
assignments. This feature makes tuple assignment quite versatile.
Naturally, the number of variables on the left and the number of values on the right have to be
the same:
>>> (one, two, three, four) = (1, 2, 3)
ValueError: need more than 3 values to unpack
Tuples as return values
Functions can always only return a single value, but by making that value a tuple, we can
effectively group together as many values as we like, and return them together. This is very
useful — we often want to know some batsman’s highest and lowest score, or we want to find
the mean and the standard deviation, or we want to know the year, the month, and the day, or if
we’re doing some some ecological modelling we may want to know the number of rabbits and
the number of wolves on an island at a given time.
For example, we could write a function that returns both the area and the circumference of a
circle of radius r:
1
2
3
4
5
def circle_stats(r):
""" Return (circumference, area) of a circle of radius
˓→r """
circumference = 2 * math.pi * r
area = math.pi * r * r
return (circumference, area)
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Composability of Data Structures
We saw in an earlier chapter that we could make a list of pairs, and we had an example where
one of the items in the tuple was itself a list:
students = [
("John", ["CompSci", "Physics"]),
("Vusi", ["Maths", "CompSci", "Stats"]),
("Jess", ["CompSci", "Accounting", "Economics",
˓→"Management"]),
("Sarah", ["InfSys", "Accounting", "Economics", "CommLaw
˓→"]),
("Zuki", ["Sociology", "Economics", "Law", "Stats",
˓→"Music"])]
Tuples items can themselves be other tuples. For example, we could improve the information
about our movie stars to hold the full date of birth rather than just the year, and we could have
a list of some of her movies and dates that they were made, and so on:
julia_more_info = ( ("Julia", "Roberts"), (8, "October",
˓→1967),
"Actress", ("Atlanta", "Georgia"),
[ ("Duplicity", 2009),
("Notting Hill", 1999),
("Pretty Woman", 1990),
("Erin Brockovich", 2000),
("Eat Pray Love", 2010),
("Mona Lisa Smile", 2003),
("Oceans Twelve", 2004) ])
Notice in this case that the tuple has just five elements — but each of those in turn can be
another tuple, a list, a string, or any other kind of Python value. This property is known as
being heterogeneous, meaning that it can be composed of elements of different types.
Glossary
data structure An organization of data for the purpose of making it easier to use.
immutable data value A data value which cannot be modified. Assignments to elements or
slices (sub-parts) of immutable values cause a runtime error.
mutable data value A data value which can be modified. The types of all mutable values are
compound types. Lists and dictionaries are mutable; strings and tuples are not.
tuple An immutable data value that contains related elements. Tuples are used to group together related data, such as a person’s name, their age, and their gender.
tuple assignment An assignment to all of the elements in a tuple using a single assignment
statement. Tuple assignment occurs simultaneously rather than in sequence, making it
useful for swapping values.
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Exercises
1. We’ve said nothing in this chapter about whether you can pass tuples as arguments to a
function. Construct a small Python example to test whether this is possible, and write up
your findings.
2. Is a pair a generalization of a tuple, or is a tuple a generalization of a pair?
3. Is a pair a kind of tuple, or is a tuple a kind of pair?
Lists
A list is an ordered collection of values. The values that make up a list are called its elements,
or its items. We will use the term element or item to mean the same thing. Lists are similar to
strings, which are ordered collections of characters, except that the elements of a list can be of
any type. Lists and strings — and other collections that maintain the order of their items — are
called sequences.
List values
There are several ways to create a new list; the simplest is to enclose the elements in square
brackets ([ and ]):
1
2
numbers = [10, 20, 30, 40]
words = ["spam", "bungee", "swallow"]
The first example is a list of four integers. The second is a list of three strings. The elements
of a list don’t have to be the same type. The following list contains a string, a float, an integer,
and (amazingly) another list:
1
stuffs = ["hello", 2.0, 5, [10, 20]]
A list within another list is said to be nested.
Finally, a list with no elements is called an empty list, and is denoted [].
We have already seen that we can assign list values to variables or pass lists as parameters to
functions:
1
2
3
4
5
>>> vocabulary = ["apple", "cheese", "dog"]
>>> numbers = [17, 123]
>>> an_empty_list = []
>>> print(vocabulary, numbers, an_empty_list)
["apple", "cheese", "dog"] [17, 123] []
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Accessing elements
The syntax for accessing the elements of a list is the same as the syntax for accessing the
characters of a string — the index operator: [] (not to be confused with an empty list). The
expression inside the brackets specifies the index. Remember that the indices start at 0:
>>> numbers[0]
17
Any expression evaluating to an integer can be used as an index:
>>> numbers[9-8]
123
>>> numbers[1.0]
Traceback (most recent call last):
File "<interactive input>", line 1, in <module>
TypeError: list indices must be integers, not float
If you try to access or assign to an element that does not exist, you get a runtime error:
>>> numbers[2]
Traceback (most recent call last):
File "<interactive input>", line 1, in <module>
IndexError: list index out of range
It is common (but wrong!) to use a loop variable as a list index.
1
horsemen = ["war", "famine", "pestilence", "death"]
2
3
4
for i in [0, 1, 2, 3]:
print(horsemen[i])
Each time through the loop, the variable i is used as an index into the list, printing the i‘th
element. This pattern of computation is called a list traversal.
The above sample doesn’t need or use the index i for anything besides getting the items from
the list, so this more direct version — where the for loop gets the items — is much more
clear!
1
horsemen = ["war", "famine", "pestilence", "death"]
2
3
4
for h in horsemen:
print(h)
List length
The function len returns the length of a list, which is equal to the number of its elements. If
you are going to use an integer index to access the list, it is a good idea to use this value as the
upper bound of a loop instead of a constant. That way, if the size of the list changes, you won’t
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have to go through the program changing all the loops; they will work correctly for any size
list:
1
horsemen = ["war", "famine", "pestilence", "death"]
2
3
4
for i in range(len(horsemen)):
print(horsemen[i])
The last time the body of the loop is executed, i is len(horsemen) - 1, which is the
index of the last element. (But the version without the index looks even better now! The
version above is not the right way to do things!)
1
horsemen = ["war", "famine", "pestilence", "death"]
2
3
4
for horseman in horsemen:
print horseman
Although a list can contain another list, the nested list still counts as a single element in its
parent list. The length of this list is 4:
>>> len(["car makers", 1, ["Ford", "Toyota", "BMW"], [1, 2,
˓→3]])
4
List membership
in and not in are Boolean operators that test membership in a sequence. We used them
previously with strings, but they also work with lists and other sequences:
>>> horsemen = ["war", "famine", "pestilence", "death"]
>>> "pestilence" in horsemen
True
>>> "debauchery" in horsemen
False
>>> "debauchery" not in horsemen
True
Using this produces a more elegant version of the nested loop program we previously used to
count the number of students doing Computer Science in the section Nested Loops for Nested
Data:
1
2
3
4
5
6
students = [
("John", ["CompSci", "Physics"]),
("Vusi", ["Maths", "CompSci", "Stats"]),
("Jess", ["CompSci", "Accounting", "Economics",
˓→"Management"]),
("Sarah", ["InfSys", "Accounting", "Economics", "CommLaw
˓→"]),
("Zuki", ["Sociology", "Economics", "Law", "Stats",
˓→"Music"])]
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7
8
9
10
11
12
# Count how many students are taking CompSci
counter = 0
for name, subjects in students:
if "CompSci" in subjects:
counter += 1
13
14
print("The number of students taking CompSci is", counter)
List operations
The + operator concatenates lists:
>>>
>>>
>>>
>>>
[1,
first_list = [1, 2, 3]
second_list = [4, 5, 6]
both_lists = first_list + second_list
both_lists
2, 3, 4, 5, 6]
Similarly, the * operator repeats a list a given number of times:
>>>
[0,
>>>
[1,
[0] * 4
0, 0, 0]
[1, 2, 3] * 3
2, 3, 1, 2, 3, 1, 2, 3]
The first example repeats [0] four times. The second example repeats the list [1, 2, 3]
three times.
List slices
The slice operations we saw previously with strings let us work with sublists:
>>> a_list = ["a", "b", "c", "d", "e", "f"]
>>> a_list[1:3]
['b', 'c']
>>> a_list[:4]
['a', 'b', 'c', 'd']
>>> a_list[3:]
['d', 'e', 'f']
>>> a_list[:]
['a', 'b', 'c', 'd', 'e', 'f']
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Lists are mutable
Unlike strings, lists are mutable, which means we can change their elements. Using the index
operator on the left side of an assignment, we can update one of the elements:
>>> fruit = ["banana", "apple", "quince"]
>>> fruit[0] = "pear"
>>> fruit[2] = "orange"
>>> fruit
['pear', 'apple', 'orange']
The bracket operator applied to a list can appear anywhere in an expression. When it appears
on the left side of an assignment, it changes one of the elements in the list, so the first element
of fruit has been changed from "banana" to "pear", and the last from "quince" to
"orange". An assignment to an element of a list is called item assignment. Item assignment
does not work for strings:
>>> my_string = "TEST"
>>> my_string[2] = "X"
Traceback (most recent call last):
File "<interactive input>", line 1, in <module>
TypeError: 'str' object does not support item assignment
but it does for lists:
>>> my_list = ["T", "E", "S", "T"]
>>> my_list[2] = "X"
>>> my_list
['T', 'E', 'X', 'T']
With the slice operator we can update a whole sublist at once:
>>> a_list = ["a", "b", "c", "d", "e", "f"]
>>> a_list[1:3] = ["x", "y"]
>>> a_list
['a', 'x', 'y', 'd', 'e', 'f']
We can also remove elements from a list by assigning an empty list to them:
>>> a_list = ["a", "b", "c", "d", "e", "f"]
>>> a_list[1:3] = []
>>> a_list
['a', 'd', 'e', 'f']
And we can add elements to a list by squeezing them into an empty slice at the desired location:
>>> a_list = ["a", "d", "f"]
>>> a_list[1:1] = ["b", "c"]
>>> a_list
['a', 'b', 'c', 'd', 'f']
>>> a_list[4:4] = ["e"]
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>>> a_list
['a', 'b', 'c', 'd', 'e', 'f']
List deletion
Using slices to delete list elements can be error-prone. Python provides an alternative that is
more readable. The del statement removes an element from a list:
>>> a = ["one", "two", "three"]
>>> del a[1]
>>> a
['one', 'three']
As you might expect, del causes a runtime error if the index is out of range.
You can also use del with a slice to delete a sublist:
>>> a_list = ["a", "b", "c", "d", "e", "f"]
>>> del a_list[1:5]
>>> a_list
['a', 'f']
As usual, the sublist selected by slice contains all the elements up to, but not including, the
second index.
Objects and references
After we execute these assignment statements
1
2
a = "banana"
b = "banana"
we know that a and b will refer to a string object with the letters "banana". But we don’t
know yet whether they point to the same string object.
There are two possible ways the Python interpreter could arrange its memory:
In one case, a and b refer to two different objects that have the same value. In the second case,
they refer to the same object.
We can test whether two names refer to the same object using the is operator:
>>> a is b
True
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This tells us that both a and b refer to the same object, and that it is the second of the two state
snapshots that accurately describes the relationship.
Since strings are immutable, Python optimizes resources by making two names that refer to the
same string value refer to the same object.
This is not the case with lists:
>>> a
>>> b
>>> a
True
>>> a
False
= [1, 2, 3]
= [1, 2, 3]
== b
is b
The state snapshot here looks like this:
a and b have the same value but do not refer to the same object.
Aliasing
Since variables refer to objects, if we assign one variable to another, both variables refer to the
same object:
>>> a = [1, 2, 3]
>>> b = a
>>> a is b
True
In this case, the state snapshot looks like this:
Because the same list has two different names, a and b, we say that it is aliased. Changes made
with one alias affect the other:
>>> b[0] = 5
>>> a
[5, 2, 3]
Although this behavior can be useful, it is sometimes unexpected or undesirable. In general, it
is safer to avoid aliasing when you are working with mutable objects (i.e. lists at this point in
our textbook, but we’ll meet more mutable objects as we cover classes and objects, dictionaries
and sets). Of course, for immutable objects (i.e. strings, tuples), there’s no problem — it is
just not possible to change something and get a surprise when you access an alias name. That’s
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why Python is free to alias strings (and any other immutable kinds of data) when it sees an
opportunity to economize.
Cloning lists
If we want to modify a list and also keep a copy of the original, we need to be able to make a
copy of the list itself, not just the reference. This process is sometimes called cloning, to avoid
the ambiguity of the word copy.
The easiest way to clone a list is to use the slice operator:
>>>
>>>
>>>
[1,
a = [1, 2, 3]
b = a[:]
b
2, 3]
Taking any slice of a creates a new list. In this case the slice happens to consist of the whole
list. So now the relationship is like this:
Now we are free to make changes to b without worrying that we’ll inadvertently be changing
a:
>>> b[0] = 5
>>> a
[1, 2, 3]
Lists and for loops
The for loop also works with lists, as we’ve already seen. The generalized syntax of a for
loop is:
for <VARIABLE> in <LIST>:
<BODY>
So, as we’ve seen
1
2
3
friends = ["Joe", "Zoe", "Brad", "Angelina", "Zuki", "Thandi
˓→", "Paris"]
for friend in friends:
print(friend)
It almost reads like English: For (every) friend in (the list of) friends, print (the name of the)
friend.
Any list expression can be used in a for loop:
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1
2
3
for number in range(20):
if number % 3 == 0:
print(number)
4
5
6
for fruit in ["banana", "apple", "quince"]:
print("I like to eat " + fruit + "s!")
The first example prints all the multiples of 3 between 0 and 19. The second example expresses
enthusiasm for various fruits.
Since lists are mutable, we often want to traverse a list, changing each of its elements. The
following squares all the numbers in the list xs:
1
xs = [1, 2, 3, 4, 5]
2
3
4
for i in range(len(xs)):
xs[i] = xs[i]**2
Take a moment to think about range(len(xs)) until you understand how it works.
In this example we are interested in both the value of an item, (we want to square that value),
and its index (so that we can assign the new value to that position). This pattern is common
enough that Python provides a nicer way to implement it:
1
xs = [1, 2, 3, 4, 5]
2
3
4
for (i, val) in enumerate(xs):
xs[i] = val**2
enumerate generates pairs of both (index, value) during the list traversal. Try this next
example to see more clearly how enumerate works:
1
2
for (i, v) in enumerate(["banana", "apple", "pear", "lemon
˓→"]):
print(i, v)
0
1
2
3
banana
apple
pear
lemon
List parameters
Passing a list as an argument actually passes a reference to the list, not a copy or clone of
the list. So parameter passing creates an alias for you: the caller has one variable referencing
the list, and the called function has an alias, but there is only one underlying list object. For
example, the function below takes a list as an argument and multiplies each element in the list
by 2:
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2
3
4
def double_stuff(stuff_list):
""" Overwrite each element in a_list with double its
˓→value. """
for (index, stuff) in enumerate(stuff_list):
stuff_list[index] = 2 * stuff
If we add the following onto our script:
1
2
3
things = [2, 5, 9]
double_stuff(things)
print(things)
When we run it we’ll get:
[4, 10, 18]
In the function above, the parameter stuff_list and the variable things are aliases for
the same object. So before any changes to the elements in the list, the state snapshot looks like
this:
Since the list object is shared by two frames, we drew it between them.
If a function modifies the items of a list parameter, the caller sees the change.
List methods
The dot operator can also be used to access built-in methods of list objects. We’ll start with the
most useful method for adding something onto the end of an existing list:
>>>
>>>
>>>
>>>
>>>
>>>
[5,
mylist = []
mylist.append(5)
mylist.append(27)
mylist.append(3)
mylist.append(12)
mylist
27, 3, 12]
append is a list method which adds the argument passed to it to the end of the list. We’ll use
it heavily when we’re creating new lists. Continuing with this example, we show several other
list methods:
>>> mylist.insert(1, 12)
˓→items up
>>> mylist
[5, 12, 27, 3, 12]
>>> mylist.count(12)
142
# Insert 12 at pos 1, shift other
# How many times is 12 in mylist?
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2
>>> mylist.extend([5, 9, 5, 11])
˓→end of mylist
>>> mylist
[5, 12, 27, 3, 12, 5, 9, 5, 11])
>>> mylist.index(9)
˓→in mylist
6
>>> mylist.reverse()
>>> mylist
[11, 5, 9, 5, 12, 3, 27, 12, 5]
>>> mylist.sort()
>>> mylist
[3, 5, 5, 5, 9, 11, 12, 12, 27]
>>> mylist.remove(12)
˓→the list
>>> mylist
[3, 5, 5, 5, 9, 11, 12, 27]
# Put whole list onto
# Find index of first 9
# Remove the first 12 in
Experiment and play with the list methods shown here, and read their documentation until you
feel confident that you understand how they work.
Pure functions and modifiers
As seen before, there is a difference between a pure function and one with side-effects. The
difference is shown below as lists have some special gotcha’s. Functions which take lists as
arguments and change them during execution are called modifiers and the changes they make
are called side effects.
A pure function does not produce side effects. It communicates with the calling program only
through parameters, which it does not modify, and a return value. Here is double_stuff
written as a pure function:
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2
3
4
5
6
7
8
def double_stuff(a_list):
""" Return a new list which contains
doubles of the elements in a_list.
"""
new_list = []
for value in a_list:
new_elem = 2 * value
new_list.append(new_elem)
9
return new_list
10
This version of double_stuff does not change its arguments:
>>> things = [2, 5, 9]
>>> more_things = double_stuff(things)
>>> things
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[2, 5, 9]
>>> more_things
[4, 10, 18]
An early rule we saw for assignment said “first evaluate the right hand side, then assign the
resulting value to the variable”. So it is quite safe to assign the function result to the same
variable that was passed to the function:
>>>
>>>
>>>
[4,
things = [2, 5, 9]
things = double_stuff(things)
things
10, 18]
Functions that produce lists
The pure version of double_stuff above made use of an important pattern for your toolbox. Whenever you need to write a function that creates and returns a list, the pattern is usually:
1
2
3
4
5
initialize a result variable to be an empty list
loop
create a new element
append it to result
return the result
Let us show another use of this pattern. Assume you already have a function is_prime(x)
that can test if x is prime. Write a function to return a list of all prime numbers less than n:
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2
3
4
5
6
7
def primes_lessthan(n):
""" Return a list of all prime numbers less than n. """
result = []
for i in range(2, n):
if is_prime(i):
result.append(i)
return result
Strings and lists
Two of the most useful methods on strings involve conversion to and from lists of substrings.
The split method (which we’ve already seen) breaks a string into a list of words. By default,
any number of whitespace characters is considered a word boundary:
>>> song = "The rain in Spain..."
>>> words = song.split()
>>> words
['The', 'rain', 'in', 'Spain...']
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An optional argument called a delimiter can be used to specify which string to use as the
boundary marker between substrings. The following example uses the string ai as the delimiter:
>>> song.split("ai")
['The r', 'n in Sp', 'n...']
Notice that the delimiter doesn’t appear in the result.
The inverse of the split method is join. You choose a desired separator string, (often
called the glue) and join the list with the glue between each of the elements:
>>> glue = ";"
>>> phrase = glue.join(words)
>>> phrase
'The;rain;in;Spain...'
The list that you glue together (words in this example) is not modified. Also, as these next
examples show, you can use empty glue or multi-character strings as glue:
>>> " --- ".join(words)
'The --- rain --- in --- Spain...'
>>> "".join(words)
'TheraininSpain...'
list and range
Python has a built-in type conversion function called list that tries to turn whatever you give
it into a list.
>>> letters = list("Crunchy Frog")
>>> letters
["C", "r", "u", "n", "c", "h", "y", " ", "F", "r", "o", "g"]
>>> "".join(letters)
'Crunchy Frog'
One particular feature of range is that it doesn’t instantly compute all its values: it “puts
off” the computation, and does it on demand, or “lazily”. We’ll say that it gives a promise to
produce the values when they are needed. This is very convenient if your computation shortcircuits a search and returns early, as in this case:
1
2
3
4
5
6
7
def f(n):
""" Find the first positive integer between 101 and less
than n that is divisible by 21
"""
for i in range(101, n):
if (i % 21 == 0):
return i
8
9
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10
11
test(f(110) == 105)
test(f(1000000000) == 105)
In the second test, if range were to eagerly go about building a list with all those elements,
you would soon exhaust your computer’s available memory and crash the program. But it is
cleverer than that! This computation works just fine, because the range object is just a promise
to produce the elements if and when they are needed. Once the condition in the if becomes
true, no further elements are generated, and the function returns. (Note: Before Python 3,
range was not lazy. If you use an earlier versions of Python, YMMV!)
YMMV: Your Mileage May Vary
The acronym YMMV stands for your mileage may vary. American car advertisements often quoted fuel consumption figures for cars, e.g. that they would get 28
miles per gallon. But this always had to be accompanied by legal small-print warning the reader that they might not get the same. The term YMMV is now used
idiomatically to mean “your results may differ”, e.g. The battery life on this phone
is 3 days, but YMMV.
You’ll sometimes find the lazy range wrapped in a call to list. This forces Python to turn
the lazy promise into an actual list:
>>> range(10)
# Create a lazy promise
range(0, 10)
>>> list(range(10))
# Call in the promise, to produce a
˓→list.
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]
Looping and lists
Computers are useful because they can repeat computation, accurately and fast. So loops are
going to be a central feature of almost all programs you encounter.
Tip: Don’t create unnecessary lists
Lists are useful if you need to keep data for later computation. But if you don’t need lists, it is
probably better not to generate them.
Here are two functions that both generate ten million random numbers, and return the sum of
the numbers. They both work.
1
2
import random
joe = random.Random()
3
4
5
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def sum1():
""" Build a list of random numbers, then sum them """
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7
8
9
xs = []
for i in range(10000000):
num = joe.randrange(1000)
˓→number
xs.append(num)
# Generate one random
# Save it in our list
10
11
12
tot = sum(xs)
return tot
13
14
15
16
17
18
19
20
def sum2():
""" Sum the random numbers as we generate them """
tot = 0
for i in range(10000000):
num = joe.randrange(1000)
tot += num
return tot
21
22
23
print(sum1())
print(sum2())
What reasons are there for preferring the second version here? (Hint: open a tool like the
Performance Monitor on your computer, and watch the memory usage. How big can you make
the list before you get a fatal memory error in sum1?)
In a similar way, when working with files, we often have an option to read the whole file
contents into a single string, or we can read one line at a time and process each line as we read
it. Line-at-a-time is the more traditional and perhaps safer way to do things — you’ll be able
to work comfortably no matter how large the file is. (And, of course, this mode of processing
the files was essential in the old days when computer memories were much smaller.) But you
may find whole-file-at-once is sometimes more convenient!
Nested lists
A nested list is a list that appears as an element in another list. In this list, the element with
index 3 is a nested list:
>>> nested = ["hello", 2.0, 5, [10, 20]]
If we output the element at index 3, we get:
>>> print(nested[3])
[10, 20]
To extract an element from the nested list, we can proceed in two steps:
>>> elem = nested[3]
>>> elem[0]
10
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Or we can combine them:
>>> nested[3][1]
20
Bracket operators evaluate from left to right, so this expression gets the 3’th element of
nested and extracts the 1’th element from it.
Matrices
Nested lists are often used to represent matrices. For example, the matrix:
might be represented as:
>>> mx = [[1, 2, 3], [4, 5, 6], [7, 8, 9]]
mx is a list with three elements, where each element is a row of the matrix. We can select an
entire row from the matrix in the usual way:
>>> mx[1]
[4, 5, 6]
Or we can extract a single element from the matrix using the double-index form:
>>> mx[1][2]
6
The first index selects the row, and the second index selects the column. Although this way
of representing matrices is common, it is not the only possibility. A small variation is to use
a list of columns instead of a list of rows. Later we will see a more radical alternative using a
dictionary.
Glossary
aliases Multiple variables that contain references to the same object.
clone To create a new object that has the same value as an existing object. Copying a reference
to an object creates an alias but doesn’t clone the object.
delimiter A character or string used to indicate where a string should be split.
element One of the values in a list (or other sequence). The bracket operator selects elements
of a list. Also called item.
immutable data value A data value which cannot be modified. Assignments to elements or
slices (sub-parts) of immutable values cause a runtime error.
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index An integer value that indicates the position of an item in a list. Indexes start from 0.
item See element.
list A collection of values, each in a fixed position within the list. Like other types str, int,
float, etc. there is also a list type-converter function that tries to turn whatever
argument you give it into a list.
list traversal The sequential accessing of each element in a list.
modifier A function which changes its arguments inside the function body. Only mutable
types can be changed by modifiers.
mutable data value A data value which can be modified. The types of all mutable values are
compound types. Lists and dictionaries are mutable; strings and tuples are not.
nested list A list that is an element of another list.
object A thing to which a variable can refer.
pattern A sequence of statements, or a style of coding something that has general applicability
in a number of different situations. Part of becoming a mature Computer Scientist is to
learn and establish the patterns and algorithms that form your toolkit. Patterns often
correspond to your “mental chunking”.
promise An object that promises to do some work or deliver some values if they’re eventually
needed, but it lazily puts off doing the work immediately. Calling range produces a
promise.
pure function A function which has no side effects. Pure functions only make changes to the
calling program through their return values.
sequence Any of the data types that consist of an ordered collection of elements, with each
element identified by an index.
side effect A change in the state of a program made by calling a function. Side effects can
only be produced by modifiers.
step size The interval between successive elements of a linear sequence. The third (and optional argument) to the range function is called the step size. If not specified, it defaults
to 1.
Exercises
1. What is the Python interpreter’s response to the following?
>>> list(range(10, 0, -2))
The three arguments to the range function are start, stop, and step, respectively. In this
example, start is greater than stop. What happens if start < stop and step <
0? Write a rule for the relationships among start, stop, and step.
2. Consider this fragment of code:
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import turtle
2
3
4
5
tess = turtle.Turtle()
alex = tess
alex.color("hotpink")
Does this fragment create one or two turtle instances? Does setting the color of alex
also change the color of tess? Explain in detail.
3. Draw a state snapshot for a and b before and after the third line of the following Python
code is executed:
1
2
3
a = [1, 2, 3]
b = a[:]
b[0] = 5
4. What will be the output of the following program?
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2
3
4
5
this = ["I", "am", "not", "a", "crook"]
that = ["I", "am", "not", "a", "crook"]
print("Test 1: {0}".format(this is that))
that = this
print("Test 2: {0}".format(this is that))
Provide a detailed explanation of the results.
5. Lists can be used to represent mathematical vectors. In this exercise and several that
follow you will write functions to perform standard operations on vectors. Create a script
named vectors.py and write Python code to pass the tests in each case.
Write a function add_vectors(vector1,vector2) that takes two lists of numbers of the same length, and returns a new list containing the sums of the corresponding
elements of each:
1
2
3
add_vectors([1, 1], [1, 1]) == [2, 2]
add_vectors([1, 2], [1, 4]) == [2, 6]
add_vectors([1, 2, 1], [1, 4, 3]) == [2, 6, 4]
6. Write a function scalar_mult(scalar, vector) that takes a number, scalar,
and a list, vector and returns the scalar multiple of vector by scalar. :
1
2
3
scalar_mult(5, [1, 2]) == [5, 10]
scalar_mult(3, [1, 0, -1]) == [3, 0, -3]
scalar_mult(7, [3, 0, 5, 11, 2]) == [21, 0, 35, 77, 14]
7. Write a function dot_product(vec1,vec2) that takes two lists of numbers of the
same length, and returns the sum of the products of the corresponding elements of each
(the dot_product).
1
2
3
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dot_product([1, 1], [1, 1]) == 2
dot_product([1, 2], [1, 4]) == 9
dot_product([1, 2, 1], [1, 4, 3]) == 12
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8. Extra challenge for the mathematically inclined:
Write a function
cross_product(vec1, vec2) that takes two lists of numbers of length 3
and returns their cross product. You should write your own tests.
9. Describe the relationship between " ".join(song.split()) and song in the
fragment of code below. Are they the same for all strings assigned to song? When
would they be different?
1
song = "The rain in Spain..."
10. Write a function replace(s, old, new) that replaces all occurrences of old with
new in a string s:
1
replace("Mississippi", "i", "I") == "MIssIssIppI"
2
3
4
5
song = "I love spom! Spom is my favorite food. Spom,
˓→spom, yum!"
replace(s, "om", "am") ==
"I love spam! Spam is my favorite food. Spam, spam,
˓→yum!"
6
7
8
replace(s, "o", "a") ==
"I lave spam! Spam is my favarite faad. Spam, spam,
˓→yum!"
Hint: use the split and join methods.
Dictionaries
All of the compound data types we have studied in detail so far — strings, lists, and tuples —
are sequence types, which use integers as indices to access the values they contain within them.
Dictionaries are yet another kind of compound type. They are Python’s built-in mapping
type. They map keys, which can be any immutable type, to values, which can be any type
(heterogeneous), just like the elements of a list or tuple. In other languages, they are called
associative arrays since they associate a key with a value.
As an example, we will create a dictionary to translate English words into Spanish. For this
dictionary, the keys are strings.
One way to create a dictionary is to start with the empty dictionary and add key:value pairs.
The empty dictionary is denoted {}:
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>>> english_spanish = {}
>>> english_spanish["one"] = "uno"
>>> english_spanish["two"] = "dos"
The first assignment creates a dictionary named english_spanish; the other assignments
add new key:value pairs to the dictionary. We can print the current value of the dictionary in
the usual way:
>>> print(english_spanish)
{"two": "dos", "one": "uno"}
The key:value pairs of the dictionary are separated by commas. Each pair contains a key and a
value separated by a colon.
Hashing
The order of the pairs may not be what was expected. Python uses complex algorithms, designed for very fast access, to determine where the key:value pairs are stored in a dictionary.
For our purposes we can think of this ordering as unpredictable.
You also might wonder why we use dictionaries at all when the same concept of mapping a key
to a value could be implemented using a list of tuples:
>>> {"apples": 430, "bananas": 312, "oranges": 525, "pears": 217}
{'pears': 217, 'apples': 430, 'oranges': 525, 'bananas': 312}
>>> [('apples', 430), ('bananas', 312), ('oranges', 525), ('pears',
˓→217)]
[('apples', 430), ('bananas', 312), ('oranges', 525), ('pears',
˓→217)]
The reason is dictionaries are very fast, implemented using a technique called hashing, which
allows us to access a value very quickly. By contrast, the list of tuples implementation is slow.
If we wanted to find a value associated with a key, we would have to iterate over every tuple,
checking the 0th element. What if the key wasn’t even in the list? We would have to get to the
end of it to find out.
Another way to create a dictionary is to provide a list of key:value pairs using the same syntax
as the previous output:
>>> english_spanish = {"one": "uno", "two": "dos", "three":
˓→"tres"}
It doesn’t matter what order we write the pairs. The values in a dictionary are accessed with
keys, not with indices, so there is no need to care about ordering.
Here is how we use a key to look up the corresponding value:
>>> print(english_spanish["two"])
'dos'
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The key "two" yields the value "dos".
Lists, tuples, and strings have been called sequences, because their items occur in order. The
dictionary is the first compound type that we’ve seen that is not a sequence, so we can’t index
or slice a dictionary.
Dictionary operations
The del statement removes a key:value pair from a dictionary. For example, the following
dictionary contains the names of various fruits and the number of each fruit in stock:
>>> inventory = {"apples": 430, "bananas": 312, "oranges":
˓→525, "pears": 217}
>>> print(inventory)
{'pears': 217, 'apples': 430, 'oranges': 525, 'bananas':
˓→312}
If someone buys all of the bananas, we can remove the entry from the dictionary:
>>> del inventory["bananas"]
>>> print(inventory)
{'apples': 430, 'oranges': 525, 'pears': 217}
If we then try to see how many bananas we have, we get an error (because, yes, we have no
bananas). (Try this!)
Or if we’re expecting more bananas soon, we might just change the value associated with
bananas:
>>> inventory["bananas"] = 0
>>> print(inventory)
{'pears': 217, 'apples': 430, 'oranges': 525, 'bananas': 0}
A new shipment of bananas arriving could be handled like this:
>>> inventory["bananas"] += 200
>>> print(inventory)
{'pears': 0, 'apples': 430, 'oranges': 525, 'bananas': 512}
The len function also works on dictionaries; it returns the number of key:value pairs:
>>> len(inventory)
4
Dictionary methods
Dictionaries have a number of useful built-in methods.
The keys method returns what Python 3 calls a view of its underlying keys. A view object
has some similarities to the range object we saw earlier — it is a lazy promise, to deliver its
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elements when they’re needed by the rest of the program. We can iterate over the view, or turn
the view into a list like this:
1
2
for key in english_spanish.keys():
# The order of the k's
˓→is not defined
print("Got key", key, "which maps to value", english_
˓→spanish[key])
3
4
5
keys = list(english_spanish.keys())
print(keys)
This produces this output:
Got key three which maps to value tres
Got key two which maps to value dos
Got key one which maps to value uno
['three', 'two', 'one']
It is so common to iterate over the keys in a dictionary that we can omit the keys method call
in the for loop — iterating over a dictionary implicitly iterates over its keys:
1
2
for key in english_spanish:
print("Got key", key)
The values method is similar; it returns a view object which can be turned into a list:
>>> list(english_spanish.values())
['tres', 'dos', 'uno']
The items method also returns a view, which promises a list of tuples — one tuple for each
key:value pair:
>>> list(english_spanish.items())
[('three', 'tres'), ('two', 'dos'), ('one', 'uno')]
Tuples are often useful for getting both the key and the value at the same time while we are
looping:
1
2
for (key,value) in english_spanish.items():
print("Got",key,"that maps to",value)
This produces:
Got three that maps to tres
Got two that maps to dos
Got one that maps to uno
The in and not in operators can test if a key is in the dictionary:
>>> "one" in english_spanish
True
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>>> "six" in english_spanish
False
>>> "tres" in english_spanish
˓→keys, not values.
False
# Note that 'in' tests
This method can be very useful, since looking up a non-existent key in a dictionary causes a
runtime error:
>>> english_spanish["dog"]
Traceback (most recent call last):
...
KeyError: 'dog'
Aliasing and copying
As in the case of lists, because dictionaries are mutable, we need to be aware of aliasing.
Whenever two variables refer to the same object, changes to one affect the other.
If we want to modify a dictionary and keep a copy of the original, use the copy method. For
example, opposites is a dictionary that contains pairs of opposites:
>>> opposites = {"up": "down", "right": "wrong", "yes": "no
˓→"}
>>> alias = opposites
>>> copy = opposites.copy() # Shallow copy
alias and opposites refer to the same object; copy refers to a fresh copy of the same
dictionary. If we modify alias, opposites is also changed:
>>> alias["right"] = "left"
>>> opposites["right"]
'left'
If we modify copy, opposites is unchanged:
>>> copy["right"] = "privilege"
>>> opposites["right"]
'left'
Counting letters
In the exercises in Chapter 8 (Strings) we wrote a function that counted the number of occurrences of a letter in a string. A more general version of this problem is to form a frequency
table of the letters in the string, that is, how many times each letter appears.
Such a frequency table might be useful for compressing a text file. Because different letters
appear with different frequencies, we can compress a file by using shorter codes for common
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letters and longer codes for letters that appear less frequently.
Dictionaries provide an elegant way to generate a frequency table:
>>> letter_counts = {}
>>> for letter in "Mississippi":
...
letter_counts[letter] = letter_counts.get(letter,
˓→0) + 1
...
>>> letter_counts
{'M': 1, 's': 4, 'p': 2, 'i': 4}
We start with an empty dictionary. For each letter in the string, we find the current count
(possibly zero) and increment it. At the end, the dictionary contains pairs of letters and their
frequencies.
It might be more appealing to display the frequency table in alphabetical order. We can do that
with the items and sort methods (more precisely, sort orders lexicographically):
>>> letter_items = list(letter_counts.items())
>>> letter_items.sort()
>>> print(letter_items)
[('M', 1), ('i', 4), ('p', 2), ('s', 4)]
Notice in the first line we had to call the type conversion function list. That turns the promise
we get from items into a list, a step that is needed before we can use the list’s sort method.
Glossary
call graph A graph consisting of nodes which represent function frames (or invocations), and
directed edges (lines with arrows) showing which frames gave rise to other frames.
dictionary A collection of key:value pairs that maps from keys to values. The keys can be any
immutable value, and the associated value can be of any type.
immutable data value A data value which cannot be modified. Assignments to elements or
slices (sub-parts) of immutable values cause a runtime error.
key A data item that is mapped to a value in a dictionary. Keys are used to look up values in a
dictionary. Each key must be unique across the dictionary.
key:value pair One of the pairs of items in a dictionary. Values are looked up in a dictionary
by key.
mapping type A mapping type is a data type comprised of a collection of keys and associated
values. Python’s only built-in mapping type is the dictionary. Dictionaries implement the
associative array abstract data type.
memo Temporary storage of precomputed values to avoid duplicating the same computation.
mutable data value A data value which can be modified. The types of all mutable values are
compound types. Lists and dictionaries are mutable; strings and tuples are not.
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Exercises
1. Write a program that reads a string and returns a table of the letters of the alphabet in
alphabetical order which occur in the string together with the number of times each letter
occurs. Case should be ignored. A sample output of the program when the user enters
the data “ThiS is String with Upper and lower case Letters”, would look this this:
a
c
d
e
g
h
i
l
n
o
p
r
s
t
u
w
2
1
1
5
1
2
4
2
2
1
2
4
5
5
1
2
2. Give the Python interpreter’s response to each of the following from a continuous interpreter session:
(a) >>> dictionary = {"apples": 15, "bananas": 35, "grapes": 12}
>>> dictionary["bananas"]
(b) >>> dictionary["oranges"] = 20
>>> len(dictionary)
(c) >>> "grapes" in dictionary
(d) >>> dictionary["pears"]
(e) >>> dictionary.get("pears", 0)
(f) >>> fruits = list(dictionary.keys())
>>> fruits.sort()
>>> print(fruits)
(g) >>> del dictionary["apples"]
>>> "apples" in dictionary
Be sure you understand why you get each result. Then apply what you have learned to
fill in the body of the function below:
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1
2
def add_fruit(inventory, fruit, quantity=0):
return None
3
4
5
6
7
8
9
10
# Make these tests work...
new_inventory = {}
add_fruit(new_inventory, "strawberries", 10)
test("strawberries" in new_inventory)
test(new_inventory["strawberries"] == 10)
add_fruit(new_inventory, "strawberries", 25)
test(new_inventory["strawberries"] == 35)
3. Write a program called alice_words.py that creates a text file named
alice_words.txt containing an alphabetical listing of all the words, and the number of times each occurs, in the text version of Alice’s Adventures in Wonderland.
(You can obtain a free plain text version of the book, along with many others, from
http://www.gutenberg.org.) The first 10 lines of your output file should look something
like this:
Word
Count
=======================
a
631
a-piece
1
abide
1
able
1
about
94
above
3
absence
1
absurd
2
How many times does the word alice occur in the book?
4. What is the longest word in Alice in Wonderland? How many characters does it have?
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CHAPTER 6
Numpy
The standard Python data types are not very suited for mathematical operations. For example,
suppose we have the list a = [2, 3, 8]. If we multiply this list by an integer, we get:
>>> a = [2, 3, 8]
>>> 2 * a
[2, 3, 8, 2, 3, 8]
And float‘s are not even allowed:
>>> a = [2, 3, 8]
>>> 2 * a
>>> 2.1 * a
TypeError: can't multiply sequence by non-int of type 'float
˓→'
In order to solve this using Python lists, we would have to do something like:
values = [2, 3, 8]
result = []
for x in values:
result.append(2.1 * x)
This is not very elegant, is it? This is because Python list‘s are not designed as mathematical
objects. Rather, they are purely a collection of items. In order to get a type of list which behaves
like a mathematical array or matrix, we use Numpy.
>>> import numpy as np
>>> a = np.array([2, 3, 8])
>>> 2.1 * a
array([ 4.2,
6.3, 16.8])
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As we can see, this worked the way we expected it to. We note a couple of things: - We
abbreviated numpy to np, this is conventional. - np.array takes a Python list as argument.
- The list [2, 3, 8] contains int‘s, yet the result contains float‘s. This means numpy
changed the data type automatically for us.
Now let’s take it a step further and see what happens when we multiply together array’s.
>>> import numpy as np
>>> a = np.array([2, 3, 8])
>>> a * a
array([ 4, 9, 64])
>>> a**2
array([ 4, 9, 64])
This has nicely squared the array element-wise.
Note: Those in the know might be a bit surprised by this. After all, if a is a vector,
shouldn’t a**2 be the dot product of the two vectors, ⃗ · ⃗? Well, numpy arrays
are not vectors in the algebraic sense. Arithmetic operations between arrays are
performed element-wise, not on the arrays as a whole.
To tell numpy we want the dot product we simply use the np.dot function:
>>> a = np.array([2, 3, 8])
>>> np.dot(a,a)
77
Furthermore, if you pass 2D arrays to np.dot it will behave like matrix multiplication. Several other similar NumPy algebraic functions are available (like
np.cross, np.outer, etc.)
Bottom line: when you want to treat numpy array operations as vector or matrix
operations, make use of the specialized functions to this end.
Shape
One of the most important properties an array is its shape. We have already seen 1 dimensional
(1D) arrays, but arrays can have any dimensions you like. Images for example, consist of a 2D
array of pixels. But in color images every pixel is an RGB tuple: the intensity in red, green and
blue. Every pixel itself is therefore an array as well. This makes a color image 3D overall.
To get the shape of an array, we use shape:
>>> import numpy as np
>>> a = np.array([2, 3, 8])
>>> a.shape
(3,)
Something slightly more interesting:
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>>> b = np.array([
[2, 3, 8],
[4, 5, 6],
])
>>> b.shape
(2, 3)
Slicing
Just like with lists, we might want to select certain values from an array. For 1D arrays it works
just like for normal python lists:
>>> a = np.array([2, 3, 8])
>>> a[2]
8
>>> a[1:]
np.array([3, 8])
However, when dealing with higher dimensional arrays something else happens:
>>> b = np.array([
[2, 3, 8],
[4, 5, 6],
])
>>> b[1]
array([4, 5, 6])
>>> b[1][2]
6
We see that using b[1] returns the 1th row along the first dimenion, which is still an array.
After that, we can select individual items from that. This can be abbreviated to:
>>> b[1, 2]
6
But what if I wanted the 1th column instead of the first row? Then we use : to select all items
along the first dimension, and then a 1:
>>> b[:, 1]
array([3, 5])
By comparing with the definition of b, we see that this is the column we were looking for.
Note: Instead of first, I write 1th on purpose to signify the existence of a 0th element. Remember that in Python, as in any self-respecting programming language,
we start counting at zero.
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Find out more about advanced slicing at the Numpy indexing documentation page.
Masking
This is perhaps the single most powerful feature of Numpy. Suppose we have an array, and we
want to throw away all values above a certain cutoff:
>>> a = np.array([230, 10, 284, 39, 76])
>>> cutoff = 200
>>> a > cutoff
np.array([True, False, True, False, False])
Simply using the larger than operator lets us know in which cases the test was positive. Now
we set all the values above 200 to zero:
>>> a = np.array([230, 10, 284, 39, 76])
>>> cutoff = 200
>>> a[a > cutoff] = 0
>>> a
np.array([0, 10, 0, 39, 76])
The crucial line is a[a > cutoff] = 0. This selects all the points in the array where the
test was positive and assigns 0 to that position. Without knowing this trick we would have had
to loop over the array:
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
a = np.array([230, 10, 284, 39, 76])
cutoff = 200
new_a = []
for x in a:
if x > cutoff:
new_a.append(0)
else:
new_a.append(x)
a = np.array(new_a)
Looks rather silly now, doesn’t it? When working with images this becomes even more obvious,
because there we might have to loop over three dimensions before we can use the if/else. Can
you imagine the mess?
Broadcasting
Another powerful feature of Numpy is broadcasting. Broadcasting takes place when you perform operations between arrays of different shapes. For instance
>>> a = np.array([
[0, 1],
[2, 3],
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[4, 5],
])
>>> b = np.array([10, 100])
>>> a * b
array([[ 0, 100],
[ 20, 300],
[ 40, 500]])
The shapes of a and b don’t match. In order to proceed, Numpy will stretch b into a second
dimension, as if it were stacked three times upon itself. The operation then takes place elementwise.
One of the rules of broadcasting is that only dimensions of size 1 can be stretched (if an array
only has one dimension, all other dimensions are considered for broadcasting purposes to have
size 1). In the example above b is 1D, and has shape (2,). For broadcasting with a, which has
two dimensions, Numpy adds another dimension of size 1 to b. b now has shape (1, 2). This
new dimension can now be stretched three times so that b‘s shape matches a‘s shape of (3, 2).
The other rule is that dimensions are compared from the last to the first. Any dimensions that
do not match must be stretched to become equally sized. However, according to the previous
rule, only dimensions of size 1 can stretch. This means that some shapes cannot broadcast and
Numpy will give you an error:
>>> c = np.array([
[0, 1, 2],
[3, 4, 5],
])
>>> b = np.array([10, 100])
>>> c * b
ValueError: operands could not be broadcast together with
˓→shapes (2,3) (2,)
What happens here is that Numpy, again, adds a dimension to b, making it of shape (1, 2). The
sizes of the last dimensions of b and c (2 and 3, respectively) are then compared and found to
differ. Since none of these dimensions is of size 1 (therefore, unstretchable) Numpy gives up
and produces an error.
The solution to multiplying c and b above is to specifically tell Numpy that it must add that
extra dimension as the second dimension of b. This is done by using None to index that second
dimension. The shape of b then becomes (2, 1), which is compatible for broadcasting with c:
>>> c = np.array([
[0, 1, 2],
[3, 4, 5],
])
>>> b = np.array([10, 100])
>>> c * b[:, None]
array([[ 0, 10, 20],
[300, 400, 500]])
A good visual description of these rules, together with some advanced broadcasting applica6.4. Broadcasting
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tions can be found in this tutorial of Numpy broadcasting rules.
dtype
A commonly used term in working with numpy is dtype - short for data type. This is typically
int or float, followed by some number, e.g. int8. This means the value is integer with a size
of 8 bits. As an example, let’s discuss the properties of an int8.
Each bit is either 0 or 1. With 8 of them, we have 28 = 256 possible values. Since we also have
to count zero itself, the largest possible value is 255. The data type we have now described is
called uint8, where the u stands for unsigned: only positive values are allowed. If we want
to allow negative numbers we use int8. The range then shifts to -128 to +127.
The same holds for bigger numbers. An int64 for example is a 64 bit unsigned integer with
a range of -9223372036854775808 to 9223372036854775807. It is also the standard type on a
64 bits machine. You might think bigger is better. You’d be wrong. If you know the elements
of your array are never going to be bigger than 100, why waste all the memory space? You
might be better off setting your array to uint8 to conserve memory. In general however, the
default setting is fine. Only when you run into memory related problems should you remember
this comment.
What happens when you set numbers bigger than the maximum value of your dtype?
>>> import numpy as np
>>> a = np.array([200], dtype='uint8')
>>> a + a
array([144], dtype=uint8)
That doesn’t seem right, does it? If you add two uint8, the result of 200 + 200 cannot be
400, because that doesn’t fit in a uint8. In standard Python, Python does a lot of magic in
the background to make sure the result is the 400 you would expect. But numpy doesn’t, and
will return 144. Why 144 is left as an exercise. To fix this, you should make sure that your
numbers where not stored as uint8, but as something larger; uint16 for example. That way
the resulting 400 will fit.
>>> import numpy as np
>>> a = np.array([200], dtype='uint16')
>>> a + a
array([400], dtype=uint16)
By now you must be thinking: so bigger is better after all! Just use the biggest possible int all
the time, and you’ll be fine! Apart from the fact that there is no biggest int, there is a bigger
problem. If you work with images, each pixel from that image is stored as an RGB tuple: the
intensity in red, green and blue. Each of these is a uint8 value for most standard formats
such as .jpg and .png. For example, (0, 0, 0) will be black, and (255, 0, 0) is red. This means
that when you load an image from your hard drive this dtype is selected for you, and if you are
not aware of this, what will happen when you add an image to itself? (In other words, place
two copies on top of each other) You might expect that everything will become more dense.
Instead, you’ll get noise because of the effect we just talked about.
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Changing dtype
To change the dtype of an existing array, you can use the astype method:
>>> import numpy as np
>>> a = np.array([200], dtype='uint8')
>>> a.astype('uint64')
Advanced Usage
Numpy has vast capabilities. It has way too many options to discuss here. More information
can be found in
1. the Quickstart Numpy Tutorial;
2. the Numpy indexing documentation (for advanced slicing and indexing);
3. and the Numpy broadcasting rules (for what happens when performing operations between arrays of different shapes and sizes).
Excercises
1. Make an array with dtype = uint8 and elements of your choosing. Keep adding to
it until (one of) the items go over 255. What happens? Hint: make an array, and just add
a constant to it. The constant will be added to all the items of the array element-wise.
2. Use a mask to multiply all values below 100 in the following list by 2:
>>> a = np.array([230, 10, 284, 39, 76])
Repeat this until all values are above 100. (Not manually, but by looping)
Then, select all values between 150 < a < 200.
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CHAPTER 7
Files
About files
While a program is running, its data is stored in random access memory (RAM). RAM is
fast and inexpensive, but it is also volatile, which means that when the program ends, or the
computer shuts down, data in RAM disappears. To make data available the next time the
computer is turned on and the program is started, it has to be written to a non-volatile storage
medium, such a hard drive, usb drive, or CD-RW.
Data on non-volatile storage media is stored in named locations called files. By reading and
writing files, programs can save information between program runs.
Working with files is a lot like working with a notebook. To use a notebook, it has to be
opened. When done, it has to be closed. While the notebook is open, it can either be read from
or written to. In either case, the notebook holder knows where they are. They can read the
whole notebook in its natural order or they can skip around.
All of this applies to files as well. To open a file, we specify its name and indicate whether we
want to read or write.
Writing our first file
Let’s begin with a simple program that writes three lines of text into a file:
1
2
3
4
with open("test.txt", "w") as myfile:
myfile.write("My first file written from Python\n")
myfile.write("---------------------------------\n")
myfile.write("Hello, world!\n")
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Opening a file creates what we call a file handle. In this example, the variable myfile refers
to the new handle object. Our program calls methods on the handle, and this makes changes to
the actual file which is usually located on our disk.
On line 1, the open function takes two arguments. The first is the name of the file, and the
second is the mode. Mode "w" means that we are opening the file for writing.
With mode "w", if there is no file named test.txt on the disk, it will be created. If there
already is one, it will be replaced by the file we are writing.
To put data in the file we invoke the write method on the handle, shown in lines 2, 3 and 4
above. In bigger programs, lines 2–4 will usually be replaced by a loop that writes many more
lines into the file.
The file is closed after line 4, at the end of the with block. A with block make sure that the
file get close even if an error occurs (power outages excluded).
A handle is somewhat like a TV remote control
We’re all familiar with a remote control for a TV. We perform operations on the
remote control — switch channels, change the volume, etc. But the real action
happens on the TV. So, by simple analogy, we’d call the remote control our handle
to the underlying TV.
Sometimes we want to emphasize the difference — the file handle is not the same
as the file, and the remote control is not the same as the TV. But at other times
we prefer to treat them as a single mental chunk, or abstraction, and we’ll just say
“close the file”, or “flip the TV channel”.
Reading a file line-at-a-time
Now that the file exists on our disk, we can open it, this time for reading, and read all the lines
in the file, one at a time. This time, the mode argument is "r" for reading:
1
2
3
4
5
with open("test.txt", "r") as my_new_handle:
for the_line in my_new_handle:
# Do something with the line we just read.
# Here we just print it.
print(the_line, end="")
This is a handy pattern for our toolbox. In bigger programs, we’d squeeze more extensive logic
into the body of the loop at line 5 — for example, if each line of the file contained the name
and email address of one of our friends, perhaps we’d split the line into some pieces and call a
function to send the friend a party invitation.
On line 5 we suppress the newline character that print usually appends to our strings with
end="". Why? This is because the string already has its own newline: the for statement in
line 2 reads everything up to and including the newline character.
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If we try to open a file that doesn’t exist, we get an error:
>>> mynewhandle = open("wharrah.txt", "r")
FileNotFoundError: [Errno 2] No such file or directory:
˓→"wharrah.txt"
Turning a file into a list of lines
It is often useful to fetch data from a disk file and turn it into a list of lines. Suppose we have a
file containing our friends and their email addresses, one per line in the file. But we’d like the
lines sorted into alphabetical order. A good plan is to read everything into a list of lines, then
sort the list, and then write the sorted list back to another file:
1
2
with open("friends.txt", "r") as input_file:
all_lines = input_file.readlines()
3
4
all_lines.sort()
5
6
7
8
with open("sortedfriends.txt", "w") as output_file:
for line in all_lines:
outut_file.write(line)
The readlines method in line 2 reads all the lines and returns a list of the strings.
We could have used the template from the previous section to read each line one-at-a-time, and
to build up the list ourselves, but it is a lot easier to use the method that the Python implementors
gave us!
Reading the whole file at once
Another way of working with text files is to read the complete contents of the file into a string,
and then to use our string-processing skills to work with the contents.
We’d normally use this method of processing files if we were not interested in the line structure
of the file. For example, we’ve seen the split method on strings which can break a string
into words. So here is how we might count the number of words in a file:
1
2
3
4
with open("somefile.txt") as f:
content = f.read()
words = content.split()
print("There are {0} words in the file.".format(len(words)))
Notice here that we left out the "r" mode in line 1. By default, if we don’t supply the mode,
Python opens the file for reading.
Your file paths may need to be explicitly named.
7.4. Turning a file into a list of lines
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In the above example, we’re assuming that the file somefile.txt is in the same directory
as your Python source code. If this is not the case, you may need to provide a full or a relative
path to the file. On Windows, a full path could look like "C:\\temp\\somefile.txt",
while on a Unix system the full path could be "/home/jimmy/somefile.txt".
We’ll return to this later in this chapter.
An example
Many useful line-processing programs will read a text file line-at-a-time and do some minor
processing as they write the lines to an output file. They might number the lines in the output
file, or insert extra blank lines after every 60 lines to make it convenient for printing on sheets
of paper, or extract some specific columns only from each line in the source file, or only print
lines that contain a specific substring. We call this kind of program a filter.
Here is a filter that copies one file to another, omitting any lines that begin with #:
1
2
3
4
5
6
def filter(oldfile, newfile):
with open(oldfile, "r") as infile, open(newfile, "w")
˓→as outfile:
for line in infile:
# Put any processing logic here
if not line.startswith('#'):
outfile.write(line)
On line 2, we open two files: the file to read, and the file to write. From line 3, we read the
input file line by line. We write the line in the output file only if the condition on line 5 is true.
Directories
Files on non-volatile storage media are organized by a set of rules known as a file system.
File systems are made up of files and directories, which are containers for both files and other
directories.
When we create a new file by opening it and writing, the new file goes in the current directory
(wherever we were when we ran the program). Similarly, when we open a file for reading,
Python looks for it in the current directory.
If we want to open a file somewhere else, we have to specify the path to the file, which is the
name of the directory (or folder) where the file is located:
>>> wordsfile = open("/usr/share/dict/words", "r")
>>> wordlist = wordsfile.readlines()
>>> print(wordlist[:6])
['\n', 'A\n', "A's\n", 'AOL\n', "AOL's\n", 'Aachen\n']
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This (Unix) example opens a file named words that resides in a directory named dict, which
resides in share, which resides in usr, which resides in the top-level directory of the system,
called /. It then reads in each line into a list using readlines, and prints out the first 5
elements from that list.
A Windows path might be "c:/temp/words.txt" or "c:\\temp\\words.txt". Because backslashes are used to escape things like newlines and tabs, we need to write two backslashes in a literal string to get one! So the length of these two strings is the same!
We cannot use / or \ as part of a filename; they are reserved as a delimiter between directory
and filenames.
The file /usr/share/dict/words should exist on Unix-based systems, and contains a
list of words in alphabetical order.
What about fetching something from the web?
The Python libraries are pretty messy in places. But here is a very simple example that copies
the contents at some web URL to a local file.
1
import urllib.request
2
3
4
url = "http://xml.resource.org/public/rfc/txt/rfc793.txt"
destination_filename = "rfc793.txt"
5
6
urllib.request.urlretrieve(url, destination_filename)
The urlretrieve function — just one call — could be used to download any kind of content
from the Internet.
We’ll need to get a few things right before this works:
• The resource we’re trying to fetch must exist! Check this using a browser.
• We’ll need permission to write to the destination filename, and the file will be created in the “current directory” - i.e. the same folder that the Python program is
saved in.
• If we are behind a proxy server that requires authentication, (as some students are),
this may require some more special handling to work around our proxy. Use a local
resource for the purpose of this demonstration!
Here is a slightly different example using the requests module. This module is not part of
the standard library distributed with python, however it is easier to use and significantly more
potent than the urllib module distributed with python. Read requests documentation on http:
//docs.python-requests.org to learn how to install and use the module. Here, rather than save
the web resource to our local disk, we read it directly into a string, and we print that string:
1
import requests
2
3
url = "http://xml.resource.org/public/rfc/txt/rfc793.txt"
7.8. What about fetching something from the web?
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4
5
response = requests.get(url)
print(response.text)
Opening the remote URL returns the response from the server. That response contains several
types of information, and the requests module allows us to access them in various ways. On
line 5, we get the downloaded document as a single string. We could also read it line by line as
follows:
1
import requests
2
3
4
5
6
url = "http://xml.resource.org/public/rfc/txt/rfc793.txt"
response = requests.get(url)
for line in response:
print(line)
Glossary
delimiter A sequence of one or more characters used to specify the boundary between separate
parts of text.
directory A named collection of files, also called a folder. Directories can contain files and
other directories, which are referred to as subdirectories of the directory that contains
them.
file A named entity, usually stored on a hard drive, floppy disk, or CD-ROM, that contains a
stream of characters.
file system A method for naming, accessing, and organizing files and the data they contain.
handle An object in our program that is connected to an underlying resource (e.g. a file). The
file handle lets our program manipulate/read/write/close the actual file that is on our disk.
mode A distinct method of operation within a computer program. Files in Python can be
opened in one of four modes: read ("r"), write ("w"), append ("a"), and read and
write ("+").
non-volatile memory Memory that can maintain its state without power. Hard drives, flash
drives, and rewritable compact disks (CD-RW) are each examples of non-volatile memory.
path A sequence of directory names that specifies the exact location of a file.
text file A file that contains printable characters organized into lines separated by newline
characters.
volatile memory Memory which requires an electrical current to maintain state. The main
memory or RAM of a computer is volatile. Information stored in RAM is lost when the
computer is turned off.
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Exercises
1. Write a program that reads a file and writes out a new file with the lines in reversed order
(i.e. the first line in the old file becomes the last one in the new file.)
2. Write a program that reads a file and prints only those lines that contain the substring
snake.
3. Write a program that reads a text file and produces an output file which is a copy of the
file, except the first five columns of each line contain a four digit line number, followed by
a space. Start numbering the first line in the output file at 1. Ensure that every line number
is formatted to the same width in the output file. Use one of your Python programs as
test data for this exercise: your output should be a printed and numbered listing of the
Python program.
4. Write a program that undoes the numbering of the previous exercise: it should read a file
with numbered lines and produce another file without line numbers.
5. Write a program that takes the dictionary used above, and returns some of the words
using 1337sp34k
7.10. Exercises
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CHAPTER 8
Modules
A module is a file containing Python definitions and statements intended for use in other Python
programs. There are many Python modules that come with Python as part of the standard
library. We have seen at least two of these already, the turtle module and the string
module.
We have also shown you how to access help. The help system contains a listing of all the
standard modules that are available with Python. Play with help!
Random numbers
We often want to use random numbers in programs, here are a few typical uses:
• To play a game of chance where the computer needs to throw some dice, pick a number,
or flip a coin,
• To shuffle a deck of playing cards randomly,
• To allow/make an enemy spaceship appear at a random location and start shooting at the
player,
• To simulate possible rainfall when we make a computerized model for estimating the
environmental impact of building a dam,
• For encrypting banking sessions on the Internet.
Python provides a module random that helps with tasks like this. You can look it up using
help, but here are the key things we’ll do with it:
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import random
2
3
# Create a black box object that generates random numbers
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rng = random.Random()
5
6
7
dice_throw = rng.randrange(1,7)
# Return an int, one of 1,
˓→2,3,4,5,6
delay_in_seconds = rng.random() * 5.0
The randrange method call generates an integer between its lower and upper argument,
using the same semantics as range — so the lower bound is included, but the upper bound is
excluded. All the values have an equal probability of occurring (i.e. the results are uniformly
distributed). Like range, randrange can also take an optional step argument. So let’s
assume we needed a random odd number less than 100, we could say:
1
random_odd = rng.randrange(1, 100, 2)
Other methods can also generate other distributions e.g. a bell-shaped, or “normal” distribution
might be more appropriate for estimating seasonal rainfall, or the concentration of a compound
in the body after taking a dose of medicine.
The random method returns a floating point number in the interval [0.0, 1.0) — the square
bracket means “closed interval on the left” and the round parenthesis means “open interval on
the right”. In other words, 0.0 is possible, but all returned numbers will be strictly less than
1.0. It is usual to scale the results after calling this method, to get them into an interval suitable
for your application. In the case shown here, we’ve converted the result of the method call to
a number in the interval [0.0, 5.0). Once more, these are uniformly distributed numbers —
numbers close to 0 are just as likely to occur as numbers close to 0.5, or numbers close to 1.0.
This example shows how to shuffle a list. (shuffle cannot work directly with a lazy promise,
so notice that we had to convert the range object using the list type converter first.)
1
cards = list(range(52))
2
3
rng.shuffle(cards)
# Generate ints [0 .. 51]
#
representing a pack of cards.
# Shuffle the pack
Repeatability and Testing
Random number generators are based on a deterministic algorithm — repeatable and predictable. So they’re called pseudo-random generators — they are not genuinely random. They
start with a seed value. Each time you ask for another random number, you’ll get one based
on the current seed attribute, and the state of the seed (which is one of the attributes of the
generator) will be updated.
For debugging and for writing unit tests, it is convenient to have repeatability — programs that
do the same thing every time they are run. We can arrange this by forcing the random number
generator to be initialized with a known seed every time. (Often this is only wanted during
testing — playing a game of cards where the shuffled deck was always in the same order as last
time you played would get boring very rapidly!)
1
176
drng = random.Random(123)
˓→starting state
# Create generator with known
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This alternative way of creating a random number generator gives an explicit seed value to
the object. Without this argument, the system probably uses something based on the time. So
grabbing some random numbers from drng today will give you precisely the same random
sequence as it will tomorrow!
Picking balls from bags, throwing dice, shuffling a pack of cards
Here is an example to generate a list containing n random ints between a lower and an upper
bound:
1
import random
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3
4
5
6
7
8
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10
11
12
def make_random_ints(num, lower_bound, upper_bound):
"""
Generate a list containing num random ints between
˓→lower_bound
and upper_bound. upper_bound is an open bound.
"""
rng = random.Random() # Create a random number
˓→generator
result = []
for i in range(num):
result.append(rng.randrange(lower_bound, upper_bound))
return result
>>> make_random_ints(5, 1, 13)
˓→numbers
[8, 1, 8, 5, 6]
# Pick 5 random month
Notice that we got a duplicate in the result. Often this is wanted, e.g. if we throw a die five
times, we would expect some duplicates.
But what if you don’t want duplicates? If you wanted 5 distinct months, then this algorithm is
wrong. In this case a good algorithm is to generate the list of possibilities, shuffle it, and slice
off the number of elements you want:
1
2
3
4
xs = list(range(1,13))
˓→duplicates)
rng = random.Random()
rng.shuffle(xs)
result = xs[:5]
# Make list 1..12
(there are no
# Make a random number generator
# Shuffle the list
# Take the first five elements
In statistics courses, the first case — allowing duplicates — is usually described as pulling
balls out of a bag with replacement — you put the drawn ball back in each time, so it can occur
again. The latter case, with no duplicates, is usually described as pulling balls out of the bag
without replacement. Once the ball is drawn, it doesn’t go back to be drawn again. TV lotto
games work like this.
The second “shuffle and slice” algorithm would not be so great if you only wanted a few
elements, but from a very large domain. Suppose I wanted five numbers between one and ten
8.1. Random numbers
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million, without duplicates. Generating a list of ten million items, shuffling it, and then slicing
off the first five would be a performance disaster! So let us have another try:
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import random
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5
6
7
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10
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12
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15
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def make_random_ints_no_dups(num, lower_bound, upper_bound):
"""
Generate a list containing num random ints between
lower_bound and upper_bound. upper_bound is an open
˓→bound.
The result list cannot contain duplicates.
"""
result = []
rng = random.Random()
for i in range(num):
while True:
candidate = rng.randrange(lower_bound, upper_
˓→bound)
if candidate not in result:
break
result.append(candidate)
return result
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19
20
xs = make_random_ints_no_dups(5, 1, 10000000)
print(xs)
This agreeably produces 5 random numbers, without duplicates:
[3344629, 1735163, 9433892, 1081511, 4923270]
Even this function has its pitfalls. Can you spot what is going to happen in this case?
1
xs = make_random_ints_no_dups(10, 1, 6)
The time module
As we start to work with more sophisticated algorithms and bigger programs, a natural concern
is “is our code efficient?” One way to experiment is to time how long various operations
take. The time module has a function called clock that is recommended for this purpose.
Whenever clock is called, it returns a floating point number representing how many seconds
have elapsed since your program started running.
The way to use it is to call clock and assign the result to a variable, say t0, just before you
start executing the code you want to measure. Then after execution, call clock again, (this
time we’ll save the result in variable t1). The difference t1-t0 is the time elapsed, and is a
measure of how fast your program is running.
Let’s try a small example. Python has a built-in sum function that can sum the elements in a
list. We can also write our own. How do we think they would compare for speed? We’ll try to
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do the summation of a list [0, 1, 2 ...] in both cases, and compare the results:
1
import time
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3
4
5
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7
def do_my_sum(xs):
sum = 0
for v in xs:
sum += v
return sum
8
9
10
sz = 10000000
# Lets have 10 million elements in the
˓→list
testdata = range(sz)
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13
14
15
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t0 = time.clock()
my_result = do_my_sum(testdata)
t1 = time.clock()
print("my_result
= {0} (time taken = {1:.4f} seconds)"
.format(my_result, t1-t0))
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20
21
22
t2 = time.clock()
their_result = sum(testdata)
t3 = time.clock()
print("their_result = {0} (time taken = {1:.4f} seconds)"
.format(their_result, t3-t2))
On a reasonably modest laptop, we get these results:
my_sum
= 49999995000000 (time taken = 1.5567 seconds)
their_sum = 49999995000000 (time taken = 0.9897 seconds)
So our function runs about 57% slower than the built-in one. Generating and summing up ten
million elements in under a second is not too shabby!
The math module
The math module contains the kinds of mathematical functions you’d typically find on your
calculator (sin, cos, sqrt, asin, log, log10) and some mathematical constants like pi
and e:
>>> import math
>>> math.pi
3.141592653589793
>>> math.e
2.718281828459045
>>> math.sqrt(2.0)
1.4142135623730951
>>> math.radians(90)
1.5707963267948966
8.3. The math module
# Constant pi
# Constant natural log base
# Square root function
# Convert 90 degrees to radians
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>>> math.sin(math.radians(90)) # Find sin of 90 degrees
1.0
>>> math.asin(1.0) * 2
# Double the arcsin of 1.0 to
˓→get pi
3.141592653589793
Like almost all other programming languages, angles are expressed in radians rather than degrees. There are two functions radians and degrees to convert between these two popular
ways of measuring angles.
Notice another difference between this module and our use of random and turtle: in
random and turtle we create objects and we call methods on the object. This is because
objects have state — a turtle has a color, a position, a heading, etc., and every random number
generator has a seed value that determines its next result.
Mathematical functions are “pure” and don’t have any state — calculating the square root of
2.0 doesn’t depend on any kind of state or history about what happened in the past. So the
functions are not methods of an object — they are simply functions that are grouped together
in a module called math.
Creating your own modules
All we need to do to create our own modules is to save our script as a file with a .py extension.
Suppose, for example, this script is saved as a file named seqtools.py:
1
2
def remove_at(pos, seq):
return seq[:pos] + seq[pos+1:]
We can now use our module, both in scripts we write, or in the interactive Python interpreter.
To do so, we must first import the module.
>>> import seqtools
>>> s = "A string!"
>>> seqtools.remove_at(4, s)
'A sting!'
We do not include the .py file extension when importing. Python expects the file names of
Python modules to end in .py, so the file extension is not included in the import statement.
The use of modules makes it possible to break up very large programs into manageable sized
parts, and to keep related parts together.
Namespaces
A namespace is a collection of identifiers that belong to a module, or to a function, (and as we
will see soon, in classes too). Generally, we like a namespace to hold “related” things, e.g. all
the math functions, or all the typical things we’d do with random numbers.
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Each module has its own namespace, so we can use the same identifier name in multiple modules without causing an identification problem.
1
# module1.py
2
4
question = "What is the meaning of Life, the Universe, and
˓→Everything?"
answer = 42
1
# module2.py
3
2
3
4
question = "What is your quest?"
answer = "To seek the holy grail."
We can now import both modules and access question and answer in each:
1
2
import module1
import module2
3
4
5
6
7
print(module1.question)
print(module2.question)
print(module1.answer)
print(module2.answer)
will output the following:
What is the meaning of Life, the Universe, and Everything?
What is your quest?
42
To seek the holy grail.
Functions also have their own namespaces:
1
2
3
def f():
n = 7
print("printing n inside of f:", n)
4
5
6
7
def g():
n = 42
print("printing n inside of g:", n)
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9
10
11
12
13
14
n = 11
print("printing n before calling f:", n)
f()
print("printing n after calling f:", n)
g()
print("printing n after calling g:", n)
Running this program produces the following output:
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printing
printing
printing
printing
printing
n
n
n
n
n
before calling f: 11
inside of f: 7
after calling f: 11
inside of g: 42
after calling g: 11
The three n‘s here do not collide since they are each in a different namespace — they are three
names for three different variables, just like there might be three different instances of people,
all called “Bruce”.
Namespaces permit several programmers to work on the same project without having naming
collisions.
How are namespaces, files and modules related?
Python has a convenient and simplifying one-to-one mapping, one module per file,
giving rise to one namespace. Also, Python takes the module name from the file
name, and this becomes the name of the namespace. math.py is a filename, the
module is called math, and its namespace is math. So in Python the concepts are
more or less interchangeable.
But you will encounter other languages (e.g. C#), that allow one module to span
multiple files, or one file to have multiple namespaces, or many files to all share
the same namespace. So the name of the file doesn’t need to be the same as the
namespace.
So a good idea is to try to keep the concepts distinct in your mind.
Files and directories organize where things are stored in our computer. On the
other hand, namespaces and modules are a programming concept: they help us
organize how we want to group related functions and attributes. They are not about
“where” to store things, and should not have to coincide with the file and directory
structures.
So in Python, if you rename the file math.py, its module name also changes, your
import statements would need to change, and your code that refers to functions
or attributes inside that namespace would also need to change.
In other languages this is not necessarily the case. So don’t blur the concepts, just
because Python blurs them!
Scope and lookup rules
The scope of an identifier is the region of program code in which the identifier can be accessed,
or used.
There are three important scopes in Python:
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• Local scope refers to identifiers declared within a function. These identifiers are kept in
the namespace that belongs to the function, and each function has its own namespace.
• Global scope refers to all the identifiers declared within the current module, or file.
• Built-in scope refers to all the identifiers built into Python — those like range and min
that can be used without having to import anything, and are (almost) always available.
Python can help you by telling you what is in which scope. Use the functions locals,
globals, and dir to see for yourself!
Python (like most other computer languages) uses precedence rules: the same name could occur
in more than one of these scopes, but the innermost, or local scope, will always take precedence
over the global scope, and the global scope always gets used in preference to the built-in scope.
Let’s start with a simple example:
1
2
def range(n):
return 123*n
3
4
print(range(10))
What gets printed? We’ve defined our own function called range, so there is now a potential
ambiguity. When we use range, do we mean our own one, or the built-in one? Using the
scope lookup rules determines this: our own range function, not the built-in one, is called,
because our function range is in the global namespace, which takes precedence over the builtin names.
So although names likes range and min are built-in, they can be “hidden” from your use if
you choose to define your own variables or functions that reuse those names. (It is a confusing
practice to redefine built-in names — so to be a good programmer you need to understand the
scope rules and understand that you can do nasty things that will cause confusion, and then you
avoid doing them!)
Now, a slightly more complex example:
1
2
3
4
5
n = 10
m = 3
def f(n):
m = 7
return 2*n+m
6
7
print(f(5), n, m)
This prints 17 10 3. The reason is that the two variables m and n in lines 1 and 2 are outside the
function in the global namespace. Inside the function, new variables called n and m are created
just for the duration of the execution of f. These are created in the local namespace of function
f. Within the body of f, the scope lookup rules determine that we use the local variables m
and n. By contrast, after we’ve returned from f, the n and m arguments to the print function
refer to the original variables on lines 1 and 2, and these have not been changed in any way by
executing function f.
Notice too that the def puts name f into the global namespace here. So it can be called on line
7.
8.6. Scope and lookup rules
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What is the scope of the variable n on line 1? Its scope — the region in which it is visible — is
lines 1, 2, 6, 7. It is hidden from view in lines 3, 4, 5 because of the local variable n.
Attributes and the dot operator
Variables defined inside a module are called attributes of the module. We’ve seen that objects
have attributes too: for example, most objects have a __doc__ attribute, some functions have
a __annotations__ attribute. Attributes are accessed using the dot operator (.). The
question attribute of module1 and module2 is accessed using module1.question
and module2.question.
Modules contain functions as well as attributes, and the dot operator is used to access them
in the same way. seqtools.remove_at refers to the remove_at function in the
seqtools module.
When we use a dotted name, we often refer to it as a fully qualified name, because we’re
saying exactly which question attribute we mean.
Three import statement variants
Here are three different ways to import names into the current namespace, and to use them:
1
2
import math
x = math.sqrt(10)
Here just the single identifier math is added to the current namespace. If you want to access
one of the functions in the module, you need to use the dot notation to get to it.
Here is a different arrangement:
1
2
from math import cos, sin, sqrt
x = sqrt(10)
The names are added directly to the current namespace, and can be used without qualification.
The name math is not itself imported, so trying to use the qualified form math.sqrt would
give an error.
Then we have a convenient shorthand:
1
from math import *
2
3
˓→namespace.
x = sqrt(10)
# Import all the identifiers from math,
#
adding them to the current
# Use them without qualification.
Of these three, the first method is generally preferred, even though it means a little more typing
each time. Although, we can make things shorter by importing a module under a different
name:
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2
3
>>> import math as m
>>> m.pi
3.141592653589793
But hey, with nice editors that do auto-completion, and fast fingers, that’s a small price!
Finally, observe this case:
1
2
3
def area(radius):
import math
return math.pi * radius * radius
4
5
x = math.sqrt(10)
# This gives an error
Here we imported math, but we imported it into the local namespace of area. So the name is
usable within the function body, but not in the enclosing script, because it is not in the global
namespace.
Glossary
attribute A variable defined inside a module (or class or instance – as we will see later).
Module attributes are accessed by using the dot operator (.).
dot operator The dot operator (.) permits access to attributes and functions of a module (or
attributes and methods of a class or instance – as we have seen elsewhere).
fully qualified name A name that is prefixed by some namespace identifier and the dot operator, or by an instance object, e.g. math.sqrt or tess.forward(10).
import statement A statement which makes the objects contained in a module available for
use within another module. There are two forms for the import statement. Using hypothetical modules named mymod1 and mymod2 each containing functions f1 and f2,
and variables v1 and v2, examples of these two forms include:
1
2
import mymod1
from mymod2 import f1, f2, v1, v2
The second form brings the imported objects into the namespace of the importing module, while the first form preserves a separate namespace for the imported module, requiring mymod1.v1 to access the v1 variable from that module.
method Function-like attribute of an object. Methods are invoked (called) on an object using
the dot operator. For example:
>>> s = "this is a string."
>>> s.upper()
'THIS IS A STRING.'
>>>
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We say that the method, upper is invoked on the string, s. s is implicitely the first
argument to upper.
module A file containing Python definitions and statements intended for use in other Python
programs. The contents of a module are made available to the other program by using
the import statement.
namespace A syntactic container providing a context for names so that the same name can
reside in different namespaces without ambiguity. In Python, modules, classes, functions
and methods all form namespaces.
naming collision A situation in which two or more names in a given namespace cannot be
unambiguously resolved. Using
1
import string
instead of
1
from string import *
prevents naming collisions.
ndard library A library is a collection of software used as tools in the development of
other software. The standard library of a programming language is the set of such tools
that are distributed with the core programming language. Python comes with an extensive
standard library.
Exercises
1. Open help for the calendar module.
(a) Try the following:
1
2
3
import calendar
cal = calendar.TextCalendar()
˓→instance
cal.pryear(2012)
˓→here?
# Create an
# What happens
(b) Observe that the week starts on Monday. An adventurous CompSci student believes
that it is better mental chunking to have his week start on Thursday, because then
there are only two working days to the weekend, and every week has a break in the
middle. Read the documentation for TextCalendar, and see how you can help him
print a calendar that suits his needs.
(c) Find a function to print just the month in which your birthday occurs this year.
(d) Try this:
1
2
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d = calendar.LocaleTextCalendar(6, "SPANISH")
d.pryear(2012)
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Try a few other languages, including one that doesn’t work, and see what happens.
(e) Experiment with calendar.isleap. What does it expect as an argument? What
does it return as a result? What kind of a function is this?
Make detailed notes about what you learned from these exercises.
2. Open help for the math module.
(a) How many functions are in the math module?
(b) What does math.ceil do? What about math.floor? (hint: both floor and
ceil expect floating point arguments.)
(c) Describe how we have been computing the same value as math.sqrt without
using the math module.
(d) What are the two data constants in the math module?
Record detailed notes of your investigation in this exercise.
3. Investigate the copy module. What does deepcopy do? In which exercises from last
chapter would deepcopy have come in handy?
4. Create a module named mymodule1.py. Add attributes myage set to your current
age, and year set to the current year. Create another module named mymodule2.py.
Add attributes myage set to 0, and year set to the year you were born. Now create a
file named namespace_test.py. Import both of the modules above and write the
following statement:
1
2
print( (mymodule2.myage - mymodule1.myage) ==
(mymodule2.year - mymodule1.year) )
When you will run namespace_test.py you will see either True or False as
output depending on whether or not you’ve already had your birthday this year.
What this example illustrates is that out different modules can both have attributes named
myage and year. Because they’re in different namespaces, they don’t clash with one
another. When we write namespace_test.py, we fully qualify exactly which variable year or myage we are referring to.
5. Add the following statement to mymodule1.py,
namespace_test.py from the previous exercise:
1
mymodule2.py,
and
print("My name is", __name__)
Run namespace_test.py. What happens? Why? Now add the following to the
bottom of mymodule1.py:
1
2
if __name__ == "__main__":
print("This won't run if I'm
imported.")
Run mymodule1.py and namespace_test.py again. In which case do you see
the new print statement?
6. In a Python shell / interactive interpreter, try the following:
8.10. Exercises
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>>> import this
What does Tim Peters have to say about namespaces?
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CHAPTER 9
More datatypes
You have already encountered the most important datatypes Python has to offer: bools, ints,
floats, strings, tuples, lists and dictionaries. However, there is more to it than hinted at previously. In this section, we will focus mainly on tuples and lists, and introduce sets and frozensets.
Mutable versus immutable and aliasing
Some datatypes in Python are mutable. This means their contents can be changed after they
have been created. Lists and dictionaries are good examples of mutable datatypes.
>>>
>>>
>>>
[9,
my_list = [2, 4, 5, 3, 6, 1]
my_list[0] = 9
my_list
4, 5, 3, 6, 1]
Tuples and strings are examples of immutable datatypes, their contents can not be changed after
they have been created:
>>> my_tuple = (2, 5, 3, 1)
>>> my_tuple[0] = 9
Traceback (most recent call last):
File "<interactive input>", line 2, in <module>
TypeError: 'tuple' object does not support item assignment
>>>
Mutability is usually useful, but it may lead to something called aliasing. In this case, two
variables refer to the same object and mutating one will also change the other:
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>>>
>>>
>>>
>>>
[1,
list_one = [1, 2, 3, 4, 6]
list_two = list_one
list_two[-1] = 5
list_one
2, 3, 4, 5]
This happens, because both list_one and list_two refer to the same memory address containing
the actual list. You can check this using the built-in function id:
>>> list_one = [1, 2, 3, 4, 6]
>>> list_two = list_one
>>> id(list_one) == id(list_two)
True
You can escape this problem by making a copy of the list:
>>> list_one = [1, 2, 3, 4, 6]
>>> list_two = list_one[:]
>>> id(list_one) == id(list_two)
False
>>> list_two[-1] = 5
>>> list_two
[1, 2, 3, 4, 5]
>>> list_one
[1, 2, 3, 4, 6]
However, this will not work for nested lists because of the same reason. The module copy
provides functions to solve this.
Sets and frozensets
Given that tuples and lists are ordered, and dictionaries are unordered, we can construct the
following table.
Mutable
Immutable
Ordered
list
tuple
Unordered
dict
This reveals an empty spot: we don’t know any immutable, unordered datatypes yet. Additionally, you can argue that a dictionary doesn’t belong in this table, since it is a mapping
type whilst lists and tuples are not: a dictionary maps keys to values. This is where sets and
frozensets come in. A set is an unordered, mutable datatype; and a frozenset is an unordered,
immutable datatype.
Mutable
Immutable
Ordered
list
tuple
Unordered
set
frozenset
Since sets and frozensets are unordered, they share some properties with dictionaries: for example, it’s elements are unique. Creating a set, and adding elements to it is simple.
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>>>
>>>
{1,
>>>
>>>
{1,
my_set = set([1, 4, 2, 3, 4])
my_set
2, 3, 4}
my_set.add(13)
my_set
2, 3, 4, 13}
Sets may seem sorted in the example above, but this is completely coincidental. Sets also
support common operations such as membership testing (3 in my_set); and iteration (for
x in my_set:). Additionally, you can add and substract sets from eachother:
1
2
3
4
5
6
7
set1 = set([1, 2, 3])
set2 = set([4, 5, 6])
print(set1 | set2) # {1, 2, 3, 4, 5, 6}
print(set1 & set2) # set()
set2 = set([2, 3, 4, 5])
print(set1 & set2) # {2, 3}
print(set1 - set2) # {1}
Frozensets are mostly the same as set, other then that they can not be modified; i.e. you can’t
add or remove items. See also the documentation online.
More exotic data types - such as queues, stacks and ordered dictionaries - are provided in
Python’s collections module. You can find the documentation here.
9.2. Sets and frozensets
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CHAPTER 10
Recursion
Recursion means “defining something in terms of itself” usually at some smaller scale, perhaps
multiple times, to achieve your objective. For example, we might say “A human being is
someone whose mother is a human being”, or “a directory is a structure that holds files and
(smaller) directories”, or “a family tree starts with a couple who have children, each with their
own family sub-trees”.
Programming languages generally support recursion, which means that, in order to solve a
problem, functions can call themselves to solve smaller subproblems.
Any problem that can be solved iteratively (with a for or while loop) can also be solved recursively. However, recursion takes a while wrap your head around, and because of this, it is
generally only used in specific cases, where either your problem is recursive in nature, or your
data is recursive.
Drawing Fractals
For our purposes, a fractal is a drawing which also has self-similar structure, where it can be
defined in terms of itself. This is a typical example of a problem which is recursive in nature.
Let us start by looking at the famous Koch fractal. An order 0 Koch fractal is simply a straight
line of a given size.
An order 1 Koch fractal is obtained like this: instead of drawing just one line, draw instead four
smaller segments, in the pattern shown here:
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Now what would happen if we repeated this Koch pattern again on each of the order 1 segments? We’d get this order 2 Koch fractal:
Repeating our pattern again gets us an order 3 Koch fractal:
Now let us think about it the other way around. To draw a Koch fractal of order 3, we can
simply draw four order 2 Koch fractals. But each of these in turn needs four order 1 Koch
fractals, and each of those in turn needs four order 0 fractals. Ultimately, the only drawing that
will take place is at order 0. This is very simple to code up in Python:
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2
3
4
5
def koch(tortoise, order, size):
"""
Make turtle tortoise draw a Koch fractal of 'order' and 'size
˓→'.
Leave the turtle facing the same direction.
"""
6
if order == 0:
# The base case is just a straight line
tortoise.forward(size)
else:
koch(tortoise, order-1, size/3)
# Go 1/3 of the way
tortoise.left(60)
koch(tortoise, order-1, size/3)
tortoise.right(120)
koch(tortoise, order-1, size/3)
tortoise.left(60)
koch(tortoise, order-1, size/3)
7
8
9
10
11
12
13
14
15
16
The key thing that is new here is that if order is not zero, koch calls itself recursively to get its
job done.
Let’s make a simple observation and tighten up this code. Remember that turning right by 120
is the same as turning left by -120. So with a bit of clever rearrangement, we can use a loop
instead of lines 10-16:
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1
2
3
4
5
6
7
def koch(tortoise, order, size):
if order == 0:
tortoise.forward(size)
else:
for angle in [60, -120, 60, 0]:
koch(tortoise, order-1, size/3)
tortoise.left(angle)
The final turn is 0 degrees — so it has no effect. But it has allowed us to find a pattern and
reduce seven lines of code to three, which will make things easier for our next observations.
Recursion, the high-level view
One way to think about this is to convince yourself that the function works correctly when you
call it for an order 0 fractal. Then do a mental leap of faith, saying “the fairy godmother (or
Python, if you can think of Python as your fairy godmother) knows how to handle the recursive
level 0 calls for me on lines 11, 13, 15, and 17, so I don’t need to think about that detail!” All
I need to focus on is how to draw an order 1 fractal if I can assume the order 0 one is already
working.
You’re practicing mental abstraction — ignoring the subproblem while you solve the big problem.
If this mode of thinking works (and you should practice it!), then take it to the next level. Aha!
now can I see that it will work when called for order 2 under the assumption that it is already
working for level 1.
And, in general, if I can assume the order n-1 case works, can I just solve the level n problem?
Students of mathematics who have played with proofs of induction should see some very strong
similarities here.
Recursion, the low-level operational view
Another way of trying to understand recursion is to get rid of it! If we had separate functions to
draw a level 3 fractal, a level 2 fractal, a level 1 fractal and a level 0 fractal, we could simplify
the above code, quite mechanically, to a situation where there was no longer any recursion, like
this:
1
2
def koch_0(tortoise, size):
tortoise.forward(size)
3
4
5
6
7
def koch_1(tortoise, size):
for angle in [60, -120, 60, 0]:
koch_0(tortoise, size/3)
tortoise.left(angle)
8
9
10
11
def koch_2(tortoise, size):
for angle in [60, -120, 60, 0]:
koch_1(tortoise, size/3)
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tortoise.left(angle)
12
13
14
15
16
17
def koch_3(tortoise, size):
for angle in [60, -120, 60, 0]:
koch_2(tortoise, size/3)
tortoise.left(angle)
This trick of “unrolling” the recursion gives us an operational view of what happens. You can
trace the program into koch_3, and from there, into koch_2, and then into koch_1, etc., all
the way down the different layers of the recursion.
This might be a useful hint to build your understanding. The mental goal is, however, to be
able to do the abstraction!
Recursive data structures
Most of the Python data types we have seen can be grouped inside lists and tuples in a variety
of ways. Lists and tuples can also be nested, providing many possibilities for organizing data.
The organization of data for the purpose of making it easier to use is called a data structure.
It’s election time and we are helping to compute the votes as they come in. Votes arriving from
individual wards, precincts, municipalities, counties, and states are sometimes reported as a
sum total of votes and sometimes as a list of subtotals of votes. After considering how best to
store the tallies, we decide to use a nested number list, which we define as follows:
A nested number list is a list whose elements are either:
1. numbers
2. nested number lists
Notice that the term, nested number list is used in its own definition. Recursive definitions
like this are quite common in mathematics and computer science. They provide a concise and
powerful way to describe recursive data structures that are partially composed of smaller and
simpler instances of themselves. The definition is not circular, since at some point we will
reach a list that does not have any lists as elements.
Now suppose our job is to write a function that will sum all of the values in a nested number
list. Python has a built-in function which finds the sum of a sequence of numbers:
>>> sum([1, 2, 8])
11
For our nested number list, however, sum will not work:
>>> sum([1, 2, [11, 13], 8])
Traceback (most recent call last):
File "<interactive input>", line 1, in <module>
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TypeError: unsupported operand type(s) for +: 'int' and 'list'
>>>
The problem is that the third element of this list, [11, 13], is itself a list, so it cannot just be
added to 1, 2, and 8.
Processing recursive number lists
To sum all the numbers in our recursive nested number list we need to traverse the list, visiting
each of the elements within its nested structure, adding any numeric elements to our sum, and
recursively repeating the summing process with any elements which are themselves sub-lists.
Thanks to recursion, the Python code needed to sum the values of a nested number list is
surprisingly short:
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2
3
4
5
6
7
8
9
def recursive_sum(nested_number_list):
"""Returns the total sum of all elements in nested_number_list""
˓→"
total = 0
for element in nested_number_list:
if type(element) is list:
total += recursive_sum(element)
else:
total += element
return total
The body of recursive_sum consists mainly of a for loop that traverses
nested_number_list. If element is a numerical value (the else branch), it is simply added to total. If element is a list, then recursive_sum is called again, with the
element as an argument. The statement inside the function definition in which the function calls
itself is known as the recursive call.
The example above has a base case (on line 13) which does not lead to a recursive call: the
case where the element is not a (sub-) list. Without a base case, you’ll have infinite recursion,
and your program will not work.
An alternative solution, completely recursive, would be the following. Notice that this implementation does not contain a for loop!
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2
3
4
5
6
7
def recursive_sum(nested_number_list):
"""Returns the total sum of all elements in nested_number_list""
˓→"
if len(nested_number_list) == 0:
return 0
head, *tail = nested_number_list #Assign the first element of
˓→nested_number_list to head, and the rest to tail.
if isinstance(head, list): # If head is a list....
return recursive_sum(head) + recursive_sum(tail)
10.3. Processing recursive number lists
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else:
return head + recursive_sum(tail)
8
9
Recursion is truly one of the most beautiful and elegant tools in computer science.
A slightly more complicated problem is finding the largest value in our nested number list:
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2
3
4
5
6
7
8
9
10
11
12
13
def recursive_max(nested_list):
"""
Find the maximum in a recursive structure of lists
within other lists.
Precondition: No lists or sublists are empty.
"""
largest = None
first_time = True
for element in nested_list:
if type(element) is list:
value = recursive_max(element)
else:
value = element
14
if first_time or value > largest:
largest = value
first_time = False
15
16
17
18
return largest
19
The added twist to this problem is finding a value for initializing largest. We can’t just use
nested_list[0], since that could be either a element or a list. To solve this problem (at
every recursive call) we initialize a Boolean flag (at line 8). When we’ve found the value of
interest, (at line 15) we check to see whether this is the initializing (first) value for largest,
or a value that could potentially change largest.
Again here we have a base case at line 13. If we don’t supply a base case, Python stops after
reaching a maximum recursion depth and returns a runtime error. See how this happens, by
running this little script which we will call infinite_recursion.py:
1
2
3
def recursion_depth(number):
print("{0}, ".format(number), end="")
recursion_depth(number + 1)
4
5
recursion_depth(0)
After watching the messages flash by, you will be presented with the end of a long traceback
that ends with a message like the following:
RuntimeError: maximum recursion depth exceeded ...
We would certainly never want something like this to happen to a user of one of our programs,
so in another appendix we’ll see how errors, any kinds of errors, are handled in Python.
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Case study: Fibonacci numbers
The famous Fibonacci sequence 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 134, ... was devised
by Fibonacci (1170-1250), who used this to model the breeding of (pairs) of rabbits. If, in
generation 7 you had 21 pairs in total, of which 13 were adults, then next generation the adults
will all have bred new children, and the previous children will have grown up to become adults.
So in generation 8 you’ll have 13+21=34, of which 21 are adults.
This model to explain rabbit breeding made the simplifying assumption that rabbits never died.
Scientists often make (unrealistic) simplifying assumptions and restrictions to make some headway with the problem.
If we number the terms of the sequence from 0, we can describe each term recursively as the
sum of the previous two terms:
fib(0) = 0
fib(1) = 1
fib(n) = fib(n-1) + fib(n-2)
for n >= 2
This translates very directly into some Python:
1
2
3
4
5
def fib(n):
if n <= 1:
return n
t = fib(n-1) + fib(n-2)
return t
This is a particularly inefficient algorithm, and this could be solved far more efficient iteratively:
1
2
3
4
5
import time
t0 = time.clock()
n = 35
result = fib(n)
t1 = time.clock()
6
7
print("fib({0}) = {1}, ({2:.2f} secs)".format(n, result, t1-t0))
We get the correct result, but an exploding amount of work!
fib(35) = 9227465, (10.54 secs)
Example with recursive directories and files
The following program lists the contents of a directory and all its subdirectories.
1
import os
2
3
4
def get_dirlist(path):
"""
10.4. Case study: Fibonacci numbers
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Return a sorted list of all entries in path.
This returns just the names, not the full path to the names.
"""
dirlist = os.listdir(path)
dirlist.sort()
return dirlist
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6
7
8
9
10
11
12
13
14
15
16
def print_files(path, prefix = ""):
""" Print recursive listing of contents of path """
if prefix == "": # Detect outermost call, print a heading
print("Folder listing for", path)
prefix = "| "
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18
19
20
21
22
23
dirlist = get_dirlist(path)
for file in dirlist:
print(prefix+file)
# Print the line
fullname = os.path.join(path, file)
# Turn name into full
˓→pathname
if os.path.isdir(fullname):
# If a directory,
˓→recurse.
print_files(fullname, prefix + "| ")
Calling the function print_files with some folder name will produce output similar to
this:
Folder listing for c:\python31\Lib\site-packages\pygame\examples
| __init__.py
| aacircle.py
| aliens.py
| arraydemo.py
| blend_fill.py
| blit_blends.py
| camera.py
| chimp.py
| cursors.py
| data
| | alien1.png
| | alien2.png
| | alien3.png
...
Note that something similar is already implemented in the os module: os.walk.
An animated fractal, using PyGame
Here we have a tree fractal pattern of order 8. We’ve labelled some of the edges, showing the
depth of the recursion at which each edge was drawn.
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In the tree above, the angle of deviation from the trunk is 30 degrees. Varying that angle gives
other interesting shapes, for example, with the angle at 90 degrees we get this:
An interesting animation occurs if we generate and draw trees very rapidly, each time varying
the angle a little. Although the Turtle module can draw trees like this quite elegantly, we could
struggle for good frame rates. So we’ll use PyGame instead, with a few embellishments and
observations. (Once again, we suggest you cut and paste this code into your Python environment.)
1
2
import pygame, math
pygame.init()
# prepare the pygame module for use
3
4
# Create a new surface and window.
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6
7
surface_size = 1024
main_surface = pygame.display.set_mode((surface_size,surface_size))
my_clock = pygame.time.Clock()
8
9
10
def draw_tree(order, theta, size, position, heading, color=(0,0,0),
˓→depth=0):
11
12
13
14
15
16
17
18
trunk_ratio = 0.29
# How big is the trunk relative to
˓→whole tree?
trunk = size * trunk_ratio # length of trunk
delta_x = trunk * math.cos(heading)
delta_y = trunk * math.sin(heading)
(u, v) = position
newposition = (u + delta_x, v + delta_y)
pygame.draw.line(main_surface, color, position, newposition)
19
if order > 0:
20
# Draw another layer of subtrees
21
22
23
24
25
26
27
28
29
30
# These next six lines are a simple hack to make the two
˓→major halves
# of the recursion different colors. Fiddle here to change
˓→colors
# at other depths, or when depth is even, or odd, etc.
if depth == 0:
color1 = (255, 0, 0)
color2 = (0, 0, 255)
else:
color1 = color
color2 = color
31
32
33
34
˓→
35
˓→
# make the recursive calls to draw the two subtrees
newsize = size*(1 - trunk_ratio)
draw_tree(order-1, theta, newsize, newposition, heading-theta,
color1, depth+1)
draw_tree(order-1, theta, newsize, newposition, heading+theta,
color2, depth+1)
36
37
38
def gameloop():
39
theta = 0
while True:
40
41
42
# Handle evente from keyboard, mouse, etc.
event = pygame.event.poll()
if event.type == pygame.QUIT:
break;
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45
46
47
# Updates - change the angle
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theta += 0.01
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50
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53
# Draw everything
main_surface.fill((255, 255, 0))
draw_tree(9, theta, surface_size*0.9, (surface_size//2,
˓→surface_size-50), -math.pi/2)
54
pygame.display.flip()
my_clock.tick(120)
55
56
57
58
59
60
gameloop()
pygame.quit()
• The math library works with angles in radians rather than degrees.
• Lines 14 and 15 uses some high school trigonmetry. From the length of the desired line
(trunk), and its desired angle, cos and sin help us to calculate the x and y distances
we need to move.
• Lines 22-30 are unnecessary, except if we want a colorful tree.
• In the main game loop at line 49 we change the angle on every frame, and redraw the
new tree.
• Line 18 shows that PyGame can also draw lines, and plenty more. Check out the documentation. For example, drawing a small circle at each branch point of the tree can be
accomplished by adding this line directly below line 18:
1
pygame.draw.circle(main_surface, color, (int(position[0]),
˓→int(position[1])), 3)
Another interesting effect — instructive too, if you wish to reinforce the idea of different instances of the function being called at different depths of recursion — is to create a list of
colors, and let each recursive depth use a different color for drawing. (Use the depth of the
recursion to index the list of colors.)
Mutual Recursion
In addition to a function calling just itself, it is also possible to make multiple functions that
call eachother. This is rarely really usefull, but it can be used to make state machines.
1
2
3
4
5
def function_a(n): # Do things associated with state A
if n == 0:
return
print('a')
function_b(n - 1) # Proceed to state B
6
7
8
def function_b(n):
# Do things associated with state B
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print('b')
function_a(n - 1)
9
10
# Proceed to state A
Glossary
base case A branch of the conditional statement in a recursive function that does not give rise
to further recursive calls.
infinite recursion A function that calls itself recursively without ever reaching any base case.
Eventually, infinite recursion causes a runtime error.
recursion The process of calling a function that is already executing.
recursive call The statement that calls an already executing function. Recursion can also be
indirect — function f can call g which calls h, and h could make a call back to f.
recursive definition A definition which defines something in terms of itself. To be useful it
must include base cases which are not recursive. In this way it differs from a circular
definition. Recursive definitions often provide an elegant way to express complex data
structures, like a directory that can contain other directories, or a menu that can contain
other menus.
Exercises
1. Modify the Koch fractal program so that it draws a Koch snowflake, like this:
2.
(a) Draw a Cesaro torn line fractal, of the order given by the user. We show four
different lines of orders 0,1,2,3. In this example, the angle of the tear is 10 degrees.
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(b) Four lines make a square. Use the code in part a) to draw cesaro squares. Varying
the angle gives interesting effects — experiment a bit, or perhaps let the user input
the angle of the tear.
(a) (For the mathematically inclined). In the squares shown here, the higher-order
drawings become a little larger. (Look at the bottom lines of each square - they’re
not aligned.) This is because we just halved the drawn part of the line for each recursive subproblem. So we’ve “grown” the overall square by the width of the tear(s).
Can you solve the geometry problem so that the total size of the subproblem case
(including the tear) remains exactly the same size as the original?
3. A Sierpinski triangle of order 0 is an equilateral triangle. An order 1 triangle can be
drawn by drawing 3 smaller triangles (shown slightly disconnected here, just to help our
understanding). Higher order 2 and 3 triangles are also shown. Draw Sierpinski triangles
of any order input by the user.
4. Adapt the above program to change the color of its three sub-triangles at some depth
of recursion. The illustration below shows two cases: on the left, the color is changed
at depth 0 (the outmost level of recursion), on the right, at depth 2. If the user supplies a negative depth, the color never changes. (Hint: add a new optional parameter
colorChangeDepth (which defaults to -1), and make this one smaller on each recur10.9. Exercises
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sive subcall. Then, in the section of code before you recurse, test whether the parameter
is zero, and change color.)
5. Write a function, recursive_min, that returns the smallest value in a nested number
list. Assume there are no empty lists or sublists:
6. Write a function count that returns the number of occurrences of target in a nested
list:
7. Write a function flatten that returns a simple list containing all the values in a nested
list:
8. Rewrite the fibonacci algorithm without using recursion. Can you find bigger terms of
the sequence? Can you find fib(200)?
9. Use help to find out what sys.getrecursionlimit() and sys.
setrecursionlimit(n) do. Create several experiments similar to what was
done in infinite_recursion.py to test your understanding of how these module functions
work.
10. Write a program that walks a directory structure (as in the last section of this chapter), but
instead of printing filenames, it returns a list of all the full paths of files in the directory
or the subdirectories. (Don’t include directories in this list — just files.) For example,
the output list might have elements like this:
["C:\Python31\Lib\site-packages\pygame\docs\ref\mask.html",
"C:\Python31\Lib\site-packages\pygame\docs\ref\midi.html",
...
"C:\Python31\Lib\site-packages\pygame\examples\aliens.py",
...
"C:\Python31\Lib\site-packages\pygame\examples\data\boom.wav",
... ]
11. Write a program named litter.py that creates an empty file named trash.txt in
each subdirectory of a directory tree given the root of the tree as an argument (or the
current directory as a default). Now write a program named cleanup.py that removes
all these files.
Hint #1: Use the program from the example in the last section of this chapter as a basis
for these two recursive programs. Because you’re going to destroy files on your disks,
you better get this right, or you risk losing files you care about. So excellent advice is
that initially you should fake the deletion of the files — just print the full path names
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of each file that you intend to delete. Once you’re happy that your logic is correct, and
you can see that you’re not deleting the wrong things, you can replace the print statement
with the real thing.
Hint #2: Look in the os module for a function that removes files.
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CHAPTER 11
Classes and Objects
Classes and Objects — the Basics
Object-oriented programming
Python is an object-oriented programming language, which means that it provides features
that support object-oriented programming (OOP).
Object-oriented programming has its roots in the 1960s, but it wasn’t until the mid 1980s that it
became the main programming paradigm used in the creation of new software. It was developed
as a way to handle the rapidly increasing size and complexity of software systems, and to make
it easier to modify these large and complex systems over time.
Up to now, most of the programs we have been writing use a procedural programming
paradigm. In procedural programming the focus is on writing functions or procedures which
operate on data. In object-oriented programming the focus is on the creation of objects which
contain both data and functionality together. (We have seen turtle objects, string objects, and
random number generators, to name a few places where we’ve already worked with objects.)
Usually, each object definition corresponds to some object or concept in the real world, and the
functions that operate on that object correspond to the ways real-world objects interact.
User-defined compound data types
We’ve already seen classes like str, int, float and Turtle. We are now ready to create
our own user-defined class: the Point.
Consider the concept of a mathematical point. In two dimensions, a point is two numbers
(coordinates) that are treated collectively as a single object. Points are often written in between
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parentheses with a comma separating the coordinates. For example, (0, 0) represents the
origin, and (x, y) represents the point x units to the right and y units up from the origin.
Some of the typical operations that one associates with points might be calculating the distance
of a point from the origin, or from another point, or finding a midpoint of two points, or asking
if a point falls within a given rectangle or circle. We’ll shortly see how we can organize these
together with the data.
A natural way to represent a point in Python is with two numeric values. The question, then, is
how to group these two values into a compound object. The quick and dirty solution is to use a
tuple, and for some applications that might be a good choice.
An alternative is to define a new class. This approach involves a bit more effort, but it has
advantages that will be apparent soon. We’ll want our points to each have an x and a y attribute,
so our first class definition looks like this:
1
2
class Point:
""" Point class represents and manipulates x,y coords. "
˓→""
3
4
5
6
7
def __init__(self):
""" Create a new point at the origin """
self.x = 0
self.y = 0
Class definitions can appear anywhere in a program, but they are usually near the beginning
(after the import statements). Some programmers and languages prefer to put every class in a
module of its own — we won’t do that here. The syntax rules for a class definition are the same
as for other compound statements. There is a header which begins with the keyword, class,
followed by the name of the class, and ending with a colon. Indentation levels tell us where the
class ends.
If the first line after the class header is a string, it becomes the docstring of the class, and will
be recognized by various tools. (This is also the way docstrings work in functions.)
Every class should have a method with the special name __init__. This initializer method
is automatically called whenever a new instance of Point is created. It gives the programmer the opportunity to set up the attributes required within the new instance by giving them
their initial state/values. The self parameter (we could choose any other name, but self
is the convention) is automatically set to reference the newly created object that needs to be
initialized.
So let’s use our new Point class now:
1
2
p = Point()
q = Point()
# Instantiate an object of type Point
# Make a second point
3
4
print(p.x, p.y, q.x, q.y)
˓→x and y
# Each point object has its own
This program prints:
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0 0 0 0
because during the initialization of the objects, we created two attributes called x and y for
each, and gave them both the value 0.
This should look familiar — we’ve used classes before to create more than one object:
1
from turtle import Turtle
2
3
4
tess = Turtle()
alex = Turtle()
# Instantiate objects of type Turtle
The variables p and q are assigned references to two new Point objects. A function like
Turtle or Point that creates a new object instance is called a constructor, and every class
automatically provides a constructor function which is named the same as the class.
It may be helpful to think of a class as a factory for making objects. The class itself isn’t an
instance of a point, but it contains the machinery to make point instances. Every time we call
the constructor, we’re asking the factory to make us a new object. As the object comes off the
production line, its initialization method is executed to get the object properly set up with its
factory default settings.
The combined process of “make me a new object” and “get its settings initialized to the factory
default settings” is called instantiation.
Attributes
Like real world objects, object instances have both attributes and methods.
We can modify the attributes in an instance using dot notation:
>>> p.x = 3
>>> p.y = 4
Both modules and instances create their own namespaces, and the syntax for accessing names
contained in each, called attributes, is the same. In this case the attribute we are selecting is a
data item from an instance.
The following state diagram shows the result of these assignments:
The variable p refers to a Point object, which contains two attributes. Each attribute refers to
a number.
We can access the value of an attribute using the same syntax:
>>> print(p.y)
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>>> x = p.x
>>> print(x)
3
The expression p.x means, “Go to the object p refers to and get the value of x”. In this case,
we assign that value to a variable named x. There is no conflict between the variable x (in the
global namespace here) and the attribute x (in the namespace belonging to the instance). The
purpose of dot notation is to fully qualify which variable we are referring to unambiguously.
We can use dot notation as part of any expression, so the following statements are legal:
1
2
print("(x={0}, y={1})".format(p.x, p.y))
distance_squared_from_origin = p.x * p.x + p.y * p.y
The first line outputs (x=3, y=4). The second line calculates the value 25.
Improving our initializer
To create a point at position (7, 6) currently needs three lines of code:
1
2
3
p = Point()
p.x = 7
p.y = 6
We can make our class constructor more general by placing extra parameters into the
__init__ method, as shown in this example:
1
2
class Point:
""" Point class represents and manipulates x,y coords. "
˓→""
3
def __init__(self, x=0, y=0):
""" Create a new point at x, y """
self.x = x
self.y = y
4
5
6
7
8
9
# Other statements outside the class continue below here.
The x and y parameters here are both optional. If the caller does not supply arguments, they’ll
get the default values of 0. Here is our improved class in action:
>>>
>>>
>>>
>>>
4 3
p = Point(4, 2)
q = Point(6, 3)
r = Point()
# r represents the origin (0, 0)
print(p.x, q.y, r.x)
0
Technically speaking ...
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If we are really fussy, we would argue that the __init__ method’s docstring is inaccurate.
__init__ doesn’t create the object (i.e. set aside memory for it), — it just initializes the
object to its factory-default settings after its creation.
But tools like PyScripter understand that instantiation — creation and initialization — happen together, and they choose to display the initializer’s docstring as the tooltip to guide the
programmer that calls the class constructor.
So we’re writing the docstring so that it makes the most sense when it pops up to help the
programmer who is using our Point class:
Adding other methods to our class
The key advantage of using a class like Point rather than a simple tuple (6, 7) now becomes apparent. We can add methods to the Point class that are sensible operations for points,
but which may not be appropriate for other tuples like (25, 12) which might represent, say,
a day and a month, e.g. Christmas day. So being able to calculate the distance from the origin
is sensible for points, but not for (day, month) data. For (day, month) data, we’d like different
operations, perhaps to find what day of the week it will fall on in 2020.
Creating a class like Point brings an exceptional amount of “organizational power” to our
programs, and to our thinking. We can group together the sensible operations, and the kinds of
data they apply to, and each instance of the class can have its own state.
A method behaves like a function but it is invoked on a specific instance, e.g. tess.
right(90). Like a data attribute, methods are accessed using dot notation.
Let’s add another method, distance_from_origin, to see better how methods work:
1
2
class Point:
""" Create a new Point, at coordinates x, y """
3
4
5
6
7
def __init__(self, x=0, y=0):
""" Create a new point at x, y """
self.x = x
self.y = y
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9
10
11
def distance_from_origin(self):
""" Compute my distance from the origin """
return ((self.x ** 2) + (self.y ** 2)) ** 0.5
Let’s create a few point instances, look at their attributes, and call our new method on them:
(We must run our program first, to make our Point class available to the interpreter.)
>>> p = Point(3, 4)
>>> p.x
3
>>> p.y
4
>>> p.distance_from_origin()
5.0
>>> q = Point(5, 12)
>>> q.x
5
>>> q.y
12
>>> q.distance_from_origin()
13.0
>>> r = Point()
>>> r.x
0
>>> r.y
0
>>> r.distance_from_origin()
0.0
When defining a method, the first parameter refers to the instance being manipulated. As
already noted, it is customary to name this parameter self.
Notice that the caller of distance_from_origin does not explicitly supply an argument
to match the self parameter — this is done for us, behind our back.
Instances as arguments and parameters
We can pass an object as an argument in the usual way. We’ve already seen this in some
of the turtle examples, where we passed the turtle to some function like draw_bar in the
chapter titled Conditionals, so that the function could control and use whatever turtle instance
we passed to it.
Be aware that our variable only holds a reference to an object, so passing tess into a function
creates an alias: both the caller and the called function now have a reference, but there is only
one turtle!
Here is a simple function involving our new Point objects:
1
2
214
def print_point(pt):
print("({0}, {1})".format(pt.x, pt.y))
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print_point takes a point as an argument and formats the output in whichever way we
choose. If we call print_point(p) with point p as defined previously, the output is (3,
4).
Converting an instance to a string
Most object-oriented programmers probably would not do what we’ve just done in
print_point. When we’re working with classes and objects, a preferred alternative is to
add a new method to the class. And we don’t like chatterbox methods that call print. A
better approach is to have a method so that every instance can produce a string representation
of itself. Let’s initially call it to_string:
1
2
class Point:
# ...
3
4
5
def to_string(self):
return "({0}, {1})".format(self.x, self.y)
Now we can say:
>>> p = Point(3, 4)
>>> print(p.to_string())
(3, 4)
But don’t we already have a str type converter that can turn our object into a string? Yes! And
doesn’t print automatically use this when printing things? Yes again! But these automatic
mechanisms do not yet do exactly what we want:
>>> str(p)
'<__main__.Point object at 0x01F9AA10>'
>>> print(p)
'<__main__.Point object at 0x01F9AA10>'
Python has a clever trick up its sleeve to fix this. If we call our new method __str__ instead
of to_string, the Python interpreter will use our code whenever it needs to convert a Point
to a string. Let’s re-do this again, now:
1
2
class Point:
# ...
3
4
5
def __str__(self):
# All we have done is renamed
˓→the method
return "({0}, {1})".format(self.x, self.y)
and now things are looking great!
>>> str(p)
˓→wrote.
(3, 4)
# Python now uses the __str__ method that we
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>>> print(p)
(3, 4)
Instances as return values
Functions and methods can return instances. For example, given two Point objects, find their
midpoint. First we’ll write this as a regular function:
1
2
3
4
5
def midpoint(p1, p2):
""" Return the midpoint of points p1 and p2 """
mx = (p1.x + p2.x)/2
my = (p1.y + p2.y)/2
return Point(mx, my)
The function creates and returns a new Point object:
>>> p
>>> q
>>> r
>>> r
(4.0,
= Point(3, 4)
= Point(5, 12)
= midpoint(p, q)
8.0)
Now let us do this as a method instead. Suppose we have a point object, and wish to find the
midpoint halfway between it and some other target point:
1
2
class Point:
# ...
3
4
5
6
7
8
def halfway(self, target):
""" Return the halfway point between myself and the
˓→target """
mx = (self.x + target.x)/2
my = (self.y + target.y)/2
return Point(mx, my)
This method is identical to the function, aside from some renaming. It’s usage might be like
this:
>>> p
>>> q
>>> r
>>> r
(4.0,
= Point(3, 4)
= Point(5, 12)
= p.halfway(q)
8.0)
While this example assigns each point to a variable, this need not be done. Just as function
calls are composable, method calls and object instantiation are also composable, leading to this
alternative that uses no variables:
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>>> print(Point(3, 4).halfway(Point(5, 12)))
(4.0, 8.0)
A change of perspective
The original syntax for a function call, print_time(current_time), suggests that the
function is the active agent. It says something like, “Hey, print_time! Here’s an object for you
to print.”
In object-oriented programming, the objects are considered the active agents. An invocation
like current_time.print_time() says “Hey current_time! Please print yourself!”
In our early introduction to turtles, we used an object-oriented style, so that we said tess.
forward(100), which asks the turtle to move itself forward by the given number of steps.
This change in perspective might be more polite, but it may not initially be obvious that it
is useful. But sometimes shifting responsibility from the functions onto the objects makes it
possible to write more versatile functions, and makes it easier to maintain and reuse code.
The most important advantage of the object-oriented style is that it fits our mental chunking and
real-life experience more accurately. In real life our cook method is part of our microwave
oven — we don’t have a cook function sitting in the corner of the kitchen, into which we pass
the microwave! Similarly, we use the cellphone’s own methods to send an sms, or to change its
state to silent. The functionality of real-world objects tends to be tightly bound up inside the
objects themselves. OOP allows us to accurately mirror this when we organize our programs.
Objects can have state
Objects are most useful when we also need to keep some state that is updated from time to
time. Consider a turtle object. Its state consists of things like its position, its heading, its color,
and its shape. A method like left(90) updates the turtle’s heading, forward changes its
position, and so on.
For a bank account object, a main component of the state would be the current balance, and
perhaps a log of all transactions. The methods would allow us to query the current balance,
deposit new funds, or make a payment. Making a payment would include an amount, and a
description, so that this could be added to the transaction log. We’d also want a method to show
the transaction log.
Glossary
attribute One of the named data items that makes up an instance.
class A user-defined compound type. A class can also be thought of as a template for the
objects that are instances of it. (The iPhone is a class. By December 2010, estimates are
that 50 million instances had been sold!)
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constructor Every class has a “factory”, called by the same name as the class, for making new
instances. If the class has an initializer method, this method is used to get the attributes
(i.e. the state) of the new object properly set up.
initializer method A special method in Python (called __init__) that is invoked automatically to set a newly created object’s attributes to their initial (factory-default) state.
instance An object whose type is of some class. Instance and object are used interchangeably.
instantiate To create an instance of a class, and to run its initializer.
method A function that is defined inside a class definition and is invoked on instances of that
class.
object A compound data type that is often used to model a thing or concept in the real world.
It bundles together the data and the operations that are relevant for that kind of data.
Instance and object are used interchangeably.
object-oriented programming A powerful style of programming in which data and the operations that manipulate it are organized into objects.
object-oriented language A language that provides features, such as user-defined classes and
inheritance, that facilitate object-oriented programming.
Exercises
1. Rewrite the distance function from the chapter titled Fruitful functions so that it takes
two Points as parameters instead of four numbers.
2. Add a method reflect_x to Point which returns a new Point, one which is the
reflection of the point about the x-axis. For example, Point(3, 5).reflect_x()
is (3, -5)
3. Add a method slope_from_origin which returns the slope of the line joining the
origin to the point. For example,
>>> Point(4, 10).slope_from_origin()
2.5
What cases will cause this method to fail?
4. The equation of a straight line is “y = ax + b”, (or perhaps “y = mx + c”). The coefficients
a and b completely describe the line. Write a method in the Point class so that if a point
instance is given another point, it will compute the equation of the straight line joining
the two points. It must return the two coefficients as a tuple of two values. For example,
>>> print(Point(4, 11).get_line_to(Point(6, 15)))
>>> (2, 3)
This tells us that the equation of the line joining the two points is “y = 2x + 3”. When
will this method fail?
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5. Given four points that fall on the circumference of a circle, find the midpoint of the circle.
When will this function fail?
Hint: You must know how to solve the geometry problem before you think of going
anywhere near programming. You cannot program a solution to a problem if you don’t
understand what you want the computer to do!
6. Create a new class, SMS_store. The class will instantiate SMS_store objects, similar to
an inbox or outbox on a cellphone:
my_inbox = SMS_store()
This store can hold multiple SMS messages (i.e. its internal state will just be a list of
messages). Each message will be represented as a tuple:
(has_been_viewed, from_number, time_arrived, text_of_SMS)
The inbox object should provide these methods:
my_inbox.add_new_arrival(from_number, time_arrived, text_of_SMS)
# Makes new SMS tuple, inserts it after other messages
# in the store. When creating this message, its
# has_been_viewed status is set False.
my_inbox.message_count()
# Returns the number of sms messages in my_inbox
my_inbox.get_unread_indexes()
# Returns list of indexes of all not-yet-viewed SMS messages
my_inbox.get_message(i)
# Return (from_number, time_arrived, text_of_sms) for
˓→message[i]
# Also change its state to "has been viewed".
# If there is no message at position i, return None
my_inbox.delete(i)
my_inbox.clear()
# Delete the message at index i
# Delete all messages from inbox
Write the class, create a message store object, write tests for these methods, and implement the methods.
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Classes and Objects — Digging a little deeper
Rectangles
Let’s say that we want a class to represent a rectangle which is located somewhere in the XY
plane. The question is, what information do we have to provide in order to specify such a
rectangle? To keep things simple, assume that the rectangle is oriented either vertically or
horizontally, never at an angle.
There are a few possibilities: we could specify the center of the rectangle (two coordinates)
and its size (width and height); or we could specify one of the corners and the size; or we could
specify two opposing corners. A conventional choice is to specify the upper-left corner of the
rectangle, and the size.
Again, we’ll define a new class, and provide it with an initializer and a string converter method:
1
2
class Rectangle:
""" A class to manufacture rectangle objects """
3
4
5
6
7
8
def __init__(self, posn, w, h):
""" Initialize rectangle at posn, with width w,
˓→height h """
self.corner = posn
self.width = w
self.height = h
9
10
11
12
def __str__(self):
return "({0}, {1}, {2})"
.format(self.corner, self.width, self.
˓→height)
13
14
15
16
17
box = Rectangle(Point(0, 0), 100, 200)
bomb = Rectangle(Point(100, 80), 5, 10)
˓→game
print("box: ", box)
print("bomb: ", bomb)
# In my video
To specify the upper-left corner, we have embedded a Point object (as we used it in the previous chapter) within our new Rectangle object! We create two new Rectangle objects,
and then print them producing:
box: ((0, 0), 100, 200)
bomb: ((100, 80), 5, 10)
The dot operator composes. The expression box.corner.x means, “Go to the object that
box refers to and select its attribute named corner, then go to that object and select its
attribute named x”.
The figure shows the state of this object:
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Objects are mutable
We can change the state of an object by making an assignment to one of its attributes. For
example, to grow the size of a rectangle without changing its position, we could modify the
values of width and height:
box.width += 50
box.height += 100
Of course, we’d probably like to provide a method to encapsulate this inside the class. We will
also provide another method to move the position of the rectangle elsewhere:
1
2
class Rectangle:
# ...
3
4
5
6
7
def grow(self, delta_width, delta_height):
""" Grow (or shrink) this object by the deltas """
self.width += delta_width
self.height += delta_height
8
9
10
11
12
def move(self, dx, dy):
""" Move this object by the deltas """
self.corner.x += dx
self.corner.y += dy
Let us try this:
>>> r = Rectangle(Point(10,5), 100, 50)
>>> print(r)
((10, 5), 100, 50)
>>> r.grow(25, -10)
>>> print(r)
((10, 5), 125, 40)
>>> r.move(-10, 10)
print(r)
((0, 15), 125, 40)
Sameness
The meaning of the word “same” seems perfectly clear until we give it some thought, and then
we realize there is more to it than we initially expected.
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For example, if we say, “Alice and Bob have the same car”, we mean that her car and his are
the same make and model, but that they are two different cars. If we say, “Alice and Bob have
the same mother”, we mean that her mother and his are the same person.
When we talk about objects, there is a similar ambiguity. For example, if two Points are the
same, does that mean they contain the same data (coordinates) or that they are actually the same
object?
We’ve already seen the is operator in the chapter on lists, where we talked about aliases: it
allows us to find out if two references refer to the same object:
>>> p1 = Point(3, 4)
>>> p2 = Point(3, 4)
>>> p1 is p2
False
Even though p1 and p2 contain the same coordinates, they are not the same object. If we
assign p1 to p3, then the two variables are aliases of the same object:
>>> p3 = p1
>>> p1 is p3
True
This type of equality is called shallow equality because it compares only the references, not
the contents of the objects.
To compare the contents of the objects — deep equality — we can write a function called
same_coordinates:
1
2
def same_coordinates(p1, p2):
return (p1.x == p2.x) and (p1.y == p2.y)
Now if we create two different objects that contain the same data, we can use same_point
to find out if they represent points with the same coordinates.
>>> p1 = Point(3, 4)
>>> p2 = Point(3, 4)
>>> same_coordinates(p1, p2)
True
Of course, if the two variables refer to the same object, they have both shallow and deep equality.
Beware of ==
“When I use a word,” Humpty Dumpty said, in a rather scornful tone, “it means just what I
choose it to mean — neither more nor less.” Alice in Wonderland
Python has a powerful feature that allows a designer of a class to decide what an operation like
== or < should mean. (We’ve just shown how we can control how our own objects are converted
to strings, so we’ve already made a start!) We’ll cover more detail later. But sometimes the
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implementors will attach shallow equality semantics, and sometimes deep equality, as shown
in this little experiment:
1
2
3
4
p = Point(4, 2)
s = Point(4, 2)
print("== on Points returns", p == s)
# By default, == on Point objects does a shallow equality
˓→test
5
6
7
8
9
a = [2,3]
b = [2,3]
print("== on lists returns", a == b)
# But by default, == does a deep equality test on lists
This outputs:
== on Points returns False
== on lists returns True
So we conclude that even though the two lists (or tuples, etc.) are distinct objects with different
memory addresses, for lists the == operator tests for deep equality, while in the case of points
it makes a shallow test.
Copying
Aliasing can make a program difficult to read because changes made in one place might have
unexpected effects in another place. It is hard to keep track of all the variables that might refer
to a given object.
Copying an object is often an alternative to aliasing. The copy module contains a function
called copy that can duplicate any object:
>>> import copy
>>> p1 = Point(3, 4)
>>> p2 = copy.copy(p1)
>>> p1 is p2
False
>>> same_coordinates(p1, p2)
True
Once we import the copy module, we can use the copy function to make a new Point. p1
and p2 are not the same point, but they contain the same data.
To copy a simple object like a Point, which doesn’t contain any embedded objects, copy is
sufficient. This is called shallow copying.
For something like a Rectangle, which contains a reference to a Point, copy doesn’t do
quite the right thing. It copies the reference to the Point object, so both the old Rectangle
and the new one refer to a single Point.
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If we create a box, b1, in the usual way and then make a copy, b2, using copy, the resulting
state diagram looks like this:
This is almost certainly not what we want. In this case, invoking grow on one of the
Rectangle objects would not affect the other, but invoking move on either would affect
both! This behavior is confusing and error-prone. The shallow copy has created an alias to the
Point that represents the corner.
Fortunately, the copy module contains a function named deepcopy that copies not only the
object but also any embedded objects. It won’t be surprising to learn that this operation is called
a deep copy.
>>> b2 = copy.deepcopy(b1)
Now b1 and b2 are completely separate objects.
Glossary
deep copy To copy the contents of an object as well as any embedded objects, and any objects
embedded in them, and so on; implemented by the deepcopy function in the copy
module.
deep equality Equality of values, or two references that point to objects that have the same
value.
shallow copy To copy the contents of an object, including any references to embedded objects;
implemented by the copy function in the copy module.
shallow equality Equality of references, or two references that point to the same object.
Exercises
1. Add a method area to the Rectangle class that returns the area of any instance:
r = Rectangle(Point(0, 0), 10, 5)
test(r.area() == 50)
2. Write a perimeter method in the Rectangle class so that we can find the perimeter
of any rectangle instance:
r = Rectangle(Point(0, 0), 10, 5)
test(r.perimeter() == 30)
3. Write a flip method in the Rectangle class that swaps the width and the height of
any rectangle instance:
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r = Rectangle(Point(100, 50), 10, 5)
test(r.width == 10 and r.height == 5)
r.flip()
test(r.width == 5 and r.height == 10)
4. Write a new method in the Rectangle class to test if a Point falls within the rectangle. For this exercise, assume that a rectangle at (0,0) with width 10 and height 5 has
open upper bounds on the width and height, i.e. it stretches in the x direction from [0 to
10), where 0 is included but 10 is excluded, and from [0 to 5) in the y direction. So it
does not contain the point (10, 2). These tests should pass:
r = Rectangle(Point(0, 0), 10, 5)
test(r.contains(Point(0, 0)))
test(r.contains(Point(3, 3)))
test(not r.contains(Point(3, 7)))
test(not r.contains(Point(3, 5)))
test(r.contains(Point(3, 4.99999)))
test(not r.contains(Point(-3, -3)))
5. In games, we often put a rectangular “bounding box” around our sprites. (A sprite is an
object that can move about in the game, as we will see shortly.) We can then do collision
detection between, say, bombs and spaceships, by comparing whether their rectangles
overlap anywhere.
Write a function to determine whether two rectangles collide. Hint: this might be quite a
tough exercise! Think carefully about all the cases before you code.
Even more OOP
MyTime
As another example of a user-defined type, we’ll define a class called MyTime that records the
time of day. We’ll provide an __init__ method to ensure that every instance is created with
appropriate attributes and initialization. The class definition looks like this:
1
class MyTime:
2
3
4
˓→
5
6
7
def __init__(self, hrs=0, mins=0, secs=0):
""" Create a MyTime object initialized to hrs, mins,
secs """
self.hours = hrs
self.minutes = mins
self.seconds = secs
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We can instantiate a new MyTime object:
1
tim1 = MyTime(11, 59, 30)
The state diagram for the object looks like this:
We’ll leave it as an exercise for the readers to add a __str__ method so that MyTime objects
can print themselves decently.
Pure functions
In the next few sections, we’ll write two versions of a function called add_time, which
calculates the sum of two MyTime objects. They will demonstrate two kinds of functions:
pure functions and modifiers.
The following is a rough version of add_time:
1
2
3
4
5
6
def add_time(t1, t2):
h = t1.hours + t2.hours
m = t1.minutes + t2.minutes
s = t1.seconds + t2.seconds
sum_t = MyTime(h, m, s)
return sum_t
The function creates a new MyTime object and returns a reference to the new object. This is
called a pure function because it does not modify any of the objects passed to it as parameters
and it has no side effects, such as updating global variables, displaying a value, or getting user
input.
Here is an example of how to use this function. We’ll create two MyTime objects:
current_time, which contains the current time; and bread_time, which contains the
amount of time it takes for a breadmaker to make bread. Then we’ll use add_time to figure
out when the bread will be done.
>>> current_time = MyTime(9, 14, 30)
>>> bread_time = MyTime(3, 35, 0)
>>> done_time = add_time(current_time, bread_time)
>>> print(done_time)
12:49:30
The output of this program is 12:49:30, which is correct. On the other hand, there are cases
where the result is not correct. Can you think of one?
The problem is that this function does not deal with cases where the number of seconds or
minutes adds up to more than sixty. When that happens, we have to carry the extra seconds into
the minutes column or the extra minutes into the hours column.
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Here’s a better version of the function:
1
def add_time(t1, t2):
2
3
4
5
h = t1.hours + t2.hours
m = t1.minutes + t2.minutes
s = t1.seconds + t2.seconds
6
7
8
9
if s >= 60:
s -= 60
m += 1
10
11
12
13
if m >= 60:
m -= 60
h += 1
14
15
16
sum_t = MyTime(h, m, s)
return sum_t
This function is starting to get bigger, and still doesn’t work for all possible cases. Later we
will suggest an alternative approach that yields better code.
Modifiers
There are times when it is useful for a function to modify one or more of the objects it gets as
parameters. Usually, the caller keeps a reference to the objects it passes, so any changes the
function makes are visible to the caller. Functions that work this way are called modifiers.
increment, which adds a given number of seconds to a MyTime object, would be written
most naturally as a modifier. A rough draft of the function looks like this:
1
2
def increment(t, secs):
t.seconds += secs
3
4
5
6
if t.seconds >= 60:
t.seconds -= 60
t.minutes += 1
7
8
9
10
if t.minutes >= 60:
t.minutes -= 60
t.hours += 1
The first line performs the basic operation; the remainder deals with the special cases we saw
before.
Is this function correct? What happens if the parameter seconds is much greater than sixty?
In that case, it is not enough to carry once; we have to keep doing it until seconds is less than
sixty. One solution is to replace the if statements with while statements:
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1
2
def increment(t, seconds):
t.seconds += seconds
3
4
5
6
while t.seconds >= 60:
t.seconds -= 60
t.minutes += 1
7
8
9
10
while t.minutes >= 60:
t.minutes -= 60
t.hours += 1
This function is now correct when seconds is not negative, and when hours does not exceed 23,
but it is not a particularly good solution.
Converting increment to a method
Once again, OOP programmers would prefer to put functions that work with MyTime objects
into the MyTime class, so let’s convert increment to a method. To save space, we will leave
out previously defined methods, but you should keep them in your version:
1
2
class MyTime:
# Previous method definitions here...
3
4
5
def increment(self, seconds):
self.seconds += seconds
6
7
8
9
while self.seconds >= 60:
self.seconds -= 60
self.minutes += 1
10
11
12
13
while self.minutes >= 60:
self.minutes -= 60
self.hours += 1
The transformation is purely mechanical — we move the definition into the class definition and
(optionally) change the name of the first parameter to self, to fit with Python style conventions.
Now we can invoke increment using the syntax for invoking a method.
1
current_time.increment(500)
Again, the object on which the method is invoked gets assigned to the first parameter, self.
The second parameter, seconds gets the value 500.
An “Aha!” insight
Often a high-level insight into the problem can make the programming much easier.
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In this case, the insight is that a MyTime object is really a three-digit number in base 60! The
second component is the ones column, the minute component is the sixties column, and the
hour component is the thirty-six hundreds column.
When we wrote add_time and increment, we were effectively doing addition in base 60,
which is why we had to carry from one column to the next.
This observation suggests another approach to the whole problem — we can convert a MyTime
object into a single number and take advantage of the fact that the computer knows how to do
arithmetic with numbers. The following method is added to the MyTime class to convert any
instance into a corresponding number of seconds:
1
2
class MyTime:
# ...
3
4
5
6
7
8
def to_seconds(self):
""" Return the number of seconds represented
by this instance
"""
return self.hours * 3600 + self.minutes * 60 + self.
˓→seconds
Now, all we need is a way to convert from an integer back to a MyTime object. Supposing we
have tsecs seconds, some integer division and mod operators can do this for us:
1
2
3
4
hrs = tsecs // 3600
leftoversecs = tsecs % 3600
mins = leftoversecs // 60
secs = leftoversecs % 60
You might have to think a bit to convince yourself that this technique to convert from one base
to another is correct.
In OOP we’re really trying to wrap together the data and the operations that apply to it. So
we’d like to have this logic inside the MyTime class. A good solution is to rewrite the class
initializer so that it can cope with initial values of seconds or minutes that are outside the
normalized values. (A normalized time would be something like 3 hours 12 minutes and 20
seconds. The same time, but unnormalized could be 2 hours 70 minutes and 140 seconds.)
Let’s rewrite a more powerful initializer for MyTime:
1
2
class MyTime:
# ...
3
4
5
6
7
8
def __init__(self, hrs=0, mins=0, secs=0):
""" Create a new MyTime object initialized to hrs,
˓→mins, secs.
The values of mins and secs may be outside the
˓→range 0-59,
but the resulting MyTime object will be
˓→normalized.
"""
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9
# Calculate total seconds to represent
totalsecs = hrs*3600 + mins*60 + secs
self.hours = totalsecs // 3600
# Split in h,
10
11
12
˓→
13
14
15
m, s
leftoversecs = totalsecs % 3600
self.minutes = leftoversecs // 60
self.seconds = leftoversecs % 60
Now we can rewrite add_time like this:
1
2
3
def add_time(t1, t2):
secs = t1.to_seconds() + t2.to_seconds()
return MyTime(0, 0, secs)
This version is much shorter than the original, and it is much easier to demonstrate or reason
that it is correct.
Generalization
In some ways, converting from base 60 to base 10 and back is harder than just dealing with
times. Base conversion is more abstract; our intuition for dealing with times is better.
But if we have the insight to treat times as base 60 numbers and make the investment of writing
the conversions, we get a program that is shorter, easier to read and debug, and more reliable.
It is also easier to add features later. For example, imagine subtracting two MyTime objects to
find the duration between them. The naive approach would be to implement subtraction with
borrowing. Using the conversion functions would be easier and more likely to be correct.
Ironically, sometimes making a problem harder (or more general) makes the programming
easier, because there are fewer special cases and fewer opportunities for error.
Specialization versus Generalization
Computer Scientists are generally fond of specializing their types, while mathematicians often
take the opposite approach, and generalize everything.
What do we mean by this?
If we ask a mathematician to solve a problem involving weekdays, days of the century, playing
cards, time, or dominoes, their most likely response is to observe that all these objects can
be represented by integers. Playing cards, for example, can be numbered from 0 to 51. Days
within the century can be numbered. Mathematicians will say “These things are enumerable —
the elements can be uniquely numbered (and we can reverse this numbering to get back to the
original concept). So let’s number them, and confine our thinking to integers. Luckily, we have
powerful techniques and a good understanding of integers, and so our abstractions — the way
we tackle and simplify these problems — is to try to reduce them to problems about integers.”
Computer Scientists tend to do the opposite. We will argue that there are many integer operations that are simply not meaningful for dominoes, or for days of the century. So we’ll often
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define new specialized types, like MyTime, because we can restrict, control, and specialize the
operations that are possible. Object-oriented programming is particularly popular because it
gives us a good way to bundle methods and specialized data into a new type.
Both approaches are powerful problem-solving techniques. Often it may help to try to think
about the problem from both points of view — “What would happen if I tried to reduce everything to very few primitive types?”, versus “What would happen if this thing had its own
specialized type?”
Another example
The after function should compare two times, and tell us whether the first time is strictly
after the second, e.g.
>>> t1 = MyTime(10, 55, 12)
>>> t2 = MyTime(10, 48, 22)
>>> after(t1, t2)
True
# Is t1 after t2?
This is slightly more complicated because it operates on two MyTime objects, not just one.
But we’d prefer to write it as a method anyway — in this case, a method on the first argument:
1
2
class MyTime:
# Previous method definitions here...
3
4
5
6
7
8
9
def after(self, time2):
""" Return True if I am strictly greater than time2
˓→"""
if self.hours > time2.hours:
return True
if self.hours < time2.hours:
return False
10
11
12
13
14
15
16
if self.minutes > time2.minutes:
return True
if self.minutes < time2.minutes:
return False
if self.seconds > time2.seconds:
return True
17
18
return False
We invoke this method on one object and pass the other as an argument:
1
2
if current_time.after(done_time):
print("The bread will be done before it starts!")
We can almost read the invocation like English: If the current time is after the done time, then...
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The logic of the if statements deserve special attention here. Lines 11-18 will only be reached
if the two hour fields are the same. Similarly, the test at line 16 is only executed if both times
have the same hours and the same minutes.
Could we make this easier by using our “Aha!” insight and extra work from earlier, and reducing both times to integers? Yes, with spectacular results!
1
2
class MyTime:
# Previous method definitions here...
3
4
5
6
def after(self, time2):
""" Return True if I am strictly greater than time2
˓→"""
return self.to_seconds() > time2.to_seconds()
This is a great way to code this: if we want to tell if the first time is after the second time, turn
them both into integers and compare the integers.
Operator overloading
Some languages, including Python, make it possible to have different meanings for the same
operator when applied to different types. For example, + in Python means quite different things
for integers and for strings. This feature is called operator overloading.
It is especially useful when programmers can also overload the operators for their own userdefined types.
For example, to override the addition operator +, we can provide a method named __add__:
1
2
class MyTime:
# Previously defined methods here...
3
4
5
def __add__(self, other):
return MyTime(0, 0, self.to_seconds() + other.to_
˓→seconds())
As usual, the first parameter is the object on which the method is invoked. The second parameter is conveniently named other to distinguish it from self. To add two MyTime objects,
we create and return a new MyTime object that contains their sum.
Now, when we apply the + operator to MyTime objects, Python invokes the __add__ method
that we have written:
>>> t1 = MyTime(1, 15, 42)
>>> t2 = MyTime(3, 50, 30)
>>> t3 = t1 + t2
>>> print(t3)
05:06:12
The expression t1 + t2 is equivalent to t1.__add__(t2), but obviously more elegant.
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As an exercise, add a method __sub__(self, other) that overloads the subtraction operator, and try it out.
For the next couple of exercises we’ll go back to the Point class defined in our first chapter about objects, and overload some of its operators. Firstly, adding two points adds their
respective (x, y) coordinates:
1
2
class Point:
# Previously defined methods here...
3
4
5
def __add__(self, other):
return Point(self.x + other.x,
self.y + other.y)
There are several ways to override the behavior of the multiplication operator: by defining a
method named __mul__, or __rmul__, or both.
If the left operand of * is a Point, Python invokes __mul__, which assumes that the other
operand is also a Point. It computes the dot product of the two Points, defined according to
the rules of linear algebra:
1
2
def __mul__(self, other):
return self.x * other.x + self.y * other.y
If the left operand of * is a primitive type and the right operand is a Point, Python invokes
__rmul__, which performs scalar multiplication:
1
2
def __rmul__(self, other):
return Point(other * self.x,
other * self.y)
The result is a new Point whose coordinates are a multiple of the original coordinates. If
other is a type that cannot be multiplied by a floating-point number, then __rmul__ will
yield an error.
This example demonstrates both kinds of multiplication:
>>> p1 = Point(3, 4)
>>> p2 = Point(5, 7)
>>> print(p1 * p2)
43
>>> print(2 * p2)
(10, 14)
What happens if we try to evaluate p2 * 2? Since the first parameter is a Point, Python
invokes __mul__ with 2 as the second argument. Inside __mul__, the program tries to
access the x coordinate of other, which fails because an integer has no attributes:
>>> print(p2 * 2)
AttributeError: 'int' object has no attribute 'x'
Unfortunately, the error message is a bit opaque. This example demonstrates some of the
difficulties of object-oriented programming. Sometimes it is hard enough just to figure out
what code is running.
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Polymorphism
Most of the methods we have written only work for a specific type. When we create a new
object, we write methods that operate on that type.
But there are certain operations that we will want to apply to many types, such as the arithmetic
operations in the previous sections. If many types support the same set of operations, we can
write functions that work on any of those types.
For example, the multadd operation (which is common in linear algebra) takes three parameters; it multiplies the first two and then adds the third. We can write it in Python like this:
1
2
def multadd (x, y, z):
return x * y + z
This function will work for any values of x and y that can be multiplied and for any value of z
that can be added to the product.
We can invoke it with numeric values:
>>> multadd (3, 2, 1)
7
Or with Points:
>>> p1 = Point(3,
>>> p2 = Point(5,
>>> print(multadd
(11, 15)
>>> print(multadd
44
4)
7)
(2, p1, p2))
(p1, p2, 1))
In the first case, the Point is multiplied by a scalar and then added to another Point. In
the second case, the dot product yields a numeric value, so the third parameter also has to be a
numeric value.
A function like this that can take arguments with different types is called polymorphic.
As another example, consider the function front_and_back, which prints a list twice, forward and backward:
1
2
3
4
5
def front_and_back(front):
import copy
back = copy.copy(front)
back.reverse()
print(str(front) + str(back))
Because the reverse method is a modifier, we make a copy of the list before reversing it.
That way, this function doesn’t modify the list it gets as a parameter.
Here’s an example that applies front_and_back to a list:
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>>> my_list = [1, 2, 3, 4]
>>> front_and_back(my_list)
[1, 2, 3, 4][4, 3, 2, 1]
Of course, we intended to apply this function to lists, so it is not surprising that it works. What
would be surprising is if we could apply it to a Point.
To determine whether a function can be applied to a new type, we apply Python’s fundamental
rule of polymorphism, called the duck typing rule: If all of the operations inside the function
can be applied to the type, the function can be applied to the type. The operations in the
front_and_back function include copy, reverse, and print.
Not all programming languages define polymorphism in this way. Look up duck typing, and
see if you can figure out why it has this name.
copy works on any object, and we have already written a __str__ method for Point objects, so all we need is a reverse method in the Point class:
1
2
def reverse(self):
(self.x , self.y) = (self.y, self.x)
Then we can pass Points to front_and_back:
>>> p = Point(3, 4)
>>> front_and_back(p)
(3, 4)(4, 3)
The most interesting polymorphism is the unintentional kind, where we discover that a function
we have already written can be applied to a type for which we never planned.
Glossary
dot product An operation defined in linear algebra that multiplies two Points and yields a
numeric value.
functional programming style A style of program design in which the majority of functions
are pure.
modifier A function or method that changes one or more of the objects it receives as parameters. Most modifier functions are void (do not return a value).
normalized Data is said to be normalized if it fits into some reduced range or set of rules.
We usually normalize our angles to values in the range [0..360). We normalize minutes
and seconds to be values in the range [0..60). And we’d be surprised if the local store
advertised its cold drinks at “One dollar, two hundred and fifty cents”.
operator overloading Extending built-in operators ( +, -, *, >, <, etc.) so that they do
different things for different types of arguments. We’ve seen early in the book how + is
overloaded for numbers and strings, and here we’ve shown how to further overload it for
user-defined types.
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polymorphic A function that can operate on more than one type. Notice the subtle distinction:
overloading has different functions (all with the same name) for different types, whereas
a polymorphic function is a single function that can work for a range of types.
pure function A function that does not modify any of the objects it receives as parameters.
Most pure functions are fruitful rather than void.
scalar multiplication An operation defined in linear algebra that multiplies each of the coordinates of a Point by a numeric value.
Exercises
1. Write a Boolean function between that takes two MyTime objects, t1 and t2, as
arguments, and returns True if the invoking object falls between the two times. Assume
t1 <= t2, and make the test closed at the lower bound and open at the upper bound,
i.e. return True if t1 <= obj < t2.
2. Turn the above function into a method in the MyTime class.
3. Overload the necessary operator(s) so that instead of having to write
if t1.after(t2): ...
we can use the more convenient
if t1 > t2: ...
4. Rewrite increment as a method that uses our “Aha” insight.
5. Create some test cases for the increment method. Consider specifically the case where
the number of seconds to add to the time is negative. Fix up increment so that it
handles this case if it does not do so already. (You may assume that you will never
subtract more seconds than are in the time object.)
6. Can physical time be negative, or must time always move in the forward direction? Some
serious physicists think this is not such a dumb question. See what you can find on the
Internet about this.
Collections of objects
Composition
By now, we have seen several examples of composition. One of the first examples was using a
method invocation as part of an expression. Another example is the nested structure of statements; we can put an if statement within a while loop, within another if statement, and so
on.
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Having seen this pattern, and having learned about lists and objects, we should not be surprised
to learn that we can create lists of objects. We can also create objects that contain lists (as
attributes); we can create lists that contain lists; we can create objects that contain objects; and
so on.
In this chapter and the next, we will look at some examples of these combinations, using Card
objects as an example.
Card objects
If you are not familiar with common playing cards, now would be a good time to get a deck, or
else this chapter might not make much sense. There are fifty-two cards in a deck, each of which
belongs to one of four suits and one of thirteen ranks. The suits are Spades, Hearts, Diamonds,
and Clubs (in descending order in bridge). The ranks are Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack,
Queen, and King. Depending on the game that we are playing, the rank of Ace may be higher
than King or lower than 2. The rank is sometimes called the face-value of the card.
If we want to define a new object to represent a playing card, it is obvious what the attributes
should be: rank and suit. It is not as obvious what type the attributes should be. One
possibility is to use strings containing words like "Spade" for suits and "Queen" for ranks.
One problem with this implementation is that it would not be easy to compare cards to see
which had a higher rank or suit.
An alternative is to use integers to encode the ranks and suits. By encode, we do not mean
what some people think, which is to encrypt or translate into a secret code. What a computer
scientist means by encode is to define a mapping between a sequence of numbers and the items
I want to represent. For example:
Spades
Hearts
Diamonds
Clubs
-->
-->
-->
-->
3
2
1
0
An obvious feature of this mapping is that the suits map to integers in order, so we can compare
suits by comparing integers. The mapping for ranks is fairly obvious; each of the numerical
ranks maps to the corresponding integer, and for face cards:
Jack
Queen
King
-->
-->
-->
11
12
13
The reason we are using mathematical notation for these mappings is that they are not part of
the Python program. They are part of the program design, but they never appear explicitly in
the code. The class definition for the Card type looks like this:
1
2
3
4
class Card:
def __init__(self, suit=0, rank=0):
self.suit = suit
self.rank = rank
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As usual, we provide an initialization method that takes an optional parameter for each attribute.
To create some objects, representing say the 3 of Clubs and the Jack of Diamonds, use these
commands:
1
2
three_of_clubs = Card(0, 3)
card1 = Card(1, 11)
In the first case above, for example, the first argument, 0, represents the suit Clubs.
Save this code for later use ...
In the next chapter we assume that we have save the Cards class, and the upcoming Deck
class in a file called Cards.py.
Class attributes and the __str__ method
In order to print Card objects in a way that people can easily read, we want to map the integer
codes onto words. A natural way to do that is with lists of strings. We assign these lists to class
attributes at the top of the class definition:
1
2
3
4
class Card:
suits = ["Clubs", "Diamonds", "Hearts", "Spades"]
ranks = ["narf", "Ace", "2", "3", "4", "5", "6", "7",
"8", "9", "10", "Jack", "Queen", "King"]
5
6
7
8
def __init__(self, suit=0, rank=0):
self.suit = suit
self.rank = rank
9
10
11
def __str__(self):
return (self.ranks[self.rank] + " of " + self.
˓→suits[self.suit])
A class attribute is defined outside of any method, and it can be accessed from any of the
methods in the class.
Inside __str__, we can use suits and ranks to map the numerical values of suit and
rank to strings. For example, the expression self.suits[self.suit] means use the
attribute suit from the object self as an index into the class attribute named suits, and
select the appropriate string.
The reason for the "narf" in the first element in ranks is to act as a place keeper for the zeroeth element of the list, which will never be used. The only valid ranks are 1 to 13. This wasted
item is not entirely necessary. We could have started at 0, as usual, but it is less confusing to
encode the rank 2 as integer 2, 3 as 3, and so on.
With the methods we have so far, we can create and print cards:
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>>> card1 = Card(1, 11)
>>> print(card1)
Jack of Diamonds
Class attributes like suits are shared by all Card objects. The advantage of this is that we
can use any Card object to access the class attributes:
>>> card2 = Card(1, 3)
>>> print(card2)
3 of Diamonds
>>> print(card2.suits[1])
Diamonds
Because every Card instance references the same class attribute, we have an aliasing situation.
The disadvantage is that if we modify a class attribute, it affects every instance of the class. For
example, if we decide that Jack of Diamonds should really be called Jack of Swirly Whales,
we could do this:
>>> card1.suits[1] = "Swirly Whales"
>>> print(card1)
Jack of Swirly Whales
The problem is that all of the Diamonds just became Swirly Whales:
>>> print(card2)
3 of Swirly Whales
It is usually not a good idea to modify class attributes.
Comparing cards
For primitive types, there are six relational operators ( <, >, ==, etc.) that compare values
and determine when one is greater than, less than, or equal to another. If we want our own
types to be comparable using the syntax of these relational operators, we need to define six
corresponding special methods in our class.
We’d like to start with a single method named cmp that houses the logic of ordering. By
convention, a comparison method takes two parameters, self and other, and returns 1 if the
first object is greater, -1 if the second object is greater, and 0 if they are equal to each other.
Some types are completely ordered, which means that we can compare any two elements and
tell which is bigger. For example, the integers and the floating-point numbers are completely
ordered. Some types are unordered, which means that there is no meaningful way to say that
one element is bigger than another. For example, the fruits are unordered, which is why we
cannot compare apples and oranges, and we cannot meaningfully order a collection of images,
or a collection of cellphones.
Playing cards are partially ordered, which means that sometimes we can compare cards and
sometimes not. For example, we know that the 3 of Clubs is higher than the 2 of Clubs, and
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the 3 of Diamonds is higher than the 3 of Clubs. But which is better, the 3 of Clubs or the 2 of
Diamonds? One has a higher rank, but the other has a higher suit.
In order to make cards comparable, we have to decide which is more important, rank or suit.
To be honest, the choice is arbitrary. For the sake of choosing, we will say that suit is more
important, because a new deck of cards comes sorted with all the Clubs together, followed by
all the Diamonds, and so on.
With that decided, we can write cmp:
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2
3
4
5
6
7
8
9
def cmp(self, other):
# Check the suits
if self.suit > other.suit: return 1
if self.suit < other.suit: return -1
# Suits are the same... check ranks
if self.rank > other.rank: return 1
if self.rank < other.rank: return -1
# Ranks are the same... it's a tie
return 0
In this ordering, Aces appear lower than Deuces (2s).
Now, we can define the six special methods that do the overloading of each of the relational
operators for us:
1
2
def __eq__(self, other):
return self.cmp(other) == 0
3
4
5
def __le__(self, other):
return self.cmp(other) <= 0
6
7
8
def __ge__(self, other):
return self.cmp(other) >= 0
9
10
11
def __gt__(self, other):
return self.cmp(other) > 0
12
13
14
def __lt__(self, other):
return self.cmp(other) < 0
15
16
17
def __ne__(self, other):
return self.cmp(other) != 0
With this machinery in place, the relational operators now work as we’d like them to:
>>> card1
>>> card2
>>> card3
>>> card1
False
>>> card1
True
240
=
=
=
<
Card(1, 11)
Card(1, 3)
Card(1, 11)
card2
== card3
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Decks
Now that we have objects to represent Cards, the next logical step is to define a class to
represent a Deck. Of course, a deck is made up of cards, so each Deck object will contain a
list of cards as an attribute. Many card games will need at least two different decks — a red
deck and a blue deck.
The following is a class definition for the Deck class. The initialization method creates the
attribute cards and generates the standard pack of fifty-two cards:
1
2
3
4
5
6
class Deck:
def __init__(self):
self.cards = []
for suit in range(4):
for rank in range(1, 14):
self.cards.append(Card(suit, rank))
The easiest way to populate the deck is with a nested loop. The outer loop enumerates the suits
from 0 to 3. The inner loop enumerates the ranks from 1 to 13. Since the outer loop iterates
four times, and the inner loop iterates thirteen times, the total number of times the body is
executed is fifty-two (thirteen times four). Each iteration creates a new instance of Card with
the current suit and rank, and appends that card to the cards list.
With this in place, we can instantiate some decks:
1
2
red_deck = Deck()
blue_deck = Deck()
Printing the deck
As usual, when we define a new type we want a method that prints the contents of an instance.
To print a Deck, we traverse the list and print each Card:
1
2
3
4
5
class Deck:
...
def print_deck(self):
for card in self.cards:
print(card)
Here, and from now on, the ellipsis (...) indicates that we have omitted the other methods in
the class.
As an alternative to print_deck, we could write a __str__ method for the Deck class.
The advantage of __str__ is that it is more flexible. Rather than just printing the contents of
the object, it generates a string representation that other parts of the program can manipulate
before printing, or store for later use.
Here is a version of __str__ that returns a string representation of a Deck. To add a bit of
pizzazz, it arranges the cards in a cascade where each card is indented one space more than the
previous card:
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1
2
3
4
5
6
7
class Deck:
...
def __str__(self):
s = ""
for i in range(len(self.cards)):
s = s + " " * i + str(self.cards[i]) + "\n"
return s
This example demonstrates several features. First, instead of traversing self.cards and
assigning each card to a variable, we are using i as a loop variable and an index into the list of
cards.
Second, we are using the string multiplication operator to indent each card by one more space
than the last. The expression " " * i yields a number of spaces equal to the current value of
i.
Third, instead of using the print command to print the cards, we use the str function.
Passing an object as an argument to str is equivalent to invoking the __str__ method on
the object.
Finally, we are using the variable s as an accumulator. Initially, s is the empty string. Each
time through the loop, a new string is generated and concatenated with the old value of s to get
the new value. When the loop ends, s contains the complete string representation of the Deck,
which looks like this:
>>> red_deck = Deck()
>>> print(red_deck)
Ace of Clubs
2 of Clubs
3 of Clubs
4 of Clubs
5 of Clubs
6 of Clubs
7 of Clubs
8 of Clubs
9 of Clubs
10 of Clubs
Jack of Clubs
Queen of Clubs
King of Clubs
Ace of Diamonds
2 of Diamonds
...
And so on. Even though the result appears on 52 lines, it is one long string that contains
newlines.
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Shuffling the deck
If a deck is perfectly shuffled, then any card is equally likely to appear anywhere in the deck,
and any location in the deck is equally likely to contain any card.
To shuffle the deck, we will use the randrange function from the random module. With
two integer arguments, a and b, randrange chooses a random integer in the range a <= x
< b. Since the upper bound is strictly less than b, we can use the length of a list as the second
parameter, and we are guaranteed to get a legal index. For example, if rng has already been
instantiated as a random number source, this expression chooses the index of a random card in
a deck:
1
rng.randrange(0, len(self.cards))
An easy way to shuffle the deck is by traversing the cards and swapping each card with a
randomly chosen one. It is possible that the card will be swapped with itself, but that is fine. In
fact, if we precluded that possibility, the order of the cards would be less than entirely random:
1
2
3
4
5
6
7
8
9
class Deck:
...
def shuffle(self):
import random
rng = random.Random()
# Create a random
˓→generator
num_cards = len(self.cards)
for i in range(num_cards):
j = rng.randrange(i, num_cards)
(self.cards[i], self.cards[j]) = (self.cards[j],
˓→ self.cards[i])
Rather than assume that there are fifty-two cards in the deck, we get the actual length of the list
and store it in num_cards.
For each card in the deck, we choose a random card from among the cards that haven’t been
shuffled yet. Then we swap the current card (i) with the selected card (j). To swap the cards
we use a tuple assignment:
1
(self.cards[i], self.cards[j]) = (self.cards[j], self.
˓→cards[i])
While this is a good shuffling method, a random number generator object also has a shuffle
method that can shuffle elements in a list, in place. So we could rewrite this function to use the
one provided for us:
1
2
3
4
5
6
class Deck:
...
def shuffle(self):
import random
rng = random.Random()
˓→generator
rng.shuffle(self.cards)
˓→method
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# Create a random
# uUse its shuffle
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Removing and dealing cards
Another method that would be useful for the Deck class is remove, which takes a card as a
parameter, removes it, and returns True if the card was in the deck and False otherwise:
1
2
3
4
5
6
7
8
class Deck:
...
def remove(self, card):
if card in self.cards:
self.cards.remove(card)
return True
else:
return False
The in operator returns True if the first operand is in the second. If the first operand is
an object, Python uses the object’s __eq__ method to determine equality with items in the
list. Since the __eq__ we provided in the Card class checks for deep equality, the remove
method checks for deep equality.
To deal cards, we want to remove and return the top card. The list method pop provides a
convenient way to do that:
1
2
3
4
class Deck:
...
def pop(self):
return self.cards.pop()
Actually, pop removes the last card in the list, so we are in effect dealing from the bottom of
the deck.
One more operation that we are likely to want is the Boolean function is_empty, which
returns True if the deck contains no cards:
1
2
3
4
class Deck:
...
def is_empty(self):
return self.cards == []
Glossary
encode To represent one type of value using another type of value by constructing a mapping
between them.
class attribute A variable that is defined inside a class definition but outside any method.
Class attributes are accessible from any method in the class and are shared by all instances
of the class.
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accumulator A variable used in a loop to accumulate a series of values, such as by concatenating them onto a string or adding them to a running sum.
Exercises
1. Modify cmp so that Aces are ranked higher than Kings.
Inheritance
Inheritance
The language feature most often associated with object-oriented programming is inheritance.
Inheritance is the ability to define a new class that is a modified version of an existing class.
The primary advantage of this feature is that you can add new methods to a class without
modifying the existing class. It is called inheritance because the new class inherits all of the
methods of the existing class. Extending this metaphor, the existing class is sometimes called
the parent class. The new class may be called the child class or sometimes subclass.
Inheritance is a powerful feature. Some programs that would be complicated without inheritance can be written concisely and simply with it. Also, inheritance can facilitate code reuse,
since you can customize the behavior of parent classes without having to modify them. In some
cases, the inheritance structure reflects the natural structure of the problem, which makes the
program easier to understand.
On the other hand, inheritance can make programs difficult to read. When a method is invoked,
it is sometimes not clear where to find its definition. The relevant code may be scattered among
several modules. Also, many of the things that can be done using inheritance can be done as
elegantly (or more so) without it. If the natural structure of the problem does not lend itself to
inheritance, this style of programming can do more harm than good.
In this chapter we will demonstrate the use of inheritance as part of a program that plays the
card game Old Maid. One of our goals is to write code that could be reused to implement other
card games.
A hand of cards
For almost any card game, we need to represent a hand of cards. A hand is similar to a deck,
of course. Both are made up of a set of cards, and both require operations like adding and
removing cards. Also, we might like the ability to shuffle both decks and hands.
A hand is also different from a deck. Depending on the game being played, we might want
to perform some operations on hands that don’t make sense for a deck. For example, in poker
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we might classify a hand (straight, flush, etc.) or compare it with another hand. In bridge, we
might want to compute a score for a hand in order to make a bid.
This situation suggests the use of inheritance. If Hand is a subclass of Deck, it will have all
the methods of Deck, and new methods can be added.
We add the code in this chapter to our Cards.py file from the previous chapter. In the class
definition, the name of the parent class appears in parentheses:
1
2
class Hand(Deck):
pass
This statement indicates that the new Hand class inherits from the existing Deck class.
The Hand constructor initializes the attributes for the hand, which are name and cards. The
string name identifies this hand, probably by the name of the player that holds it. The name is
an optional parameter with the empty string as a default value. cards is the list of cards in the
hand, initialized to the empty list:
1
2
3
4
class Hand(Deck):
def __init__(self, name=""):
self.cards = []
self.name = name
For just about any card game, it is necessary to add and remove cards from the deck. Removing
cards is already taken care of, since Hand inherits remove from Deck. But we have to write
add:
1
2
3
4
class Hand(Deck):
...
def add(self, card):
self.cards.append(card)
Again, the ellipsis indicates that we have omitted other methods. The list append method
adds the new card to the end of the list of cards.
Dealing cards
Now that we have a Hand class, we want to deal cards from the Deck into hands. It is not
immediately obvious whether this method should go in the Hand class or in the Deck class,
but since it operates on a single deck and (possibly) several hands, it is more natural to put it in
Deck.
deal should be fairly general, since different games will have different requirements. We may
want to deal out the entire deck at once or add one card to each hand.
deal takes two parameters, a list (or tuple) of hands and the total number of cards to deal. If
there are not enough cards in the deck, the method deals out all of the cards and stops:
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2
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3
4
5
6
7
8
9
10
def deal(self, hands, num_cards=999):
num_hands = len(hands)
for i in range(num_cards):
if self.is_empty():
break
#
˓→cards
card = self.pop()
#
˓→card
hand = hands[i % num_hands] #
˓→next?
hand.add(card)
#
˓→the hand
Break if out of
Take the top
Whose turn is
Add the card to
The second parameter, num_cards, is optional; the default is a large number, which effectively means that all of the cards in the deck will get dealt.
The loop variable i goes from 0 to num_cards-1. Each time through the loop, a card is
removed from the deck using the list method pop, which removes and returns the last item in
the list.
The modulus operator (%) allows us to deal cards in a round robin (one card at a time to each
hand). When i is equal to the number of hands in the list, the expression i % num_hands
wraps around to the beginning of the list (index 0).
Printing a Hand
To print the contents of a hand, we can take advantage of the __str__ method inherited from
Deck. For example:
>>> deck = Deck()
>>> deck.shuffle()
>>> hand = Hand("frank")
>>> deck.deal([hand], 5)
>>> print(hand)
Hand frank contains
2 of Spades
3 of Spades
4 of Spades
Ace of Hearts
9 of Clubs
It’s not a great hand, but it has the makings of a straight flush.
Although it is convenient to inherit the existing methods, there is additional information in
a Hand object we might want to include when we print one. To do that, we can provide a
__str__ method in the Hand class that overrides the one in the Deck class:
1
2
3
class Hand(Deck)
...
def __str__(self):
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4
5
6
7
8
9
s = "Hand " + self.name
if self.is_empty():
s += " is empty\n"
else:
s += " contains\n"
return s + Deck.__str__(self)
Initially, s is a string that identifies the hand. If the hand is empty, the program appends the
words is empty and returns s.
Otherwise, the program appends the word contains and the string representation of the
Deck, computed by invoking the __str__ method in the Deck class on self.
It may seem odd to send self, which refers to the current Hand, to a Deck method, until you
remember that a Hand is a kind of Deck. Hand objects can do everything Deck objects can,
so it is legal to send a Hand to a Deck method.
In general, it is always legal to use an instance of a subclass in place of an instance of a parent
class.
The CardGame class
The CardGame class takes care of some basic chores common to all games, such as creating
the deck and shuffling it:
1
2
3
4
class CardGame:
def __init__(self):
self.deck = Deck()
self.deck.shuffle()
This is the first case we have seen where the initialization method performs a significant computation, beyond initializing attributes.
To implement specific games, we can inherit from CardGame and add features for the new
game. As an example, we’ll write a simulation of Old Maid.
The object of Old Maid is to get rid of cards in your hand. You do this by matching cards by
rank and color. For example, the 4 of Clubs matches the 4 of Spades since both suits are black.
The Jack of Hearts matches the Jack of Diamonds since both are red.
To begin the game, the Queen of Clubs is removed from the deck so that the Queen of Spades
has no match. The fifty-one remaining cards are dealt to the players in a round robin. After the
deal, all players match and discard as many cards as possible.
When no more matches can be made, play begins. In turn, each player picks a card (without
looking) from the closest neighbor to the left who still has cards. If the chosen card matches a
card in the player’s hand, the pair is removed. Otherwise, the card is added to the player’s hand.
Eventually all possible matches are made, leaving only the Queen of Spades in the loser’s hand.
In our computer simulation of the game, the computer plays all hands. Unfortunately, some
nuances of the real game are lost. In a real game, the player with the Old Maid goes to some
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effort to get their neighbor to pick that card, by displaying it a little more prominently, or
perhaps failing to display it more prominently, or even failing to fail to display that card more
prominently. The computer simply picks a neighbor’s card at random.
OldMaidHand class
A hand for playing Old Maid requires some abilities beyond the general abilities of a Hand. We
will define a new class, OldMaidHand, that inherits from Hand and provides an additional
method called remove_matches:
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2
3
4
5
6
7
8
9
10
11
12
13
class OldMaidHand(Hand):
def remove_matches(self):
count = 0
original_cards = self.cards[:]
for card in original_cards:
match = Card(3 - card.suit, card.rank)
if match in self.cards:
self.cards.remove(card)
self.cards.remove(match)
print("Hand {0}: {1} matches {2}"
.format(self.name, card, match))
count += 1
return count
We start by making a copy of the list of cards, so that we can traverse the copy while removing
cards from the original. Since self.cards is modified in the loop, we don’t want to use it
to control the traversal. Python can get quite confused if it is traversing a list that is changing!
For each card in the hand, we figure out what the matching card is and go looking for it. The
match card has the same rank and the other suit of the same color. The expression 3 - card.
suit turns a Club (suit 0) into a Spade (suit 3) and a Diamond (suit 1) into a Heart (suit 2).
You should satisfy yourself that the opposite operations also work. If the match card is also in
the hand, both cards are removed.
The following example demonstrates how to use remove_matches:
>>> game = CardGame()
>>> hand = OldMaidHand("frank")
>>> game.deck.deal([hand], 13)
>>> print(hand)
Hand frank contains
Ace of Spades
2 of Diamonds
7 of Spades
8 of Clubs
6 of Hearts
8 of Spades
7 of Clubs
Queen of Clubs
7 of Diamonds
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5 of Clubs
Jack of Diamonds
10 of Diamonds
10 of Hearts
>>> hand.remove_matches()
Hand frank: 7 of Spades matches 7 of Clubs
Hand frank: 8 of Spades matches 8 of Clubs
Hand frank: 10 of Diamonds matches 10 of Hearts
>>> print(hand)
Hand frank contains
Ace of Spades
2 of Diamonds
6 of Hearts
Queen of Clubs
7 of Diamonds
5 of Clubs
Jack of Diamonds
Notice that there is no __init__ method for the OldMaidHand class. We inherit it from
Hand.
OldMaidGame class
Now we can turn our attention to the game itself. OldMaidGame is a subclass of CardGame
with a new method called play that takes a list of players as a parameter.
Since __init__ is inherited from CardGame, a new OldMaidGame object contains a new
shuffled deck:
1
2
3
4
class OldMaidGame(CardGame):
def play(self, names):
# Remove Queen of Clubs
self.deck.remove(Card(0,12))
5
6
7
8
9
# Make a hand for each player
self.hands = []
for name in names:
self.hands.append(OldMaidHand(name))
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11
12
13
14
# Deal the cards
self.deck.deal(self.hands)
print("---------- Cards have been dealt")
self.print_hands()
15
16
17
18
19
# Remove initial matches
matches = self.remove_all_matches()
print("---------- Matches discarded, play begins")
self.print_hands()
20
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22
23
24
25
26
# Play until all 50 cards are matched
turn = 0
num_hands = len(self.hands)
while matches < 25:
matches += self.play_one_turn(turn)
turn = (turn + 1) % num_hands
27
28
29
print("---------- Game is Over")
self.print_hands()
The writing of print_hands has been left as an exercise.
Some of the steps of the game have been separated into methods. remove_all_matches
traverses the list of hands and invokes remove_matches on each:
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2
3
4
5
6
7
class OldMaidGame(CardGame):
...
def remove_all_matches(self):
count = 0
for hand in self.hands:
count += hand.remove_matches()
return count
count is an accumulator that adds up the number of matches in each hand. When we’ve gone
through every hand, the total is returned (count).
When the total number of matches reaches twenty-five, fifty cards have been removed from the
hands, which means that only one card is left and the game is over.
The variable turn keeps track of which player’s turn it is. It starts at 0 and increases by one
each time; when it reaches num_hands, the modulus operator wraps it back around to 0.
The method play_one_turn takes a parameter that indicates whose turn it is. The return
value is the number of matches made during this turn:
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2
3
4
5
6
7
8
9
10
11
12
class OldMaidGame(CardGame):
...
def play_one_turn(self, i):
if self.hands[i].is_empty():
return 0
neighbor = self.find_neighbor(i)
picked_card = self.hands[neighbor].pop()
self.hands[i].add(picked_card)
print("Hand", self.hands[i].name, "picked", picked_
˓→card)
count = self.hands[i].remove_matches()
self.hands[i].shuffle()
return count
If a player’s hand is empty, that player is out of the game, so he or she does nothing and returns
0.
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Otherwise, a turn consists of finding the first player on the left that has cards, taking one card
from the neighbor, and checking for matches. Before returning, the cards in the hand are
shuffled so that the next player’s choice is random.
The method find_neighbor starts with the player to the immediate left and continues
around the circle until it finds a player that still has cards:
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2
3
4
5
6
7
8
class OldMaidGame(CardGame):
...
def find_neighbor(self, i):
num_hands = len(self.hands)
for next in range(1,num_hands):
neighbor = (i + next) % num_hands
if not self.hands[neighbor].is_empty():
return neighbor
If find_neighbor ever went all the way around the circle without finding cards, it would
return None and cause an error elsewhere in the program. Fortunately, we can prove that that
will never happen (as long as the end of the game is detected correctly).
We have omitted the print_hands method. You can write that one yourself.
The following output is from a truncated form of the game where only the top fifteen cards (tens
and higher) were dealt to three players. With this small deck, play stops after seven matches
instead of twenty-five.
>>> import cards
>>> game = cards.OldMaidGame()
>>> game.play(["Allen","Jeff","Chris"])
---------- Cards have been dealt
Hand Allen contains
King of Hearts
Jack of Clubs
Queen of Spades
King of Spades
10 of Diamonds
Hand Jeff contains
Queen of Hearts
Jack of Spades
Jack of Hearts
King of Diamonds
Queen of Diamonds
Hand Chris contains
Jack of Diamonds
King of Clubs
10 of Spades
10 of Hearts
10 of Clubs
Hand Jeff: Queen of Hearts matches Queen of Diamonds
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Hand Chris: 10 of Spades matches 10 of Clubs
---------- Matches discarded, play begins
Hand Allen contains
King of Hearts
Jack of Clubs
Queen of Spades
King of Spades
10 of Diamonds
Hand Jeff contains
Jack of Spades
Jack of Hearts
King of Diamonds
Hand Chris contains
Jack of Diamonds
King of Clubs
10 of Hearts
Hand Allen picked King of Diamonds
Hand Allen: King of Hearts matches King of Diamonds
Hand Jeff picked 10 of Hearts
Hand Chris picked Jack of Clubs
Hand Allen picked Jack of Hearts
Hand Jeff picked Jack of Diamonds
Hand Chris picked Queen of Spades
Hand Allen picked Jack of Diamonds
Hand Allen: Jack of Hearts matches Jack of Diamonds
Hand Jeff picked King of Clubs
Hand Chris picked King of Spades
Hand Allen picked 10 of Hearts
Hand Allen: 10 of Diamonds matches 10 of Hearts
Hand Jeff picked Queen of Spades
Hand Chris picked Jack of Spades
Hand Chris: Jack of Clubs matches Jack of Spades
Hand Jeff picked King of Spades
Hand Jeff: King of Clubs matches King of Spades
---------- Game is Over
Hand Allen is empty
Hand Jeff contains
Queen of Spades
Hand Chris is empty
So Jeff loses.
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Glossary
inheritance The ability to define a new class that is a modified version of a previously defined
class.
parent class The class from which a child class inherits.
child class A new class created by inheriting from an existing class; also called a subclass.
Exercises
1. Add a method, print_hands, to the OldMaidGame class which traverses self.
hands and prints each hand.
2. Define a new kind of Turtle, TurtleGTX, that comes with some extra features: it can
jump forward a given distance, and it has an odometer that keeps track of how far the
turtle has travelled since it came off the production line. (The parent class has a number
of synonyms like fd, forward, back, backward, and bk: for this exercise, just focus
on putting this functionality into the forward method.) Think carefully about how to
count the distance if the turtle is asked to move forward by a negative amount. (We
would not want to buy a second-hand turtle whose odometer reading was faked because
its previous owner drove it backwards around the block too often. Try this in a car near
you, and see if the car’s odometer counts up or down when you reverse.)
3. After travelling some random distance, your turtle should break down with a flat tyre.
After this happens, raise an exception whenever forward is called. Also provide a
change_tyre method that can fix the flat.
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CHAPTER 12
Exceptions
Catching exceptions
Whenever a runtime error occurs, it creates an exception object. The program stops running
at this point and Python prints out the traceback, which ends with a message describing the
exception that occurred.
For example, dividing by zero creates an exception:
>>> print(55/0)
Traceback (most recent call last):
File "<interactive input>", line 1, in <module>
ZeroDivisionError: integer division or modulo by zero
So does accessing a non-existent list item:
>>> a = []
>>> print(a[5])
Traceback (most recent call last):
File "<interactive input>", line 1, in <module>
IndexError: list index out of range
Or trying to make an item assignment on a tuple:
>>> tup = ("a", "b", "d", "d")
>>> tup[2] = "c"
Traceback (most recent call last):
File "<interactive input>", line 1, in <module>
TypeError: 'tuple' object does not support item assignment
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In each case, the error message on the last line has two parts: the type of error before the colon,
and specifics about the error after the colon.
Sometimes we want to execute an operation that might cause an exception, but we don’t want
the program to stop. We can handle the exception using the try statement to “wrap” a region
of code.
For example, we might prompt the user for the name of a file and then try to open it. If the file
doesn’t exist, we don’t want the program to crash; we want to handle the exception:
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2
3
4
5
filename = input("Enter a file name: ")
try:
f = open(filename, "r")
except FileNotFoundError:
print("There is no file named", filename)
The try statement has four separate clauses—or parts—introduced by the keywords try,
except, else, and finally. All clauses but the try can be omitted.
The interpretor executes the block under the try statement, and monitors for exceptions. If
one occurs, the interpretor moves to the except statement; it executes the expect block if
the exception raised match the exception requested in the except statement. If no exception
occurs, the interpretor skips the block under the except clause. A else block is executed
after the try one, if no exception occurred. A finally block is executed in any case. With
all the statements, a try clause looks like:
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2
3
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7
8
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12
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14
user_input = input('Type a number:')
try:
# Try do do something that could fail.
user_input_as_number = float(user_input)
except ValueError:
# This will be executed if a ``ValueError`` is raised.
print('You did not enter a number.')
else:
# This will be executed if not exception got raised in
˓→the
# ``try`` statement.
print('The square of your number is ', user_input_as_
˓→number**2)
finally:
# This will be executed whether or not an exception is
˓→raised.
print('Thank you')
When using a try clause, you should have as little as possible in the try block. If too many
things happen in that block, you risk handling an unexpected exception.
If the try block can fail if various way, you can handle different exceptions in the same try
clause:
It is also possible not to specify a particular exception in the except statement. In this case,
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any exception will be handled. Such bare except statement should be avoided, though, as
they can easily mask bugs.
Raising our own exceptions
Can our program deliberately cause its own exceptions? If our program detects an error condition, we can raise an exception. Here is an example that gets input from the user and checks
that the number is non-negative:
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2
3
4
5
6
7
def get_age():
age = int(input("Please enter your age: "))
if age < 0:
# Create a new instance of an exception
my_error = ValueError("{0} is not a valid age".
˓→format(age))
raise my_error
return age
Line 5 creates an exception object, in this case, a ValueError object, which encapsulates
specific information about the error. Assume that in this case function A called B which called
C which called D which called get_age. The raise statement on line 6 carries this object
out as a kind of “return value”, and immediately exits from get_age() to its caller D. Then
D again exits to its caller C, and C exits to B and so on, each returning the exception object to
their caller, until it encounters a try ... except that can handle the exception. We call
this “unwinding the call stack”.
ValueError is one of the built-in exception types which most closely matches the kind of
error we want to raise. The complete listing of built-in exceptions can be found at the Built-in
Exceptions section of the Python Library Reference , again by Python’s creator, Guido van
Rossum.
If the function that called get_age (or its caller, or their caller, ...) handles the error, then the
program can carry on running; otherwise, Python prints the traceback and exits:
>>> get_age()
Please enter your age: 42
42
>>> get_age()
Please enter your age: -2
Traceback (most recent call last):
File "<interactive input>", line 1, in <module>
File "learn_exceptions.py", line 4, in get_age
raise ValueError("{0} is not a valid age".format(age))
ValueError: -2 is not a valid age
The error message includes the exception type and the additional information that was provided
when the exception object was first created.
It is often the case that lines 5 and 6 (creating the exception object, then raising the exception)
are combined into a single statement, but there are really two different and independent things
12.2. Raising our own exceptions
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happening, so perhaps it makes sense to keep the two steps separate when we first learn to work
with exceptions. Here we show it all in a single statement:
1
raise ValueError("{0} is not a valid age".format(age))
Revisiting an earlier example
Using exception handling, we can now modify our recursion_depth example from the
previous chapter so that it stops when it reaches the maximum recursion depth allowed:
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2
3
4
5
6
def recursion_depth(number):
print("Recursion depth number", number)
try:
recursion_depth(number + 1)
except:
print("I cannot go any deeper into this wormhole.")
7
8
recursion_depth(0)
Run this version and observe the results.
The finally clause of the try statement
A common programming pattern is to grab a resource of some kind — e.g. we create a window
for turtles to draw on, or we dial up a connection to our internet service provider, or we may
open a file for writing. Then we perform some computation which may raise an exception, or
may work without any problems.
Whatever happens, we want to “clean up” the resources we grabbed — e.g. close the window,
disconnect our dial-up connection, or close the file. The finally clause of the try statement
is the way to do just this. Consider this (somewhat contrived) example:
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2
import turtle
import time
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4
5
6
7
def show_poly():
try:
win = turtle.Screen()
˓→g. a window
tess = turtle.Turtle()
# Grab/create a resource, e.
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10
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12
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# This dialog could be cancelled,
#
or the conversion to int might fail, or n might
˓→be zero.
n = int(input("How many sides do you want in your
˓→polygon?"))
angle = 360 / n
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13
14
15
16
17
18
for i in range(n):
tess.forward(10)
tess.left(angle)
time.sleep(3)
˓→seconds
finally:
win.bye()
# Draw the polygon
# Make program wait a few
# Close the turtle's window
19
20
21
22
show_poly()
show_poly()
show_poly()
In lines 20–22, show_poly is called three times. Each one creates a new window for its
turtle, and draws a polygon with the number of sides input by the user. But what if the user
enters a string that cannot be converted to an int? What if they close the dialog? We’ll
get an exception, but even though we’ve had an exception, we still want to close the turtle’s
window. Lines 17–18 does this for us. Whether we complete the statements in the try clause
successfully or not, the finally block will always be executed.
Notice that the exception is still unhandled — only an except clause can handle an exception,
so our program will still crash. But at least its turtle window will be closed before it crashes!
Glossary
exception An error that occurs at runtime.
handle an exception To prevent an exception from causing our program to crash, by wrapping
the block of code in a try ... except construct.
raise To create a deliberate exception by using the raise statement.
Exercises
1. Write a function named readposint that uses the input dialog to prompt the user
for a positive integer and then checks the input to confirm that it meets the requirements.
It should be able to handle inputs that cannot be converted to int, as well as negative
ints, and edge cases (e.g. when the user closes the dialog, or does not enter anything at
all.)
12.5. Glossary
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Chapter 12. Exceptions
CHAPTER 13
Fitting
Suppose we want to determine the gravitational acceleration. To this end, we could drop an
object from the building and measure how long it takes for the object to reach the ground with
a stopwatch. Newton’s laws predict the following model:
1
ℎ = 2
2
where ℎ is the height from which we dropped the object, and  the time it takes to hit the ground.
So from our one measurement we could now calculate the gravitational acceleration .
We can measure the height very accurately, but since we use a stopwatch to measure time,
that value is a lot less reliable because you might have started and stopped your stopwatch at
the wrong moments. Therefore, the result will not be very accurate. To make the value more
accurate we should repeat the same measurement  times to obtain an average  and use that
instead. We’ll get back to this later.
For now we are more interested in the question: is this model correct? To test this question,
we drop our object from different heights, doing multiple measurements for each height to get
reliable values. The data, obtained by simulation for health and safety reasons, are given in the
following table:
y
10
20
30
40
50
t
1.4
2.1
2.6
3.0
3.3
n
5
3
8
15
30
Since the model predicts a parabola, we want to fit the data to this model to see how good it
works. It might be a bit confusing, but ℎ is our x axis, and  is the y axis.
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We use the symfit package to do our fitting. You can find the installation instructions here.
To fit the data to the model we run the following code:
import numpy as np
from symfit.api import Variable, Parameter, Fit, sqrt
t_data = np.array([1.4, 2.1, 2.6, 3.0, 3.3])
h_data = np.array([10, 20, 30, 40, 50])
# We now define our model
h = Variable()
g = Parameter()
t_model = sqrt(2 * h / g)
fit = Fit(t_model, h_data, t_data)
fit_result = fit.execute()
print(fit_result)
Looking at these results, we see that  = 9.09 ± 0.15 for this dataset. In order to plot this result
alongside the data, we need to calculate values for the model. In the same script, we can do:
# Make an array from 0 to 50 in 1000 steps
h_range = np.linspace(0, 50, 1000)
fit_data = t_model(h=h_range, g=fit_result.params.g)
This gives the model evaluated at all the points in h_range. Making the actual plot is left to
you as an exercise. We see that we can reach the value of g by calling fit_result.params.g, this
returns 9.09.
Let’s think for a second about the implications. The value of  is  = 9.81 in the Netherlands.
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Based on our result, the textbooks should be rewritten because that value is extremely unlikely
to be true given the small standard deviation in our data. It is at this point that we remember
that our data point were not infinitely precise: we took many measurements and averaged
them. This means there is an uncertainty in each of our data points. We will now account for
this additional uncertainty and see what this does to our conclusion. To do this we first have to
describe how the fitting actually works.
How does it work?
In fitting we want to find values for the parameters such that the differences between the model
and the data are as small as possible. The differences (residuals) are easy to calculate:
 ( , ⃗) − 
Here we have written the parameters as a vector ⃗, to indicate that we can have multiple parameters.  and  are the x and y coordinates of the i’th datapoint. However, if we were to
minimize the sum over all these differences we would have a problem, because these differences can be either possitive or negative. This means there’s many ways to add these values
and get zero out of the sum. We therefore take the sum over the residuals squared:
2
 =

∑︁
( ( , ⃗) −  )2
=1
Now if we minimize 2 , we get the best possible values for our parameters. The fitting algorithm actually just takes some values for the parameters, calculates 2 , then changes the values
slightly by adding or subtracting a number, and checks if this new value is smaller than the old
one. If this is true it keeps going in the same direction until the value of 2 starts to increase.
That’s when you know you’ve hit a minimum. Of cource the trick is to do this smartly, and a
lot of algorithms have been developed in order to do this.
Propagating Uncertanties
In the example above we the fitting process assumed that every measurement was equally reliable. But this is not true. By repeating a measurement and averaging the result, we can improve
the accuracy. So in our example, we dropped our object from every height a couple of times
and took the average. Therefore, we want to assign a weight depending on how accurate the
average value for that height is. Statistically the weight  to use is  = 12 , where  is the

standard deviation for each point.
Our sum to minimize now changes to:
13.1. How does it work?
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2
 =

∑︁
2
 ( ( , ⃗) −  ) =

∑︁
( ( , ⃗) −  )2
=1
=1
2
But how do we know the standard deviation in the mean value we calculate for every height?
Suppose the standard deviation of our stopwatch is ℎ = 0.2. If we do  measurements
from the same height, the average time is found by calculating

1 ∑︁

¯ =
 =1
It can be shown that the standard deviation of the mean is now:
¯ =
ℎ
√

So we see that by increasing the amount of measurements, we can decrease the uncertainty in
¯. Our simulated data now changes to:
y
10
20
30
40
50
t
1.4
2.1
2.6
3.0
3.3
n
5
3
8
15
30

0.089
0.115
0.071
0.052
0.037
The values of  have been calculated by using the above formula. Let’s fit to this new data set
using symfit. Notice that there are some small differences to the code:
import numpy as np
from symfit.api import Variable, Parameter, Fit, sqrt
t_data = np.array([1.4, 2.1, 2.6, 3.0, 3.3])
h_data = np.array([10, 20, 30, 40, 50])
n = np.array([5, 3, 8, 15, 30])
sigma = 0.2
sigma_t = sigma / np.sqrt(n)
# We now define our model
h = Variable()
t = Variable()
g = Parameter()
t_model = {t: sqrt(2 * h / g)}
fit = Fit(t_model, h=h_data, t=t_data, sigma_t=sigma_t)
fit_result = fit.execute()
print(fit_result)
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Note: Named Models
Looking at the definition of t_model, we see it is now a dict. This has been done so
we can tell symfit which of our variables are uncertain by the name of the variable,
in this case t has an uncertainty sigma_t.
Including these uncertainties in the fit yields  = 9.10 ± 0.16. The accepted value of  = 9.81
is well outside the uncertainty in this data. Therefore the textbooks must be rewriten!
This example shows the importance of propagating your errors consistently. (And of the importance of performing the actual measurement as the author of a chapter on error propagation
so you don’t end up claiming the textbooks have to rewritten.)
More on symfit
There are a lot more features in symfit to help you on your quest to fitting the universe. You can
find the tutorial there.
It is recommended you read this as well before starting to fit your own data.
13.3. More on symfit
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CHAPTER 14
PyGame
PyGame is a package that is not part of the standard Python distribution, so if you do not already
have it installed (i.e. import pygame fails), download and install a suitable version from
http://pygame.org/download.shtml. These notes are based on PyGame 1.9.1, the most recent
version at the time of writing.
PyGame comes with a substantial set of tutorials, examples, and help, so there is ample opportunity to stretch yourself on the code. You may need to look around a bit to find these resources,
though: if you’ve installed PyGame on a Windows machine, for example, they’ll end up in a
folder like C:\Python31\Lib\site-packages\pygame\ where you will find directories for docs and
examples.
The game loop
The structure of the games we’ll consider always follows this fixed pattern:
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In every game, in the setup section we’ll create a window, load and prepare some content, and
then enter the game loop. The game loop continuously does four main things:
• it polls for events — i.e. asks the system whether events have occurred — and responds
appropriately,
• it updates whatever internal data structures or objects need changing,
• it draws the current state of the game into a (non-visible) surface,
• it puts the just-drawn surface on display.
1
import pygame
2
3
4
5
6
def main():
""" Set up the game and run the main game loop """
pygame.init()
# Prepare the pygame module for use
surface_size = 480
# Desired physical surface size,
˓→in pixels.
7
8
9
# Create surface of (width, height), and its window.
main_surface = pygame.display.set_mode((surface_size,
˓→surface_size))
10
11
# Set up some data to describe a small rectangle and
its color
small_rect = (300, 200, 150, 90)
some_color = (255, 0, 0)
# A color is a mix of
˓→(Red, Green, Blue)
˓→
12
13
14
15
16
268
while True:
event = pygame.event.poll()
# Look for any event
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17
˓→
18
if event.type == pygame.QUIT:
button clicked?
break
#
# Window close
... leave game loop
19
# Update your game objects and data structures here.
20
˓→
..
21
# We draw everything from scratch on each frame.
# So first fill everything with the background
22
23
˓→
color
main_surface.fill((0, 200, 255))
24
25
# Overpaint a smaller rectangle on the main surface
main_surface.fill(some_color, small_rect)
26
27
28
# Now the surface is ready, tell pygame to display
29
˓→
it!
pygame.display.flip()
30
31
32
pygame.quit()
˓→window.
# Once we leave the loop, close the
33
34
main()
This program pops up a window which stays there until we close it:
PyGame does all its drawing onto rectangular surfaces. After initializing PyGame at line 5, we
create a window holding our main surface. The main loop of the game extends from line 15 to
30, with the following key bits of logic:
• First (line 16) we poll to fetch the next event that might be ready for us. This step will
always be followed by some conditional statements that will determine whether any event
that we’re interested in has happened. Polling for the event consumes it, as far as PyGame
is concerned, so we only get one chance to fetch and use each event. On line 17 we test
whether the type of the event is the predefined constant called pygame.QUIT. This is the
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event that we’ll see when the user clicks the close button on the PyGame window. In
response to this event, we leave the loop.
• Once we’ve left the loop, the code at line 32 closes window, and we’ll return from function main. Your program could go on to do other things, or reinitialize pygame and
create another window, but it will usually just end too.
• There are different kinds of events — key presses, mouse motion, mouse clicks, joystick
movement, and so on. It is usual that we test and handle all these cases with new code
squeezed in before line 19. The general idea is “handle events first, then worry about the
other stuff”.
• At line 20 we’d update objects or data — for example, if we wanted to vary the color,
position, or size of the rectangle we’re about to draw, we’d re-assign some_color, and
small_rect here.
• A modern way to write games (now that we have fast computers and fast graphics cards)
is to redraw everything from scratch on every iteration of the game loop. So the first thing
we do at line 24 is fill the entire surface with a background color. The fill method of a
surface takes two arguments — the color to use for filling, and the rectangle to be filled.
But the second argument is optional, and if it is left out the entire surface is filled.
• In line 27 we fill a second rectangle, this time using some_color. The placement and
size of the rectangle are given by the tuple small_rect, a 4-element tuple (x, y,
width, height).
• It is important to understand that the origin of the PyGame’s surface is at the top left
corner (unlike the turtle module that puts its origin in the middle of the screen). So,
if you wanted the rectangle closer to the top of the window, you need to make its y
coordinate smaller.
• If your graphics display hardware tries to read from memory at the same time as the
program is writing to that memory, they will interfere with each other, causing video
noise and flicker. To get around this, PyGame keeps two buffers in the main surface —
the back buffer that the program draws to, while the front buffer is being shown to the
user. Each time the program has fully prepared its back buffer, it flips the back/front role
of the two buffers. So the drawing on lines 24 and 27 does does not change what is seen
on the screen until we flip the buffers, on line 30.
Displaying images and text
To draw an image on the main surface, we load the image, say a beach ball, into its own new
surface. The main surface has a blit method that copies pixels from the beach ball surface
into its own surface. When we call blit, we can specify where the beach ball should be placed
on the main surface. The term blit is widely used in computer graphics, and means to make a
fast copy of pixels from one area of memory to another.
So in the setup section, before we enter the game loop, we’d load the image, like this:
1
270
ball = pygame.image.load("ball.png")
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and after line 28 in the program above, we’d add this code to display our image at position
(100,120):
1
main_surface.blit(ball, (100, 120))
To display text, we need do do three things. Before we enter the game loop, we instantiate a
font object:
1
2
# Instantiate 16 point Courier font to draw text.
my_font = pygame.font.SysFont("Courier", 16)
and after line 28, again, we use the font’s render method to create a new surface containing
the pixels of the drawn text, and then, as in the case for images, we blit our new surface onto the
main surface. Notice that render takes two extra parameters — the second tells it whether to
carefully smooth edges of the text while drawing (this process is called anti-aliasing), and the
second is the color that we want the text text be. Here we’ve used (0,0,0) which is black:
1
2
the_text = my_font.render("Hello, world!", True, (0,0,0))
main_surface.blit(the_text, (10, 10))
We’ll demonstrate these two new features by counting the frames — the iterations of the game
loop — and keeping some timing information. On each frame, we’ll display the frame count,
and the frame rate. We will only update the frame rate after every 500 frames, when we’ll look
at the timing interval and can do the calculations.
1
2
import pygame
import time
3
4
def main():
5
pygame.init()
# Prepare the PyGame module for use
main_surface = pygame.display.set_mode((480, 240))
6
7
8
# Load an image to draw. Substitute your own.
# PyGame handles gif, jpg, png, etc. image types.
ball = pygame.image.load("ball.png")
9
10
11
12
# Create a font for rendering text
my_font = pygame.font.SysFont("Courier", 16)
13
14
15
frame_count = 0
frame_rate = 0
t0 = time.clock()
16
17
18
19
while True:
20
21
# Look for an event from keyboard, mouse, joystick,
22
˓→
23
24
etc.
ev = pygame.event.poll()
if ev.type == pygame.QUIT:
˓→clicked?
14.2. Displaying images and text
# Window close button
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break
25
# Leave game loop
26
# Do other bits of logic for the game here
frame_count += 1
if frame_count % 500 == 0:
t1 = time.clock()
frame_rate = 500 / (t1-t0)
t0 = t1
27
28
29
30
31
32
33
34
˓→
35
# Completely redraw the surface, starting with
background
main_surface.fill((0, 200, 255))
36
# Put a red rectangle somewhere on the surface
main_surface.fill((255,0,0), (300, 100, 150, 90))
37
38
39
# Copy our image to the surface, at this (x,y) posn
main_surface.blit(ball, (100, 120))
40
41
42
43
44
45
46
47
# Make a new surface with an image of the text
the_text = my_font.render("Frame = {0}, rate = {1:.
˓→2f} fps"
.format(frame_count, frame_rate), True,
˓→(0,0,0))
# Copy the text surface to the main surface
main_surface.blit(the_text, (10, 10))
48
# Now that everything is drawn, put it on display!
pygame.display.flip()
49
50
51
52
pygame.quit()
53
54
55
main()
The frame rate is close to ridiculous — a lot faster than one’s eye can process frames. (Commercial video games usually plan their action for 60 frames per second (fps).) Of course, our
rate will drop once we start doing something a little more strenuous inside our game loop.
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Drawing a board for the N queens puzzle
We previously solved our N queens puzzle. For the 8x8 board, one of the solutions was the
list [6,4,2,0,5,7,1,3]. Let’s use that solution as testdata, and now use PyGame to draw
that chessboard with its queens.
We’ll create a new module for the drawing code, called draw_queens.py. When we have
our test case(s) working, we can go back to our solver, import this new module, and add a call
to our new function to draw a board each time a solution is discovered.
We begin with a background of black and red squares for the board. Perhaps we could create an
image that we could load and draw, but that approach would need different background images
for different size boards. Just drawing our own red and black rectangles of the appropriate size
sounds like much more fun!
1
2
def draw_board(the_board):
""" Draw a chess board with queens, from the_board. """
3
4
5
pygame.init()
colors = [(255,0,0), (0,0,0)]
˓→black]
# Set up colors [red,
6
7
8
9
10
n = len(the_board)
# This is an NxN chess board.
surface_size = 480
# Proposed physical
˓→surface size.
square_size = surface_size // n
# sq_sz is length of
˓→a square.
surface_size = n * square_size
# Adjust to exactly
˓→fit n squares.
11
12
13
# Create the surface of (width, height), and its window.
surface = pygame.display.set_mode((surface_size,
˓→surface_size))
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Here we precompute square_size, the integer size that each square will be, so that we can
fit the squares nicely into the available window. So if we’d like the board to be 480x480, and
we’re drawing an 8x8 chessboard, then each square will need to have a size of 60 units. But we
notice that a 7x7 board cannot fit nicely into 480 — we’re going to get some ugly border that
our squares don’t fill exactly. So we recompute the surface size to exactly fit our squares before
we create the window.
Now let’s draw the squares, in the game loop. We’ll need a nested loop: the outer loop will run
over the rows of the chessboard, the inner loop over the columns:
1
2
3
4
5
6
7
8
# Draw a fresh background (a blank chess board)
for row in range(n):
# Draw each row of the board.
color_index = row % 2
# Change starting color
˓→on each row
for col in range(n):
# Run through cols drawing
˓→squares
the_square = (col*square_size, row*square_size,
˓→square_size, square_size)
surface.fill(colors[color_index], the_square)
# now flip the color index for the next square
c_index = (c_index + 1) % 2
There are two important ideas in this code: firstly, we compute the rectangle to be filled from the
row and col loop variables, multiplying them by the size of the square to get their position.
And, of course, each square is a fixed width and height. So the_square represents the
rectangle to be filled on the current iteration of the loop. The second idea is that we have to
alternate colors on every square. In the earlier setup code we created a list containing two
colors, here we manipulate color_index (which will always either have the value 0 or 1) to
start each row on a color that is different from the previous row’s starting color, and to switch
colors each time a square is filled.
This (together with the other fragments not shown to flip the surface onto the display) leads to
the pleasing backgrounds like this, for different size boards:
Now, on to drawing the queens! Recall that our solution [6,4,2,0,5,7,1,3] means that
in column 0 of the board we want a queen at row 6, at column 1 we want a queen at row 4, and
so on. So we need a loop running over each queen:
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1
2
for (col, row) in enumerate(the_board):
# draw a queen at col, row...
In this chapter we already have a beach ball image, so we’ll use that for our queens. In the
setup code before our game loop, we load the ball image (as we did before), and in the body of
the loop, we add the line:
1
surface.blit(ball, (col * square_size, row * square_size))
We’re getting there, but those queens need to be centred in their squares! Our problem arises
from the fact that both the ball and the rectangle have their upper left corner as their reference
points. If we’re going to centre this ball in the square, we need to give it an extra offset in
both the x and y direction. (Since the ball is round and the square is square, the offset in the
two directions will be the same, so we’ll just compute a single offset value, and use it in both
directions.)
The offset we need is half the (size of the square less the size of the ball). So we’ll precompute
this in the game’s setup section, after we’ve loaded the ball and determined the square size:
1
ball_offset = (square_size - ball.get_width()) // 2
Now we touch up the drawing code for the ball and we’re done:
1
surface.blit(ball, (col * square_size + ball_offset, row *
˓→square_size + ball_offset))
We might just want to think about what would happen if the ball was bigger than the square. In
that case, ball_offset would become negative. So it would still be centered in the square
- it would just spill over the boundaries, or perhaps obscure the square entirely!
Here is the complete program:
1
import pygame
2
3
4
def draw_board(the_board):
""" Draw a chess board with queens, as determined by
˓→the the_board. """
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5
6
7
pygame.init()
colors = [(255,0,0), (0,0,0)]
˓→black]
# Set up colors [red,
8
9
10
11
12
n = len(the_board)
# This is an NxN chess board.
surface_size = 480
# Proposed physical
˓→surface size.
square_size = surface_size // n
# sq_sz is length of
˓→a square.
surface_size = n * square_size
# Adjust to exactly
˓→fit n squares.
13
14
15
# Create the surface of (width, height), and its window.
surface = pygame.display.set_mode((surface_size,
˓→surface_size))
16
17
ball = pygame.image.load("ball.png")
18
19
20
21
22
# Use an extra offset to centre the ball in its square.
# If the square is too small, offset becomes negative,
#
but it will still be centered :-)
ball_offset = (square_size-ball.get_width()) // 2
23
24
while True:
25
26
27
28
29
# Look for an event from keyboard, mouse, etc.
event = pygame.event.poll()
if event.type == pygame.QUIT:
break;
30
31
32
33
34
35
36
37
38
# Draw a fresh background (a blank chess board)
for row in range(n):
# Draw each row of
˓→the board.
color_index = row % 2
# Alternate
˓→starting color
for col in range(n):
# Run through cols
˓→drawing squares
the_square = (col*square_size, row*square_
˓→size, square_size, square_size)
surface.fill(colors[color_index], the_
˓→square)
# Now flip the color index for the next
˓→square
color_index = (color_index + 1) % 2
39
40
41
42
43
276
# Now that squares are drawn, draw the queens.
for (col, row) in enumerate(the_board):
surface.blit(ball,
(col*square_size+ball_offset,row*square_
˓→size+ball_offset))
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44
pygame.display.flip()
45
46
47
48
pygame.quit()
49
50
51
52
53
54
if __name__ == "__main__":
draw_board([0, 5, 3, 1, 6, 4, 2])
# 7 x 7 to test
˓→window size
draw_board([6, 4, 2, 0, 5, 7, 1, 3])
draw_board([9, 6, 0, 3, 10, 7, 2, 4, 12, 8, 11, 5, 1])
˓→# 13 x 13
draw_board([11, 4, 8, 12, 2, 7, 3, 15, 0, 14, 10, 6, 13,
˓→ 1, 5, 9])
There is one more thing worth reviewing here. The conditional statement on line 50 tests
whether the name of the currently executing program is __main__. This allows us to distinguish whether this module is being run as a main program, or whether it has been imported
elsewhere, and used as a module. If we run this module in Python, the test cases in lines 51-54
will be executed. However, if we import this module into another program (i.e. our N queens
solver from earlier) the condition at line 50 will be false, and the statements on lines 51-54
won’t run.
Previously, our main program looked like this:
1
def main():
2
3
4
5
6
7
8
9
10
11
12
board = list(range(8))
# Generate the initial
˓→permutation
num_found = 0
tries = 0
while num_found < 10:
random.shuffle(bd)
tries += 1
if not has_clashes(bd):
print("Found solution {0} in {1} tries.".
˓→format(board, tries))
tries = 0
num_found += 1
13
14
main()
Now we just need two changes. At the top of that program, we import the module that we’ve
been working on here (assume we called it draw_queens). (You’ll have to ensure that the
two modules are saved in the same folder.) Then after line 10 here we add a call to draw the
solution that we’ve just discovered:
draw_queens.draw_board(bd)
And that gives a very satisfying combination of program that can search for solutions to the N
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queens problem, and when it finds each, it pops up the board showing the solution.
Sprites
A sprite is an object that can move about in a game, and has internal behaviour and state of its
own. For example, a spaceship would be a sprite, the player would be a sprite, and bullets and
bombs would all be sprites.
Object oriented programming (OOP) is ideally suited to a situation like this: each object can
have its own attributes and internal state, and a couple of methods. Let’s have some fun with
our N queens board. Instead of placing the queen in her final position, we’d like to drop her in
from the top of the board, and let her fall into position, perhaps bouncing along the way.
The first encapsulation we need is to turn each of our queens into an object. We’ll keep a list of
all the active sprites (i.e. a list of queen objects), and arrange two new things in our game loop:
• After handling events, but before drawing, call an update method on every sprite. This
will give each sprite a chance to modify its internal state in some way — perhaps change
its image, or change its position, or rotate itself, or make itself grow a bit bigger or a bit
smaller.
• Once all the sprites have updated themselves, the game loop can begin drawing - first the
background, and then call a draw method on each sprite in turn, and delegate (hand off)
the task of drawing to the object itself. This is in line with the OOP idea that we don’t
say “Hey, draw, show this queen!”, but we prefer to say “Hey, queen, draw yourself!”.
We start with a simple object, no movement or animation yet, just scaffolding, to see how to fit
all the pieces together:
1
class QueenSprite:
2
3
4
5
6
7
8
9
def __init__(self, img, target_posn):
""" Create and initialize a queen for this
target position on the board
"""
self.image = img
self.target_posn = target_positionn
self.position = target_position
10
11
12
def update(self):
return
# Do nothing for the moment.
13
14
15
def draw(self, target_surface):
target_surface.blit(self.image, self.position)
We’ve given the sprite three attributes: an image to be drawn, a target position, and a current
position. If we’re going to move the spite about, the current position may need to be different
from the target, which is where we want the queen finally to end up. In this code at this time
we’ve done nothing in the update method, and our draw method (which can probably remain
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this simple in future) simply draws itself at its current position on the surface that is provided
by the caller.
With its class definition in place, we now instantiate our N queens, put them into a list of sprites,
and arrange for the game loop to call the update and draw methods on each frame. The new
bits of code, and the revised game loop look like this:
1
˓→
all_sprites = []
the game
# Keep a list of all sprites in
2
3
# Create a sprite object for each queen, and populate
our list.
for (col, row) in enumerate(the_board):
a_queen = QueenSprite(ball,
(col*square_size+ball_offset, row*square_
˓→size+ball_offset))
all_sprites.append(a_queen)
˓→
4
5
6
7
8
9
10
11
12
13
while True:
# Look for an event from keyboard, mouse, etc.
event = pygame.event.poll()
if event.type == pygame.QUIT:
break;
14
15
16
17
# Ask every sprite to update itself.
for sprite in all_sprites:
sprite.update()
18
19
20
# Draw a fresh background (a blank chess board)
# ... same as before ...
21
22
23
24
# Ask every sprite to draw itself.
for sprite in all_sprites:
sprite.draw(surface)
25
26
pygame.display.flip()
This works just like it did before, but our extra work in making objects for the queens has
prepared the way for some more ambitious extensions.
Let us begin with a falling queen object. At any instant, it will have a velocity i.e. a speed,
in a certain direction. (We are only working with movement in the y direction, but use your
imagination!) So in the object’s update method, we want to change its current position by its
velocity. If our N queens board is floating in space, velocity would stay constant, but hey, here
on Earth we have gravity! Gravity changes the velocity on each time interval, so we’ll want
a ball that speeds up as it falls further. Gravity will be constant for all queens, so we won’t
keep it in the instances — we’ll just make it a variable in our module. We’ll make one other
change too: we will start every queen at the top of the board, so that it can fall towards its target
position. With these changes, we now get the following:
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1
gravity = 0.0001
2
3
class QueenSprite:
4
5
6
7
8
9
10
def __init__(self, img, target_posn):
self.image = img
self.target_position = target_position
(x, y) = target_position
self.position = (x, 0)
# Start ball at top of
˓→its column
self.y_velocity = 0
#
with zero initial
˓→velocity
11
12
13
14
15
16
def update(self):
self.y_velocity += gravity
˓→velocity
(x, y) = self.position
new_y_pos = y + self.y_velocity
˓→the ball
self.position = (x, new_y_pos)
˓→new position.
# Gravity changes
# Velocity moves
#
to this
17
18
19
def draw(self, target_surface):
# Same as before.
target_surface.blit(self.image, self.position)
Making these changes gives us a new chessboard in which each queen starts at the top of its
column, and speeds up, until it drops off the bottom of the board and disappears forever. A
good start — we have movement!
The next step is to get the ball to bounce when it reaches its own target position. It is pretty easy
to bounce something — you just change the sign of its velocity, and it will move at the same
speed in the opposite direction. Of course, if it is travelling up towards the top of the board it
will be slowed down by gravity. (Gravity always sucks down!) And you’ll find it bounces all
the way up to where it began from, reaches zero velocity, and starts falling all over again. So
we’ll have bouncing balls that never settle.
A realistic way to settle the object is to lose some energy (probably to friction) each time it
bounces — so instead of simply reversing the sign of the velocity, we multiply it by some
fractional factor — say -0.65. This means the ball only retains 65% of its energy on each
bounce, so it will, as in real life, stop bouncing after a short while, and settle on its “ground”.
The only changes are in the update method, which now looks like this:
1
2
3
4
5
6
280
def update(self):
self.y_velocity += gravity
(x, y) = self.postion
new_y_pos = y + self.y_velocity
(target_x, target_y) = self.target_posn
˓→position
dist_to_go = target_y - new_y_pos
˓→our floor?
# Unpack the
# How far to
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7
8
9
10
if dist_to_go < 0:
# Are we
˓→under floor?
self.y_velocity = -0.65 * self.y_velocity
#
˓→Bounce
new_y_pos = target_y + dist_to_go
# Move back
˓→above floor
11
12
self.position = (x, new_y_pos)
˓→new position.
# Set our
Heh, heh, heh! We’re not going to show animated screenshots, so copy the code into your
Python environment and see for yourself.
Events
The only kind of event we’re handled so far has been the QUIT event. But we can also detect
keydown and keyup events, mouse motion, and mousebutton down or up events. Consult the
PyGame documentation and follow the link to Event.
When your program polls for and receives an event object from PyGame, its event type will
determine what secondary information is available. Each event object carries a dictionary
(which you may only cover in due course in these notes). The dictionary holds certain keys and
values that make sense for the type of event.
For example, if the type of event is MOUSEMOTION, we’ll be able to find the mouse position
and information about the state of the mouse buttons in the dictionary attached to the event.
Similarly, if the event is KEYDOWN, we can learn from the dictionary which key went down,
and whether any modifier keys (shift, control, alt, etc.) are also down. You also get events when
the game window becomes active (i.e. gets focus) or loses focus.
The event object with type NOEVENT is returned if there are no events waiting. Events can be
printed, allowing you to experiment and play around. So dropping these lines of code into the
game loop directly after polling for any event is quite informative:
1
2
if event.type != pygame.NOEVENT:
˓→interesting!
print(event)
# Only print if it is
With this is place, hit the space bar and the escape key, and watch the events you get. Click
your three mouse buttons. Move your mouse over the window. (This causes a vast cascade of
events, so you may also need to filter those out of the printing.) You’ll get output that looks
something like this:
<Event(17-VideoExpose {})>
<Event(1-ActiveEvent {'state': 1, 'gain': 0})>
<Event(2-KeyDown {'scancode': 57, 'key': 32, 'unicode': ' ', 'mod':
˓→0})>
<Event(3-KeyUp {'scancode': 57, 'key': 32, 'mod': 0})>
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<Event(2-KeyDown {'scancode': 1, 'key': 27, 'unicode': '\x1b',
˓→': 0})>
<Event(3-KeyUp {'scancode': 1, 'key': 27, 'mod': 0})>
...
<Event(4-MouseMotion {'buttons': (0, 0, 0), 'pos': (323, 194),
˓→': (-3, -1)})>
<Event(4-MouseMotion {'buttons': (0, 0, 0), 'pos': (322, 193),
˓→': (-1, -1)})>
<Event(4-MouseMotion {'buttons': (0, 0, 0), 'pos': (321, 192),
˓→': (-1, -1)})>
<Event(4-MouseMotion {'buttons': (0, 0, 0), 'pos': (319, 192),
˓→': (-2, 0)})>
<Event(5-MouseButtonDown {'button': 1, 'pos': (319, 192)})>
<Event(6-MouseButtonUp {'button': 1, 'pos': (319, 192)})>
<Event(4-MouseMotion {'buttons': (0, 0, 0), 'pos': (319, 191),
˓→': (0, -1)})>
<Event(5-MouseButtonDown {'button': 2, 'pos': (319, 191)})>
<Event(5-MouseButtonDown {'button': 5, 'pos': (319, 191)})>
<Event(6-MouseButtonUp {'button': 5, 'pos': (319, 191)})>
<Event(6-MouseButtonUp {'button': 2, 'pos': (319, 191)})>
<Event(5-MouseButtonDown {'button': 3, 'pos': (319, 191)})>
<Event(6-MouseButtonUp {'button': 3, 'pos': (319, 191)})>
...
<Event(1-ActiveEvent {'state': 1, 'gain': 0})>
<Event(12-Quit {})>
'mod
'rel
'rel
'rel
'rel
'rel
So let us now make these changes to the code near the top of our game loop:
1
while True:
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
# Look for an event from keyboard,
ev = pygame.event.poll()
if event.type == pygame.QUIT:
break;
if event.type == pygame.KEYDOWN:
key = ev.dict["key"]
if key == 27:
break
˓→loop.
if key == ord("r"):
colors[0] = (255, 0, 0)
˓→black.
elif key == ord("g"):
colors[0] = (0, 255, 0)
˓→black.
elif key == ord("b"):
colors[0] = (0, 0, 255)
˓→black.
mouse, etc.
# On Escape key ...
#
leave the game
# Change to red +
# Change to green +
# Change to blue +
17
18
˓→
282
if event.type == pygame.MOUSEBUTTONDOWN: # Mouse gone
down?
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19
˓→
20
posn_of_click = event.dict["pos"]
# Get the
coordinates.
print(posn_of_click)
# Just print them.
Lines 7-16 show typical processing for a KEYDOWN event — if a key has gone down, we
test which key it is, and take some action. With this in place, we have another way to quit our
queens program — by hitting the escape key. Also, we can use keys to change the color of the
board that is drawn.
Finally, at line 20, we respond (pretty lamely) to the mouse button going down.
As a final exercise in this section, we’ll write a better response handler to mouse clicks. What
we will do is figure out if the user has clicked the mouse on one of our sprites. If there is a sprite
under the mouse when the click occurs, we’ll send the click to the sprite and let it respond in
some sensible way.
We’ll begin with some code that finds out which sprite is under the clicked position, perhaps
none! We add a method to the class, contains_point, which returns True if the point is
within the rectangle of the sprite:
1
2
3
4
5
6
7
8
def contains_point(self, point):
""" Return True if my sprite rectangle contains point
˓→pt """
(my_x, my_y) = self.position
my_width = self.image.get_width()
my_height = self.image.get_height()
(x, y) = point
return ( x >= my_x and x < my_x + my_width and
y >= my_y and y < my_y + my_height)
Now in the game loop, once we’ve seen the mouse event, we determine which queen, if any,
should be told to respond to the event:
1
2
3
4
5
6
if ev.type == pygame.MOUSEBUTTONDOWN:
posn_of_click = event.dict["pos"]
for sprite in all_sprites:
if sprite.contains_point(posn_of_click):
sprite.handle_click()
break
And the final thing is to write a new method called handle_click in the QueenSprite
class. When a sprite is clicked, we’ll just add some velocity in the up direction, i.e. kick it back
into the air.
1
2
def handle_click(self):
self.y_velocity += -0.3
# Kick it up
With these changes we have a playable game! See if you can keep all the balls on the move,
not allowing any one to settle!
14.5. Events
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A wave of animation
Many games have sprites that are animated: they crouch, jump and shoot. How do they do that?
Consider this sequence of 10 images: if we display them in quick succession, Duke will wave
at us. (Duke is a friendly visitor from the kingdom of Javaland.)
A compound image containing smaller patches which are intended for animation is called a
sprite sheet. Download this sprite sheet by right-clicking in your browser and saving it in your
working directory with the name duke_spritesheet.png.
The sprite sheet has been quite carefully prepared: each of the 10 patches are spaced exactly
50 pixels apart. So, assuming we want to draw patch number 4 (numbering from 0), we want
to draw only the rectangle that starts at x position 200, and is 50 pixels wide, within the sprite
sheet. Here we’ve shown the patches and highlighted the patch we want to draw.
The blit method we’ve been using — for copying pixels from one surface to another — can
copy a sub-rectangle of the source surface. So the grand idea here is that each time we draw
Duke, we won’t blit the whole sprite sheet. Instead we’ll provide an extra rectangle argument
that determines which portion of the sprite sheet will be blitted.
We’re going to add new code in this section to our existing N queens drawing game. What we
want is to put some instances of Duke on the chessboard somewhere. If the user clicks on one
of them, we’ll get him to respond by waving back, for one cycle of his animation.
But before we do that, we need another change. Up until now, our game loop has been running
at really fast frame rates that are unpredictable. So we’ve chosen some magic numbers for
gravity and for bouncing and kicking the ball on the basis of trial-and-error. If we’re going to
start animating more sprites, we need to tame our game loop to operate at a fixed, known frame
rate. This will allow us to plan our animation better.
PyGame gives us the tools to do this in just two lines of code. In the setup section of the game,
we instantiate a new Clock object:
1
my_clock = pygame.time.Clock()
and right at the bottom of the game loop, we call a method on this object that limits the frame
rate to whatever we specify. So let’s plan our game and animation for 60 frames per second, by
adding this line at the bottom of our game loop:
1
284
my_clock.tick(60)
˓→60 fps
# Waste time so that frame rate becomes
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You’ll find that you have to go back and adjust the numbers for gravity and kicking the ball
now, to match this much slower frame rate. When we plan an animation so that it only works
sensibly at a fixed frame rate, we say that we’ve baked the animation. In this case we’re baking
our animations for 60 frames per second.
To fit into the existing framework that we already have for our queens board, we want to create
a DukeSprite class that has all the same methods as the QueenSprite class. Then we can
add one or more Duke instances onto our list of all_sprites, and our existing game loop
will then call methods of the Duke instance. Let us start with skeleton scaffolding for the new
class:
1
class DukeSprite:
2
3
4
5
def __init__(self, img, target_position):
self.image = img
self.position = target_position
6
7
8
def update(self):
return
9
10
11
def draw(self, target_surface):
return
12
13
14
def handle_click(self):
return
15
16
17
18
def contains_point(self, pt):
# Use code from QueenSprite here
return
The only changes we’ll need to the existing game are all in the setup section. We load up the
new sprite sheet and instantiate a couple of instances of Duke, at the positions we want on the
chessboard. So before entering the game loop, we add this code:
1
2
# Load the sprite sheet
duke_sprite_sheet = pygame.image.load("duke_spritesheet.png
˓→")
3
4
5
6
# Instantiate two duke instances, put them on the
˓→chessboard
duke1 = DukeSprite(duke_sprite_sheet,(square_size*2, 0))
duke2 = DukeSprite(duke_sprite_sheet,(square_size*5, sq_sz))
7
8
9
10
# Add them to the list of sprites which our game loop
˓→manages
all_sprites.append(duke1)
all_sprites.append(duke2)
Now the game loop will test if each instance has been clicked, will call the click handler for
that instance. It will also call update and draw for all sprites. All the remaining changes we
need to make will be made in the methods of the DukeSprite class.
14.6. A wave of animation
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Let’s begin with drawing one of the patches.
We’ll introduce a new attribute
curr_patch_num into the class. It holds a value between 0 and 9, and determines which
patch to draw. So the job of the draw method is to compute the sub-rectangle of the patch to
be drawn, and to blit only that portion of the spritesheet:
1
2
3
4
def draw(self, target_surface):
patch_rect = (self.curr_patch_num * 50, 0,
50, self.image.get_height())
target_surface.blit(self.image, self.posn, patch_rect)
Now on to getting the animation to work. We need to arrange logic in update so that if we’re
busy animating, we change the curr_patch_num every so often, and we also decide when
to bring Duke back to his rest position, and stop the animation. An important issue is that the
game loop frame rate — in our case 60 fps — is not the same as the animation rate — the rate
at which we want to change Duke’s animation patches. So we’ll plan Duke wave’s animation
cycle for a duration of 1 second. In other words, we want to play out Duke’s 10 animation
patches over 60 calls to update. (This is how the baking of the animation takes place!) So
we’ll keep another animation frame counter in the class, which will be zero when we’re not
animating, and each call to update will increment the counter up to 59, and then back to
0. We can then divide that animation counter by 6, to set the curr_patch_num variable to
select the patch we want to show.
1
2
3
4
def update(self):
if self.anim_frame_count > 0:
self.anim_frame_count = (self.anim_frame_count + 1 )
˓→% 60
self.curr_patch_num = self.anim_frame_count // 6
Notice that if anim_frame_count is zero, i.e. Duke is at rest, nothing happens here.
But if we start the counter running, it will count up to 59 before settling back to zero. Notice also, that because anim_frame_count can only be a value between 0 and 59, the
curr_patch_num will always stay between 0 and 9. Just what we require!
Now how do we trigger the animation, and start it running? On the mouse click.
1
2
3
def handle_click(self):
if self.anim_frame_count == 0:
self.anim_frame_count = 5
Two things of interest here. We only start the animation if Duke is at rest. Clicks on Duke while
he is already waving get ignored. And when we do start the animation, we set the counter to
5 — this means that on the very next call to update the counter becomes 6, and the image
changes. If we had set the counter to 1, we would have needed to wait for 5 more calls to
update before anything happened — a slight lag, but enough to make things feel sluggish.
The final touch-up is to initialize our two new attributes when we instantiate the class. Here is
the code for the whole class now:
1
class DukeSprite:
2
3
286
def __init__(self, img, target_posn):
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4
5
6
7
self.image = img
self.position = target_posn
self.anim_frame_count = 0
self.curr_patch_num = 0
8
9
10
11
12
def update(self):
if self.anim_frame_count > 0:
self.anim_frame_count = (self.anim_frame_count +
˓→1 ) % 60
self.curr_patch_num = self.anim_frame_count // 6
13
14
15
16
17
def draw(self, target_surface):
patch_rect = (self.curr_patch_num * 50, 0,
50, self.image.get_height())
target_surface.blit(self.image, self.posn, patch_
˓→rect)
18
19
20
21
22
23
24
25
26
def contains_point(self, pt):
""" Return True if my sprite rectangle contains
˓→pt """
(my_x, my_y) = self.posn
my_width = self.image.get_width()
my_height = self.image.get_height()
(x, y) = pt
return ( x >= my_x and x < my_x + my_width and
y >= my_y and y < my_y + my_height)
27
28
29
30
def handle_click(self):
if self.anim_frame_count == 0:
self.anim_frame_count = 5
Now we have two extra Duke instances on our chessboard, and clicking on either causes that
instance to wave.
14.6. A wave of animation
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Aliens - a case study
Find the example games with the PyGame package, (On a windows system, something like
C:\Python3\Lib\site-packages\pygame\examples) and play the Aliens game. Then read the
code, in an editor or Python environment that shows line numbers.
It does a number of much more advanced things that we do, and relies on the PyGame framework for more of its logic. Here are some of the points to notice:
• The frame rate is deliberately constrained near the bottom of the game loop at line 311.
If we change that number we can make the game very slow or unplayably fast!
• There are different kinds of sprites: Explosions, Shots, Bombs, Aliens and a Player.
Some of these have more than one image — by swapping the images, we get animation
of the sprites, i.e. the Alien spacecraft lights change, and this is done at line 112.
• Different kinds of objects are referenced in different groups of sprites, and PyGame helps
maintain these. This lets the program check for collisions between, say, the list of shots
fired by the player, and the list of spaceships that are attacking. PyGame does a lot of the
hard work for us.
• Unlike our game, objects in the Aliens game have a limited lifetime, and have to get
killed. For example, if we shoot, a Shot object is created — if it reaches the top of the
screen without expoding against anything, it has to be removed from the game. Lines
141-142 do this. Similarly, when a falling bomb gets close to the ground (line 156), it
instantiates a new Explosion sprite, and the bomb kills itself.
• There are random timings that add to the fun — when to spawn the next Alien, when an
Alien drops the next bomb, etc.
• The game plays sounds too: a less-than-relaxing loop sound, plus sounds for the shots
and explosions.
Reflections
Object oriented programming is a good organizational tool for software. In the examples in this
chapter, we’ve started to use (and hopefully appreciate) these benefits. Here we had N queens
each with its own state, falling to its own floor level, bouncing, getting kicked, etc. We might
have managed without the organizational power of objects — perhaps we could have kept lists
of velocities for each queen, and lists of target positions, and so on — our code would likely
have been much more complicated, ugly, and a lot poorer!
Glossary
animation rate The rate at which we play back successive patches to create the illusion of
movement. In the sample we considered in this chapter, we played Duke’s 10 patches
over the duration of one second. Not the same as the frame rate.
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baked animation An animation that is designed to look good at a predetermined fixed frame
rate. This reduces the amount of computation that needs to be done when the game is
running. High-end commercial games usually bake their animations.
blit A verb used in computer graphics, meaning to make a fast copy of an image or pixels
from a sub-rectangle of one image or surface to another surface or image.
frame rate The rate at which the game loop executes and updates the display.
game loop A loop that drives the logic of a game. It will usually poll for events, then update
each of the objects in the game, then get everything drawn, and then put the newly drawn
frame on display.
pixel A single picture element, or dot, from which images are made.
poll To ask whether something like a keypress or mouse movement has happened. Game loops
usually poll to discover what events have occurred. This is different from event-driven
programs like the ones seen in the chapter titled “Events”. In those cases, the button click
or keypress event triggers the call of a handler function in your program, but this happens
behind your back.
sprite An active agent or element in a game, with its own state, position and behaviour.
surface This is PyGame’s term for what the Turtle module calls a canvas. A surface is a
rectangle of pixels used for displaying shapes and images.
Exercises
1. Have fun with Python, and with PyGame.
2. We deliberately left a bug in the code for animating Duke. If you click on one of the
chessboard squares to the right of Duke, he waves anyway. Why? Find a one-line fix for
the bug.
3. Use your preferred search engine to search their image library for “sprite sheet playing
cards”. Create a list [0..51] to represent an encoding of the 52 cards in a deck. Shuffle
the cards, slice off the top five as your hand in a poker deal. Display the hand you have
been dealt.
4. So the Aliens game is in outer space, without gravity. Shots fly away forever, and bombs
don’t speed up when they fall. Add some gravity to the game. Decide if you’re going to
allow your own shots to fall back on your head and kill you.
5. Those pesky Aliens seem to pass right through each other! Change the game so that they
collide, and destroy each other in a mighty explosion.
14.10. Exercises
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CHAPTER 15
Copyright Notice
Copyright (C) Peter Wentworth, Jeffrey Elkner, Allen B. Downey and Chris Meyers.
Edited by Martin Roelfs, Peter Kroon, Kasper Loopstra, Jonathan Barnoud,
Manuel Nuno Melo and Lourens-Jan Ugen.
Permission is granted to copy, distribute and/or modify this document
under the terms of the GNU Free Documentation License, Version 1.3
or any later version published by the Free Software Foundation;
with Invariant Sections being Foreword, Preface, and Contributor List, no
Front-Cover Texts, and no Back-Cover Texts. A copy of the license is
included in the section entitled “GNU Free Documentation License”.
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Chapter 15. Copyright Notice
CHAPTER 16
Contributions
This version is adapted the third edition of the book “How to Think Like a Computer Scientist”
by Peter Wentworth, Jeffrey Elkner, Allen B. Downey, and Chris Meyers; as found on https:
//code.launchpad.net/~thinkcspy-rle-team/thinkcspy/thinkcspy3-rle.
This book is used for the course “Programming for Life Scientists” as taught at the University
of Groningen (RuG). As such, the original book was adapted to specifically suit this course.
The main point here was to change the goal of the book from “how to think like a computer
scientist” to “how to think as a scientist with a computer”. In other words, the emphasis has
been put on learning how to use a computer (and Python) to solve everyday scientific problems.
This version of the book is available on github at https://github.com/tBuLi/HowToThink. There
you are welcome to report issues and suggest changes.
Original contributor List
Note: This is the contributor list from the original book.
To paraphrase the philosophy of the Free Software Foundation, this book is free like free
speech, but not necessarily free like free pizza. It came about because of a collaboration that
would not have been possible without the GNU Free Documentation License. So we would
like to thank the Free Software Foundation for developing this license and, of course, making
it available to us.
We would also like to thank the more than 100 sharp-eyed and thoughtful readers who have sent
us suggestions and corrections over the past few years. In the spirit of free software, we decided
to express our gratitude in the form of a contributor list. Unfortunately, this list is not complete,
but we are doing our best to keep it up to date. It was also getting too large to include everyone
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who sends in a typo or two. You have our gratitude, and you have the personal satisfaction of
making a book you found useful better for you and everyone else who uses it. New additions
to the list for the 2nd edition will be those who have made on-going contributions.
If you have a chance to look through the list, you should realize that each person here has spared
you and all subsequent readers from the confusion of a technical error or a less-than-transparent
explanation, just by sending us a note.
Impossible as it may seem after so many corrections, there may still be errors in this book.
If you should stumble across one, we hope you will take a minute to contact us. The email
address (for the Python 3 version of the book) is [email protected] . Substantial changes
made due to your suggestions will add you to the next version of the contributor list (unless
you ask to be omitted). Thank you!
Second Edition
• An email from Mike MacHenry set me straight on tail recursion. He not only pointed out
an error in the presentation, but suggested how to correct it.
• It wasn’t until 5th Grade student Owen Davies came to me in a Saturday morning Python
enrichment class and said he wanted to write the card game, Gin Rummy, in Python that
I finally knew what I wanted to use as the case study for the object oriented programming
chapters.
• A special thanks to pioneering students in Jeff’s Python Programming class at GCTAA
during the 2009-2010 school year: Safath Ahmed, Howard Batiste, Louis Elkner-Alfaro,
and Rachel Hancock. Your continual and thoughtfull feedback led to changes in most of
the chapters of the book. You set the standard for the active and engaged learners that
will help make the new Governor’s Academy what it is to become. Thanks to you this is
truly a student tested text.
• Thanks in a similar vein to the students in Jeff’s Computer Science class at the HBWoodlawn program during the 2007-2008 school year: James Crowley, Joshua Eddy,
Eric Larson, Brian McGrail, and Iliana Vazuka.
• Ammar Nabulsi sent in numerous corrections from Chapters 1 and 2.
• Aldric Giacomoni pointed out an error in our definition of the Fibonacci sequence in
Chapter 5.
• Roger Sperberg sent in several spelling corrections and pointed out a twisted piece of
logic in Chapter 3.
• Adele Goldberg sat down with Jeff at PyCon 2007 and gave him a list of suggestions and
corrections from throughout the book.
• Ben Bruno sent in corrections for chapters 4, 5, 6, and 7.
• Carl LaCombe pointed out that we incorrectly used the term commutative in chapter 6
where symmetric was the correct term.
• Alessandro Montanile sent in corrections for errors in the code examples and text in
chapters 3, 12, 15, 17, 18, 19, and 20.
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• Emanuele Rusconi found errors in chapters 4, 8, and 15.
• Michael Vogt reported an indentation error in an example in chapter 6, and sent in a
suggestion for improving the clarity of the shell vs. script section in chapter 1.
First Edition
• Lloyd Hugh Allen sent in a correction to Section 8.4.
• Yvon Boulianne sent in a correction of a semantic error in Chapter 5.
• Fred Bremmer submitted a correction in Section 2.1.
• Jonah Cohen wrote the Perl scripts to convert the LaTeX source for this book into beautiful HTML.
• Michael Conlon sent in a grammar correction in Chapter 2 and an improvement in style
in Chapter 1, and he initiated discussion on the technical aspects of interpreters.
• Benoit Girard sent in a correction to a humorous mistake in Section 5.6.
• Courtney Gleason and Katherine Smith wrote horsebet.py, which was used as a case
study in an earlier version of the book. Their program can now be found on the website.
• Lee Harr submitted more corrections than we have room to list here, and indeed he should
be listed as one of the principal editors of the text.
• James Kaylin is a student using the text. He has submitted numerous corrections.
• David Kershaw fixed the broken catTwice function in Section 3.10.
• Eddie Lam has sent in numerous corrections to Chapters 1, 2, and 3. He also fixed the
Makefile so that it creates an index the first time it is run and helped us set up a versioning
scheme.
• Man-Yong Lee sent in a correction to the example code in Section 2.4.
• David Mayo pointed out that the word unconsciously in Chapter 1 needed to be changed
to subconsciously .
• Chris McAloon sent in several corrections to Sections 3.9 and 3.10.
• Matthew J. Moelter has been a long-time contributor who sent in numerous corrections
and suggestions to the book.
• Simon Dicon Montford reported a missing function definition and several typos in Chapter 3. He also found errors in the increment function in Chapter 13.
• John Ouzts corrected the definition of return value in Chapter 3.
• Kevin Parks sent in valuable comments and suggestions as to how to improve the distribution of the book.
• David Pool sent in a typo in the glossary of Chapter 1, as well as kind words of encouragement.
• Michael Schmitt sent in a correction to the chapter on files and exceptions.
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• Robin Shaw pointed out an error in Section 13.1, where the printTime function was used
in an example without being defined.
• Paul Sleigh found an error in Chapter 7 and a bug in Jonah Cohen’s Perl script that
generates HTML from LaTeX.
• Craig T. Snydal is testing the text in a course at Drew University. He has contributed
several valuable suggestions and corrections.
• Ian Thomas and his students are using the text in a programming course. They are the
first ones to test the chapters in the latter half of the book, and they have make numerous
corrections and suggestions.
• Keith Verheyden sent in a correction in Chapter 3.
• Peter Winstanley let us know about a longstanding error in our Latin in Chapter 3.
• Chris Wrobel made corrections to the code in the chapter on file I/O and exceptions.
• Moshe Zadka has made invaluable contributions to this project. In addition to writing
the first draft of the chapter on Dictionaries, he provided continual guidance in the early
stages of the book.
• Christoph Zwerschke sent several corrections and pedagogic suggestions, and explained
the difference between gleich and selbe.
• James Mayer sent us a whole slew of spelling and typographical errors, including two in
the contributor list.
• Hayden McAfee caught a potentially confusing inconsistency between two examples.
• Angel Arnal is part of an international team of translators working on the Spanish version
of the text. He has also found several errors in the English version.
• Tauhidul Hoque and Lex Berezhny created the illustrations in Chapter 1 and improved
many of the other illustrations.
• Dr. Michele Alzetta caught an error in Chapter 8 and sent some interesting pedagogic
comments and suggestions about Fibonacci and Old Maid.
• Andy Mitchell caught a typo in Chapter 1 and a broken example in Chapter 2.
• Kalin Harvey suggested a clarification in Chapter 7 and caught some typos.
• Christopher P. Smith caught several typos and is helping us prepare to update the book
for Python 2.2.
• David Hutchins caught a typo in the Foreword.
• Gregor Lingl is teaching Python at a high school in Vienna, Austria. He is working on a
German translation of the book, and he caught a couple of bad errors in Chapter 5.
• Julie Peters caught a typo in the Preface.
• search
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APPENDIX A
Modules
A module is a file containing Python definitions and statements intended for use in other Python
programs. There are many Python modules that come with Python as part of the standard
library. We have seen at least two of these already, the turtle module and the string
module.
We have also shown you how to access help. The help system contains a listing of all the
standard modules that are available with Python. Play with help!
Random numbers
We often want to use random numbers in programs, here are a few typical uses:
• To play a game of chance where the computer needs to throw some dice, pick a number,
or flip a coin,
• To shuffle a deck of playing cards randomly,
• To allow/make an enemy spaceship appear at a random location and start shooting at the
player,
• To simulate possible rainfall when we make a computerized model for estimating the
environmental impact of building a dam,
• For encrypting banking sessions on the Internet.
Python provides a module random that helps with tasks like this. You can look it up using
help, but here are the key things we’ll do with it:
1
import random
2
3
# Create a black box object that generates random numbers
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rng = random.Random()
5
6
7
dice_throw = rng.randrange(1,7)
# Return an int, one of 1,
˓→2,3,4,5,6
delay_in_seconds = rng.random() * 5.0
The randrange method call generates an integer between its lower and upper argument,
using the same semantics as range — so the lower bound is included, but the upper bound is
excluded. All the values have an equal probability of occurring (i.e. the results are uniformly
distributed). Like range, randrange can also take an optional step argument. So let’s
assume we needed a random odd number less than 100, we could say:
1
random_odd = rng.randrange(1, 100, 2)
Other methods can also generate other distributions e.g. a bell-shaped, or “normal” distribution
might be more appropriate for estimating seasonal rainfall, or the concentration of a compound
in the body after taking a dose of medicine.
The random method returns a floating point number in the interval [0.0, 1.0) — the square
bracket means “closed interval on the left” and the round parenthesis means “open interval on
the right”. In other words, 0.0 is possible, but all returned numbers will be strictly less than
1.0. It is usual to scale the results after calling this method, to get them into an interval suitable
for your application. In the case shown here, we’ve converted the result of the method call to
a number in the interval [0.0, 5.0). Once more, these are uniformly distributed numbers —
numbers close to 0 are just as likely to occur as numbers close to 0.5, or numbers close to 1.0.
This example shows how to shuffle a list. (shuffle cannot work directly with a lazy promise,
so notice that we had to convert the range object using the list type converter first.)
1
cards = list(range(52))
2
3
rng.shuffle(cards)
# Generate ints [0 .. 51]
#
representing a pack of cards.
# Shuffle the pack
Repeatability and Testing
Random number generators are based on a deterministic algorithm — repeatable and predictable. So they’re called pseudo-random generators — they are not genuinely random. They
start with a seed value. Each time you ask for another random number, you’ll get one based
on the current seed attribute, and the state of the seed (which is one of the attributes of the
generator) will be updated.
For debugging and for writing unit tests, it is convenient to have repeatability — programs that
do the same thing every time they are run. We can arrange this by forcing the random number
generator to be initialized with a known seed every time. (Often this is only wanted during
testing — playing a game of cards where the shuffled deck was always in the same order as last
time you played would get boring very rapidly!)
1
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drng = random.Random(123)
˓→starting state
# Create generator with known
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This alternative way of creating a random number generator gives an explicit seed value to
the object. Without this argument, the system probably uses something based on the time. So
grabbing some random numbers from drng today will give you precisely the same random
sequence as it will tomorrow!
Picking balls from bags, throwing dice, shuffling a pack of cards
Here is an example to generate a list containing n random ints between a lower and an upper
bound:
1
import random
2
3
4
5
6
7
8
9
10
11
12
def make_random_ints(num, lower_bound, upper_bound):
"""
Generate a list containing num random ints between
˓→lower_bound
and upper_bound. upper_bound is an open bound.
"""
rng = random.Random() # Create a random number
˓→generator
result = []
for i in range(num):
result.append(rng.randrange(lower_bound, upper_bound))
return result
>>> make_random_ints(5, 1, 13)
˓→numbers
[8, 1, 8, 5, 6]
# Pick 5 random month
Notice that we got a duplicate in the result. Often this is wanted, e.g. if we throw a die five
times, we would expect some duplicates.
But what if you don’t want duplicates? If you wanted 5 distinct months, then this algorithm is
wrong. In this case a good algorithm is to generate the list of possibilities, shuffle it, and slice
off the number of elements you want:
1
2
3
4
xs = list(range(1,13))
˓→duplicates)
rng = random.Random()
rng.shuffle(xs)
result = xs[:5]
# Make list 1..12
(there are no
# Make a random number generator
# Shuffle the list
# Take the first five elements
In statistics courses, the first case — allowing duplicates — is usually described as pulling
balls out of a bag with replacement — you put the drawn ball back in each time, so it can occur
again. The latter case, with no duplicates, is usually described as pulling balls out of the bag
without replacement. Once the ball is drawn, it doesn’t go back to be drawn again. TV lotto
games work like this.
The second “shuffle and slice” algorithm would not be so great if you only wanted a few
elements, but from a very large domain. Suppose I wanted five numbers between one and ten
A.1. Random numbers
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million, without duplicates. Generating a list of ten million items, shuffling it, and then slicing
off the first five would be a performance disaster! So let us have another try:
1
import random
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
def make_random_ints_no_dups(num, lower_bound, upper_bound):
"""
Generate a list containing num random ints between
lower_bound and upper_bound. upper_bound is an open
˓→bound.
The result list cannot contain duplicates.
"""
result = []
rng = random.Random()
for i in range(num):
while True:
candidate = rng.randrange(lower_bound, upper_
˓→bound)
if candidate not in result:
break
result.append(candidate)
return result
18
19
20
xs = make_random_ints_no_dups(5, 1, 10000000)
print(xs)
This agreeably produces 5 random numbers, without duplicates:
[3344629, 1735163, 9433892, 1081511, 4923270]
Even this function has its pitfalls. Can you spot what is going to happen in this case?
1
xs = make_random_ints_no_dups(10, 1, 6)
The time module
As we start to work with more sophisticated algorithms and bigger programs, a natural concern
is “is our code efficient?” One way to experiment is to time how long various operations
take. The time module has a function called clock that is recommended for this purpose.
Whenever clock is called, it returns a floating point number representing how many seconds
have elapsed since your program started running.
The way to use it is to call clock and assign the result to a variable, say t0, just before you
start executing the code you want to measure. Then after execution, call clock again, (this
time we’ll save the result in variable t1). The difference t1-t0 is the time elapsed, and is a
measure of how fast your program is running.
Let’s try a small example. Python has a built-in sum function that can sum the elements in a
list. We can also write our own. How do we think they would compare for speed? We’ll try to
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do the summation of a list [0, 1, 2 ...] in both cases, and compare the results:
1
import time
2
3
4
5
6
7
def do_my_sum(xs):
sum = 0
for v in xs:
sum += v
return sum
8
9
10
sz = 10000000
# Lets have 10 million elements in the
˓→list
testdata = range(sz)
11
12
13
14
15
16
t0 = time.clock()
my_result = do_my_sum(testdata)
t1 = time.clock()
print("my_result
= {0} (time taken = {1:.4f} seconds)"
.format(my_result, t1-t0))
17
18
19
20
21
22
t2 = time.clock()
their_result = sum(testdata)
t3 = time.clock()
print("their_result = {0} (time taken = {1:.4f} seconds)"
.format(their_result, t3-t2))
On a reasonably modest laptop, we get these results:
my_sum
= 49999995000000 (time taken = 1.5567 seconds)
their_sum = 49999995000000 (time taken = 0.9897 seconds)
So our function runs about 57% slower than the built-in one. Generating and summing up ten
million elements in under a second is not too shabby!
The math module
The math module contains the kinds of mathematical functions you’d typically find on your
calculator (sin, cos, sqrt, asin, log, log10) and some mathematical constants like pi
and e:
>>> import math
>>> math.pi
3.141592653589793
>>> math.e
2.718281828459045
>>> math.sqrt(2.0)
1.4142135623730951
>>> math.radians(90)
1.5707963267948966
A.3. The math module
# Constant pi
# Constant natural log base
# Square root function
# Convert 90 degrees to radians
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>>> math.sin(math.radians(90)) # Find sin of 90 degrees
1.0
>>> math.asin(1.0) * 2
# Double the arcsin of 1.0 to
˓→get pi
3.141592653589793
Like almost all other programming languages, angles are expressed in radians rather than degrees. There are two functions radians and degrees to convert between these two popular
ways of measuring angles.
Notice another difference between this module and our use of random and turtle: in
random and turtle we create objects and we call methods on the object. This is because
objects have state — a turtle has a color, a position, a heading, etc., and every random number
generator has a seed value that determines its next result.
Mathematical functions are “pure” and don’t have any state — calculating the square root of
2.0 doesn’t depend on any kind of state or history about what happened in the past. So the
functions are not methods of an object — they are simply functions that are grouped together
in a module called math.
Creating your own modules
All we need to do to create our own modules is to save our script as a file with a .py extension.
Suppose, for example, this script is saved as a file named seqtools.py:
1
2
def remove_at(pos, seq):
return seq[:pos] + seq[pos+1:]
We can now use our module, both in scripts we write, or in the interactive Python interpreter.
To do so, we must first import the module.
>>> import seqtools
>>> s = "A string!"
>>> seqtools.remove_at(4, s)
'A sting!'
We do not include the .py file extension when importing. Python expects the file names of
Python modules to end in .py, so the file extension is not included in the import statement.
The use of modules makes it possible to break up very large programs into manageable sized
parts, and to keep related parts together.
Namespaces
A namespace is a collection of identifiers that belong to a module, or to a function, (and as we
will see soon, in classes too). Generally, we like a namespace to hold “related” things, e.g. all
the math functions, or all the typical things we’d do with random numbers.
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Each module has its own namespace, so we can use the same identifier name in multiple modules without causing an identification problem.
1
# module1.py
2
4
question = "What is the meaning of Life, the Universe, and
˓→Everything?"
answer = 42
1
# module2.py
3
2
3
4
question = "What is your quest?"
answer = "To seek the holy grail."
We can now import both modules and access question and answer in each:
1
2
import module1
import module2
3
4
5
6
7
print(module1.question)
print(module2.question)
print(module1.answer)
print(module2.answer)
will output the following:
What is the meaning of Life, the Universe, and Everything?
What is your quest?
42
To seek the holy grail.
Functions also have their own namespaces:
1
2
3
def f():
n = 7
print("printing n inside of f:", n)
4
5
6
7
def g():
n = 42
print("printing n inside of g:", n)
8
9
10
11
12
13
14
n = 11
print("printing n before calling f:", n)
f()
print("printing n after calling f:", n)
g()
print("printing n after calling g:", n)
Running this program produces the following output:
A.5. Namespaces
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printing
printing
printing
printing
printing
n
n
n
n
n
before calling f: 11
inside of f: 7
after calling f: 11
inside of g: 42
after calling g: 11
The three n‘s here do not collide since they are each in a different namespace — they are three
names for three different variables, just like there might be three different instances of people,
all called “Bruce”.
Namespaces permit several programmers to work on the same project without having naming
collisions.
How are namespaces, files and modules related?
Python has a convenient and simplifying one-to-one mapping, one module per file,
giving rise to one namespace. Also, Python takes the module name from the file
name, and this becomes the name of the namespace. math.py is a filename, the
module is called math, and its namespace is math. So in Python the concepts are
more or less interchangeable.
But you will encounter other languages (e.g. C#), that allow one module to span
multiple files, or one file to have multiple namespaces, or many files to all share
the same namespace. So the name of the file doesn’t need to be the same as the
namespace.
So a good idea is to try to keep the concepts distinct in your mind.
Files and directories organize where things are stored in our computer. On the
other hand, namespaces and modules are a programming concept: they help us
organize how we want to group related functions and attributes. They are not about
“where” to store things, and should not have to coincide with the file and directory
structures.
So in Python, if you rename the file math.py, its module name also changes, your
import statements would need to change, and your code that refers to functions
or attributes inside that namespace would also need to change.
In other languages this is not necessarily the case. So don’t blur the concepts, just
because Python blurs them!
Scope and lookup rules
The scope of an identifier is the region of program code in which the identifier can be accessed,
or used.
There are three important scopes in Python:
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• Local scope refers to identifiers declared within a function. These identifiers are kept in
the namespace that belongs to the function, and each function has its own namespace.
• Global scope refers to all the identifiers declared within the current module, or file.
• Built-in scope refers to all the identifiers built into Python — those like range and min
that can be used without having to import anything, and are (almost) always available.
Python can help you by telling you what is in which scope. Use the functions locals,
globals, and dir to see for yourself!
Python (like most other computer languages) uses precedence rules: the same name could occur
in more than one of these scopes, but the innermost, or local scope, will always take precedence
over the global scope, and the global scope always gets used in preference to the built-in scope.
Let’s start with a simple example:
1
2
def range(n):
return 123*n
3
4
print(range(10))
What gets printed? We’ve defined our own function called range, so there is now a potential
ambiguity. When we use range, do we mean our own one, or the built-in one? Using the
scope lookup rules determines this: our own range function, not the built-in one, is called,
because our function range is in the global namespace, which takes precedence over the builtin names.
So although names likes range and min are built-in, they can be “hidden” from your use if
you choose to define your own variables or functions that reuse those names. (It is a confusing
practice to redefine built-in names — so to be a good programmer you need to understand the
scope rules and understand that you can do nasty things that will cause confusion, and then you
avoid doing them!)
Now, a slightly more complex example:
1
2
3
4
5
n = 10
m = 3
def f(n):
m = 7
return 2*n+m
6
7
print(f(5), n, m)
This prints 17 10 3. The reason is that the two variables m and n in lines 1 and 2 are outside the
function in the global namespace. Inside the function, new variables called n and m are created
just for the duration of the execution of f. These are created in the local namespace of function
f. Within the body of f, the scope lookup rules determine that we use the local variables m
and n. By contrast, after we’ve returned from f, the n and m arguments to the print function
refer to the original variables on lines 1 and 2, and these have not been changed in any way by
executing function f.
Notice too that the def puts name f into the global namespace here. So it can be called on line
7.
A.6. Scope and lookup rules
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What is the scope of the variable n on line 1? Its scope — the region in which it is visible — is
lines 1, 2, 6, 7. It is hidden from view in lines 3, 4, 5 because of the local variable n.
Attributes and the dot operator
Variables defined inside a module are called attributes of the module. We’ve seen that objects
have attributes too: for example, most objects have a __doc__ attribute, some functions have
a __annotations__ attribute. Attributes are accessed using the dot operator (.). The
question attribute of module1 and module2 is accessed using module1.question
and module2.question.
Modules contain functions as well as attributes, and the dot operator is used to access them
in the same way. seqtools.remove_at refers to the remove_at function in the
seqtools module.
When we use a dotted name, we often refer to it as a fully qualified name, because we’re
saying exactly which question attribute we mean.
Three import statement variants
Here are three different ways to import names into the current namespace, and to use them:
1
2
import math
x = math.sqrt(10)
Here just the single identifier math is added to the current namespace. If you want to access
one of the functions in the module, you need to use the dot notation to get to it.
Here is a different arrangement:
1
2
from math import cos, sin, sqrt
x = sqrt(10)
The names are added directly to the current namespace, and can be used without qualification.
The name math is not itself imported, so trying to use the qualified form math.sqrt would
give an error.
Then we have a convenient shorthand:
1
from math import *
2
3
˓→namespace.
x = sqrt(10)
# Import all the identifiers from math,
#
adding them to the current
# Use them without qualification.
Of these three, the first method is generally preferred, even though it means a little more typing
each time. Although, we can make things shorter by importing a module under a different
name:
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2
3
>>> import math as m
>>> m.pi
3.141592653589793
But hey, with nice editors that do auto-completion, and fast fingers, that’s a small price!
Finally, observe this case:
1
2
3
def area(radius):
import math
return math.pi * radius * radius
4
5
x = math.sqrt(10)
# This gives an error
Here we imported math, but we imported it into the local namespace of area. So the name is
usable within the function body, but not in the enclosing script, because it is not in the global
namespace.
Glossary
attribute A variable defined inside a module (or class or instance – as we will see later).
Module attributes are accessed by using the dot operator (.).
dot operator The dot operator (.) permits access to attributes and functions of a module (or
attributes and methods of a class or instance – as we have seen elsewhere).
fully qualified name A name that is prefixed by some namespace identifier and the dot operator, or by an instance object, e.g. math.sqrt or tess.forward(10).
import statement A statement which makes the objects contained in a module available for
use within another module. There are two forms for the import statement. Using hypothetical modules named mymod1 and mymod2 each containing functions f1 and f2,
and variables v1 and v2, examples of these two forms include:
1
2
import mymod1
from mymod2 import f1, f2, v1, v2
The second form brings the imported objects into the namespace of the importing module, while the first form preserves a separate namespace for the imported module, requiring mymod1.v1 to access the v1 variable from that module.
method Function-like attribute of an object. Methods are invoked (called) on an object using
the dot operator. For example:
>>> s = "this is a string."
>>> s.upper()
'THIS IS A STRING.'
>>>
A.9. Glossary
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We say that the method, upper is invoked on the string, s. s is implicitely the first
argument to upper.
module A file containing Python definitions and statements intended for use in other Python
programs. The contents of a module are made available to the other program by using
the import statement.
namespace A syntactic container providing a context for names so that the same name can
reside in different namespaces without ambiguity. In Python, modules, classes, functions
and methods all form namespaces.
naming collision A situation in which two or more names in a given namespace cannot be
unambiguously resolved. Using
1
import string
instead of
1
from string import *
prevents naming collisions.
ndard library A library is a collection of software used as tools in the development of
other software. The standard library of a programming language is the set of such tools
that are distributed with the core programming language. Python comes with an extensive
standard library.
Exercises
1. Open help for the calendar module.
(a) Try the following:
1
2
3
import calendar
cal = calendar.TextCalendar()
˓→instance
cal.pryear(2012)
˓→here?
# Create an
# What happens
(b) Observe that the week starts on Monday. An adventurous CompSci student believes
that it is better mental chunking to have his week start on Thursday, because then
there are only two working days to the weekend, and every week has a break in the
middle. Read the documentation for TextCalendar, and see how you can help him
print a calendar that suits his needs.
(c) Find a function to print just the month in which your birthday occurs this year.
(d) Try this:
1
2
308
d = calendar.LocaleTextCalendar(6, "SPANISH")
d.pryear(2012)
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Try a few other languages, including one that doesn’t work, and see what happens.
(e) Experiment with calendar.isleap. What does it expect as an argument? What
does it return as a result? What kind of a function is this?
Make detailed notes about what you learned from these exercises.
2. Open help for the math module.
(a) How many functions are in the math module?
(b) What does math.ceil do? What about math.floor? (hint: both floor and
ceil expect floating point arguments.)
(c) Describe how we have been computing the same value as math.sqrt without
using the math module.
(d) What are the two data constants in the math module?
Record detailed notes of your investigation in this exercise.
3. Investigate the copy module. What does deepcopy do? In which exercises from last
chapter would deepcopy have come in handy?
4. Create a module named mymodule1.py. Add attributes myage set to your current
age, and year set to the current year. Create another module named mymodule2.py.
Add attributes myage set to 0, and year set to the year you were born. Now create a
file named namespace_test.py. Import both of the modules above and write the
following statement:
1
2
print( (mymodule2.myage - mymodule1.myage) ==
(mymodule2.year - mymodule1.year) )
When you will run namespace_test.py you will see either True or False as
output depending on whether or not you’ve already had your birthday this year.
What this example illustrates is that out different modules can both have attributes named
myage and year. Because they’re in different namespaces, they don’t clash with one
another. When we write namespace_test.py, we fully qualify exactly which variable year or myage we are referring to.
5. Add the following statement to mymodule1.py,
namespace_test.py from the previous exercise:
1
mymodule2.py,
and
print("My name is", __name__)
Run namespace_test.py. What happens? Why? Now add the following to the
bottom of mymodule1.py:
1
2
if __name__ == "__main__":
print("This won't run if I'm
imported.")
Run mymodule1.py and namespace_test.py again. In which case do you see
the new print statement?
6. In a Python shell / interactive interpreter, try the following:
A.10. Exercises
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>>> import this
What does Tim Peters have to say about namespaces?
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APPENDIX B
More datatypes
You have already encountered the most important datatypes Python has to offer: bools, ints,
floats, strings, tuples, lists and dictionaries. However, there is more to it than hinted at previously. In this section, we will focus mainly on tuples and lists, and introduce sets and frozensets.
Mutable versus immutable and aliasing
Some datatypes in Python are mutable. This means their contents can be changed after they
have been created. Lists and dictionaries are good examples of mutable datatypes.
>>>
>>>
>>>
[9,
my_list = [2, 4, 5, 3, 6, 1]
my_list[0] = 9
my_list
4, 5, 3, 6, 1]
Tuples and strings are examples of immutable datatypes, their contents can not be changed after
they have been created:
>>> my_tuple = (2, 5, 3, 1)
>>> my_tuple[0] = 9
Traceback (most recent call last):
File "<interactive input>", line 2, in <module>
TypeError: 'tuple' object does not support item assignment
>>>
Mutability is usually useful, but it may lead to something called aliasing. In this case, two
variables refer to the same object and mutating one will also change the other:
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>>>
>>>
>>>
>>>
[1,
list_one = [1, 2, 3, 4, 6]
list_two = list_one
list_two[-1] = 5
list_one
2, 3, 4, 5]
This happens, because both list_one and list_two refer to the same memory address containing
the actual list. You can check this using the built-in function id:
>>> list_one = [1, 2, 3, 4, 6]
>>> list_two = list_one
>>> id(list_one) == id(list_two)
True
You can escape this problem by making a copy of the list:
>>> list_one = [1, 2, 3, 4, 6]
>>> list_two = list_one[:]
>>> id(list_one) == id(list_two)
False
>>> list_two[-1] = 5
>>> list_two
[1, 2, 3, 4, 5]
>>> list_one
[1, 2, 3, 4, 6]
However, this will not work for nested lists because of the same reason. The module copy
provides functions to solve this.
Sets and frozensets
Given that tuples and lists are ordered, and dictionaries are unordered, we can construct the
following table.
Mutable
Immutable
Ordered
list
tuple
Unordered
dict
This reveals an empty spot: we don’t know any immutable, unordered datatypes yet. Additionally, you can argue that a dictionary doesn’t belong in this table, since it is a mapping
type whilst lists and tuples are not: a dictionary maps keys to values. This is where sets and
frozensets come in. A set is an unordered, mutable datatype; and a frozenset is an unordered,
immutable datatype.
Mutable
Immutable
Ordered
list
tuple
Unordered
set
frozenset
Since sets and frozensets are unordered, they share some properties with dictionaries: for example, it’s elements are unique. Creating a set, and adding elements to it is simple.
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>>>
>>>
{1,
>>>
>>>
{1,
my_set = set([1, 4, 2, 3, 4])
my_set
2, 3, 4}
my_set.add(13)
my_set
2, 3, 4, 13}
Sets may seem sorted in the example above, but this is completely coincidental. Sets also
support common operations such as membership testing (3 in my_set); and iteration (for
x in my_set:). Additionally, you can add and substract sets from eachother:
1
2
3
4
5
6
7
set1 = set([1, 2, 3])
set2 = set([4, 5, 6])
print(set1 | set2) # {1, 2, 3, 4, 5, 6}
print(set1 & set2) # set()
set2 = set([2, 3, 4, 5])
print(set1 & set2) # {2, 3}
print(set1 - set2) # {1}
Frozensets are mostly the same as set, other then that they can not be modified; i.e. you can’t
add or remove items. See also the documentation online.
More exotic data types - such as queues, stacks and ordered dictionaries - are provided in
Python’s collections module. You can find the documentation here.
B.2. Sets and frozensets
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APPENDIX C
Recursion
Recursion means “defining something in terms of itself” usually at some smaller scale, perhaps
multiple times, to achieve your objective. For example, we might say “A human being is
someone whose mother is a human being”, or “a directory is a structure that holds files and
(smaller) directories”, or “a family tree starts with a couple who have children, each with their
own family sub-trees”.
Programming languages generally support recursion, which means that, in order to solve a
problem, functions can call themselves to solve smaller subproblems.
Any problem that can be solved iteratively (with a for or while loop) can also be solved recursively. However, recursion takes a while wrap your head around, and because of this, it is
generally only used in specific cases, where either your problem is recursive in nature, or your
data is recursive.
Drawing Fractals
For our purposes, a fractal is a drawing which also has self-similar structure, where it can be
defined in terms of itself. This is a typical example of a problem which is recursive in nature.
Let us start by looking at the famous Koch fractal. An order 0 Koch fractal is simply a straight
line of a given size.
An order 1 Koch fractal is obtained like this: instead of drawing just one line, draw instead four
smaller segments, in the pattern shown here:
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Now what would happen if we repeated this Koch pattern again on each of the order 1 segments? We’d get this order 2 Koch fractal:
Repeating our pattern again gets us an order 3 Koch fractal:
Now let us think about it the other way around. To draw a Koch fractal of order 3, we can
simply draw four order 2 Koch fractals. But each of these in turn needs four order 1 Koch
fractals, and each of those in turn needs four order 0 fractals. Ultimately, the only drawing that
will take place is at order 0. This is very simple to code up in Python:
1
2
3
4
5
def koch(tortoise, order, size):
"""
Make turtle tortoise draw a Koch fractal of 'order' and 'size
˓→'.
Leave the turtle facing the same direction.
"""
6
if order == 0:
# The base case is just a straight line
tortoise.forward(size)
else:
koch(tortoise, order-1, size/3)
# Go 1/3 of the way
tortoise.left(60)
koch(tortoise, order-1, size/3)
tortoise.right(120)
koch(tortoise, order-1, size/3)
tortoise.left(60)
koch(tortoise, order-1, size/3)
7
8
9
10
11
12
13
14
15
16
The key thing that is new here is that if order is not zero, koch calls itself recursively to get its
job done.
Let’s make a simple observation and tighten up this code. Remember that turning right by 120
is the same as turning left by -120. So with a bit of clever rearrangement, we can use a loop
instead of lines 10-16:
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1
2
3
4
5
6
7
def koch(tortoise, order, size):
if order == 0:
tortoise.forward(size)
else:
for angle in [60, -120, 60, 0]:
koch(tortoise, order-1, size/3)
tortoise.left(angle)
The final turn is 0 degrees — so it has no effect. But it has allowed us to find a pattern and
reduce seven lines of code to three, which will make things easier for our next observations.
Recursion, the high-level view
One way to think about this is to convince yourself that the function works correctly when you
call it for an order 0 fractal. Then do a mental leap of faith, saying “the fairy godmother (or
Python, if you can think of Python as your fairy godmother) knows how to handle the recursive
level 0 calls for me on lines 11, 13, 15, and 17, so I don’t need to think about that detail!” All
I need to focus on is how to draw an order 1 fractal if I can assume the order 0 one is already
working.
You’re practicing mental abstraction — ignoring the subproblem while you solve the big problem.
If this mode of thinking works (and you should practice it!), then take it to the next level. Aha!
now can I see that it will work when called for order 2 under the assumption that it is already
working for level 1.
And, in general, if I can assume the order n-1 case works, can I just solve the level n problem?
Students of mathematics who have played with proofs of induction should see some very strong
similarities here.
Recursion, the low-level operational view
Another way of trying to understand recursion is to get rid of it! If we had separate functions to
draw a level 3 fractal, a level 2 fractal, a level 1 fractal and a level 0 fractal, we could simplify
the above code, quite mechanically, to a situation where there was no longer any recursion, like
this:
1
2
def koch_0(tortoise, size):
tortoise.forward(size)
3
4
5
6
7
def koch_1(tortoise, size):
for angle in [60, -120, 60, 0]:
koch_0(tortoise, size/3)
tortoise.left(angle)
8
9
10
11
def koch_2(tortoise, size):
for angle in [60, -120, 60, 0]:
koch_1(tortoise, size/3)
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tortoise.left(angle)
12
13
14
15
16
17
def koch_3(tortoise, size):
for angle in [60, -120, 60, 0]:
koch_2(tortoise, size/3)
tortoise.left(angle)
This trick of “unrolling” the recursion gives us an operational view of what happens. You can
trace the program into koch_3, and from there, into koch_2, and then into koch_1, etc., all
the way down the different layers of the recursion.
This might be a useful hint to build your understanding. The mental goal is, however, to be
able to do the abstraction!
Recursive data structures
Most of the Python data types we have seen can be grouped inside lists and tuples in a variety
of ways. Lists and tuples can also be nested, providing many possibilities for organizing data.
The organization of data for the purpose of making it easier to use is called a data structure.
It’s election time and we are helping to compute the votes as they come in. Votes arriving from
individual wards, precincts, municipalities, counties, and states are sometimes reported as a
sum total of votes and sometimes as a list of subtotals of votes. After considering how best to
store the tallies, we decide to use a nested number list, which we define as follows:
A nested number list is a list whose elements are either:
1. numbers
2. nested number lists
Notice that the term, nested number list is used in its own definition. Recursive definitions
like this are quite common in mathematics and computer science. They provide a concise and
powerful way to describe recursive data structures that are partially composed of smaller and
simpler instances of themselves. The definition is not circular, since at some point we will
reach a list that does not have any lists as elements.
Now suppose our job is to write a function that will sum all of the values in a nested number
list. Python has a built-in function which finds the sum of a sequence of numbers:
>>> sum([1, 2, 8])
11
For our nested number list, however, sum will not work:
>>> sum([1, 2, [11, 13], 8])
Traceback (most recent call last):
File "<interactive input>", line 1, in <module>
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TypeError: unsupported operand type(s) for +: 'int' and 'list'
>>>
The problem is that the third element of this list, [11, 13], is itself a list, so it cannot just be
added to 1, 2, and 8.
Processing recursive number lists
To sum all the numbers in our recursive nested number list we need to traverse the list, visiting
each of the elements within its nested structure, adding any numeric elements to our sum, and
recursively repeating the summing process with any elements which are themselves sub-lists.
Thanks to recursion, the Python code needed to sum the values of a nested number list is
surprisingly short:
1
2
3
4
5
6
7
8
9
def recursive_sum(nested_number_list):
"""Returns the total sum of all elements in nested_number_list""
˓→"
total = 0
for element in nested_number_list:
if type(element) is list:
total += recursive_sum(element)
else:
total += element
return total
The body of recursive_sum consists mainly of a for loop that traverses
nested_number_list. If element is a numerical value (the else branch), it is simply added to total. If element is a list, then recursive_sum is called again, with the
element as an argument. The statement inside the function definition in which the function calls
itself is known as the recursive call.
The example above has a base case (on line 13) which does not lead to a recursive call: the
case where the element is not a (sub-) list. Without a base case, you’ll have infinite recursion,
and your program will not work.
An alternative solution, completely recursive, would be the following. Notice that this implementation does not contain a for loop!
1
2
3
4
5
6
7
def recursive_sum(nested_number_list):
"""Returns the total sum of all elements in nested_number_list""
˓→"
if len(nested_number_list) == 0:
return 0
head, *tail = nested_number_list #Assign the first element of
˓→nested_number_list to head, and the rest to tail.
if isinstance(head, list): # If head is a list....
return recursive_sum(head) + recursive_sum(tail)
C.3. Processing recursive number lists
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else:
return head + recursive_sum(tail)
8
9
Recursion is truly one of the most beautiful and elegant tools in computer science.
A slightly more complicated problem is finding the largest value in our nested number list:
1
2
3
4
5
6
7
8
9
10
11
12
13
def recursive_max(nested_list):
"""
Find the maximum in a recursive structure of lists
within other lists.
Precondition: No lists or sublists are empty.
"""
largest = None
first_time = True
for element in nested_list:
if type(element) is list:
value = recursive_max(element)
else:
value = element
14
if first_time or value > largest:
largest = value
first_time = False
15
16
17
18
return largest
19
The added twist to this problem is finding a value for initializing largest. We can’t just use
nested_list[0], since that could be either a element or a list. To solve this problem (at
every recursive call) we initialize a Boolean flag (at line 8). When we’ve found the value of
interest, (at line 15) we check to see whether this is the initializing (first) value for largest,
or a value that could potentially change largest.
Again here we have a base case at line 13. If we don’t supply a base case, Python stops after
reaching a maximum recursion depth and returns a runtime error. See how this happens, by
running this little script which we will call infinite_recursion.py:
1
2
3
def recursion_depth(number):
print("{0}, ".format(number), end="")
recursion_depth(number + 1)
4
5
recursion_depth(0)
After watching the messages flash by, you will be presented with the end of a long traceback
that ends with a message like the following:
RuntimeError: maximum recursion depth exceeded ...
We would certainly never want something like this to happen to a user of one of our programs,
so in another appendix we’ll see how errors, any kinds of errors, are handled in Python.
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Case study: Fibonacci numbers
The famous Fibonacci sequence 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 134, ... was devised
by Fibonacci (1170-1250), who used this to model the breeding of (pairs) of rabbits. If, in
generation 7 you had 21 pairs in total, of which 13 were adults, then next generation the adults
will all have bred new children, and the previous children will have grown up to become adults.
So in generation 8 you’ll have 13+21=34, of which 21 are adults.
This model to explain rabbit breeding made the simplifying assumption that rabbits never died.
Scientists often make (unrealistic) simplifying assumptions and restrictions to make some headway with the problem.
If we number the terms of the sequence from 0, we can describe each term recursively as the
sum of the previous two terms:
fib(0) = 0
fib(1) = 1
fib(n) = fib(n-1) + fib(n-2)
for n >= 2
This translates very directly into some Python:
1
2
3
4
5
def fib(n):
if n <= 1:
return n
t = fib(n-1) + fib(n-2)
return t
This is a particularly inefficient algorithm, and this could be solved far more efficient iteratively:
1
2
3
4
5
import time
t0 = time.clock()
n = 35
result = fib(n)
t1 = time.clock()
6
7
print("fib({0}) = {1}, ({2:.2f} secs)".format(n, result, t1-t0))
We get the correct result, but an exploding amount of work!
fib(35) = 9227465, (10.54 secs)
Example with recursive directories and files
The following program lists the contents of a directory and all its subdirectories.
1
import os
2
3
4
def get_dirlist(path):
"""
C.4. Case study: Fibonacci numbers
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Return a sorted list of all entries in path.
This returns just the names, not the full path to the names.
"""
dirlist = os.listdir(path)
dirlist.sort()
return dirlist
5
6
7
8
9
10
11
12
13
14
15
16
def print_files(path, prefix = ""):
""" Print recursive listing of contents of path """
if prefix == "": # Detect outermost call, print a heading
print("Folder listing for", path)
prefix = "| "
17
18
19
20
21
22
23
dirlist = get_dirlist(path)
for file in dirlist:
print(prefix+file)
# Print the line
fullname = os.path.join(path, file)
# Turn name into full
˓→pathname
if os.path.isdir(fullname):
# If a directory,
˓→recurse.
print_files(fullname, prefix + "| ")
Calling the function print_files with some folder name will produce output similar to
this:
Folder listing for c:\python31\Lib\site-packages\pygame\examples
| __init__.py
| aacircle.py
| aliens.py
| arraydemo.py
| blend_fill.py
| blit_blends.py
| camera.py
| chimp.py
| cursors.py
| data
| | alien1.png
| | alien2.png
| | alien3.png
...
Note that something similar is already implemented in the os module: os.walk.
An animated fractal, using PyGame
Here we have a tree fractal pattern of order 8. We’ve labelled some of the edges, showing the
depth of the recursion at which each edge was drawn.
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In the tree above, the angle of deviation from the trunk is 30 degrees. Varying that angle gives
other interesting shapes, for example, with the angle at 90 degrees we get this:
An interesting animation occurs if we generate and draw trees very rapidly, each time varying
the angle a little. Although the Turtle module can draw trees like this quite elegantly, we could
struggle for good frame rates. So we’ll use PyGame instead, with a few embellishments and
observations. (Once again, we suggest you cut and paste this code into your Python environment.)
1
2
import pygame, math
pygame.init()
# prepare the pygame module for use
3
4
# Create a new surface and window.
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6
7
surface_size = 1024
main_surface = pygame.display.set_mode((surface_size,surface_size))
my_clock = pygame.time.Clock()
8
9
10
def draw_tree(order, theta, size, position, heading, color=(0,0,0),
˓→depth=0):
11
12
13
14
15
16
17
18
trunk_ratio = 0.29
# How big is the trunk relative to
˓→whole tree?
trunk = size * trunk_ratio # length of trunk
delta_x = trunk * math.cos(heading)
delta_y = trunk * math.sin(heading)
(u, v) = position
newposition = (u + delta_x, v + delta_y)
pygame.draw.line(main_surface, color, position, newposition)
19
if order > 0:
20
# Draw another layer of subtrees
21
22
23
24
25
26
27
28
29
30
# These next six lines are a simple hack to make the two
˓→major halves
# of the recursion different colors. Fiddle here to change
˓→colors
# at other depths, or when depth is even, or odd, etc.
if depth == 0:
color1 = (255, 0, 0)
color2 = (0, 0, 255)
else:
color1 = color
color2 = color
31
32
33
34
˓→
35
˓→
# make the recursive calls to draw the two subtrees
newsize = size*(1 - trunk_ratio)
draw_tree(order-1, theta, newsize, newposition, heading-theta,
color1, depth+1)
draw_tree(order-1, theta, newsize, newposition, heading+theta,
color2, depth+1)
36
37
38
def gameloop():
39
theta = 0
while True:
40
41
42
# Handle evente from keyboard, mouse, etc.
event = pygame.event.poll()
if event.type == pygame.QUIT:
break;
43
44
45
46
47
# Updates - change the angle
48
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theta += 0.01
49
50
51
52
53
# Draw everything
main_surface.fill((255, 255, 0))
draw_tree(9, theta, surface_size*0.9, (surface_size//2,
˓→surface_size-50), -math.pi/2)
54
pygame.display.flip()
my_clock.tick(120)
55
56
57
58
59
60
gameloop()
pygame.quit()
• The math library works with angles in radians rather than degrees.
• Lines 14 and 15 uses some high school trigonmetry. From the length of the desired line
(trunk), and its desired angle, cos and sin help us to calculate the x and y distances
we need to move.
• Lines 22-30 are unnecessary, except if we want a colorful tree.
• In the main game loop at line 49 we change the angle on every frame, and redraw the
new tree.
• Line 18 shows that PyGame can also draw lines, and plenty more. Check out the documentation. For example, drawing a small circle at each branch point of the tree can be
accomplished by adding this line directly below line 18:
1
pygame.draw.circle(main_surface, color, (int(position[0]),
˓→int(position[1])), 3)
Another interesting effect — instructive too, if you wish to reinforce the idea of different instances of the function being called at different depths of recursion — is to create a list of
colors, and let each recursive depth use a different color for drawing. (Use the depth of the
recursion to index the list of colors.)
Mutual Recursion
In addition to a function calling just itself, it is also possible to make multiple functions that
call eachother. This is rarely really usefull, but it can be used to make state machines.
1
2
3
4
5
def function_a(n): # Do things associated with state A
if n == 0:
return
print('a')
function_b(n - 1) # Proceed to state B
6
7
8
def function_b(n):
C.7. Mutual Recursion
# Do things associated with state B
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print('b')
function_a(n - 1)
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10
# Proceed to state A
Glossary
base case A branch of the conditional statement in a recursive function that does not give rise
to further recursive calls.
infinite recursion A function that calls itself recursively without ever reaching any base case.
Eventually, infinite recursion causes a runtime error.
recursion The process of calling a function that is already executing.
recursive call The statement that calls an already executing function. Recursion can also be
indirect — function f can call g which calls h, and h could make a call back to f.
recursive definition A definition which defines something in terms of itself. To be useful it
must include base cases which are not recursive. In this way it differs from a circular
definition. Recursive definitions often provide an elegant way to express complex data
structures, like a directory that can contain other directories, or a menu that can contain
other menus.
Exercises
1. Modify the Koch fractal program so that it draws a Koch snowflake, like this:
2.
(a) Draw a Cesaro torn line fractal, of the order given by the user. We show four
different lines of orders 0,1,2,3. In this example, the angle of the tear is 10 degrees.
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(b) Four lines make a square. Use the code in part a) to draw cesaro squares. Varying
the angle gives interesting effects — experiment a bit, or perhaps let the user input
the angle of the tear.
(a) (For the mathematically inclined). In the squares shown here, the higher-order
drawings become a little larger. (Look at the bottom lines of each square - they’re
not aligned.) This is because we just halved the drawn part of the line for each recursive subproblem. So we’ve “grown” the overall square by the width of the tear(s).
Can you solve the geometry problem so that the total size of the subproblem case
(including the tear) remains exactly the same size as the original?
3. A Sierpinski triangle of order 0 is an equilateral triangle. An order 1 triangle can be
drawn by drawing 3 smaller triangles (shown slightly disconnected here, just to help our
understanding). Higher order 2 and 3 triangles are also shown. Draw Sierpinski triangles
of any order input by the user.
4. Adapt the above program to change the color of its three sub-triangles at some depth
of recursion. The illustration below shows two cases: on the left, the color is changed
at depth 0 (the outmost level of recursion), on the right, at depth 2. If the user supplies a negative depth, the color never changes. (Hint: add a new optional parameter
colorChangeDepth (which defaults to -1), and make this one smaller on each recurC.9. Exercises
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sive subcall. Then, in the section of code before you recurse, test whether the parameter
is zero, and change color.)
5. Write a function, recursive_min, that returns the smallest value in a nested number
list. Assume there are no empty lists or sublists:
6. Write a function count that returns the number of occurrences of target in a nested
list:
7. Write a function flatten that returns a simple list containing all the values in a nested
list:
8. Rewrite the fibonacci algorithm without using recursion. Can you find bigger terms of
the sequence? Can you find fib(200)?
9. Use help to find out what sys.getrecursionlimit() and sys.
setrecursionlimit(n) do. Create several experiments similar to what was
done in infinite_recursion.py to test your understanding of how these module functions
work.
10. Write a program that walks a directory structure (as in the last section of this chapter), but
instead of printing filenames, it returns a list of all the full paths of files in the directory
or the subdirectories. (Don’t include directories in this list — just files.) For example,
the output list might have elements like this:
["C:\Python31\Lib\site-packages\pygame\docs\ref\mask.html",
"C:\Python31\Lib\site-packages\pygame\docs\ref\midi.html",
...
"C:\Python31\Lib\site-packages\pygame\examples\aliens.py",
...
"C:\Python31\Lib\site-packages\pygame\examples\data\boom.wav",
... ]
11. Write a program named litter.py that creates an empty file named trash.txt in
each subdirectory of a directory tree given the root of the tree as an argument (or the
current directory as a default). Now write a program named cleanup.py that removes
all these files.
Hint #1: Use the program from the example in the last section of this chapter as a basis
for these two recursive programs. Because you’re going to destroy files on your disks,
you better get this right, or you risk losing files you care about. So excellent advice is
that initially you should fake the deletion of the files — just print the full path names
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of each file that you intend to delete. Once you’re happy that your logic is correct, and
you can see that you’re not deleting the wrong things, you can replace the print statement
with the real thing.
Hint #2: Look in the os module for a function that removes files.
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APPENDIX D
Classes and Objects
Classes and Objects — the Basics
Object-oriented programming
Python is an object-oriented programming language, which means that it provides features
that support object-oriented programming (OOP).
Object-oriented programming has its roots in the 1960s, but it wasn’t until the mid 1980s that it
became the main programming paradigm used in the creation of new software. It was developed
as a way to handle the rapidly increasing size and complexity of software systems, and to make
it easier to modify these large and complex systems over time.
Up to now, most of the programs we have been writing use a procedural programming
paradigm. In procedural programming the focus is on writing functions or procedures which
operate on data. In object-oriented programming the focus is on the creation of objects which
contain both data and functionality together. (We have seen turtle objects, string objects, and
random number generators, to name a few places where we’ve already worked with objects.)
Usually, each object definition corresponds to some object or concept in the real world, and the
functions that operate on that object correspond to the ways real-world objects interact.
User-defined compound data types
We’ve already seen classes like str, int, float and Turtle. We are now ready to create
our own user-defined class: the Point.
Consider the concept of a mathematical point. In two dimensions, a point is two numbers
(coordinates) that are treated collectively as a single object. Points are often written in between
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parentheses with a comma separating the coordinates. For example, (0, 0) represents the
origin, and (x, y) represents the point x units to the right and y units up from the origin.
Some of the typical operations that one associates with points might be calculating the distance
of a point from the origin, or from another point, or finding a midpoint of two points, or asking
if a point falls within a given rectangle or circle. We’ll shortly see how we can organize these
together with the data.
A natural way to represent a point in Python is with two numeric values. The question, then, is
how to group these two values into a compound object. The quick and dirty solution is to use a
tuple, and for some applications that might be a good choice.
An alternative is to define a new class. This approach involves a bit more effort, but it has
advantages that will be apparent soon. We’ll want our points to each have an x and a y attribute,
so our first class definition looks like this:
1
2
class Point:
""" Point class represents and manipulates x,y coords. "
˓→""
3
4
5
6
7
def __init__(self):
""" Create a new point at the origin """
self.x = 0
self.y = 0
Class definitions can appear anywhere in a program, but they are usually near the beginning
(after the import statements). Some programmers and languages prefer to put every class in a
module of its own — we won’t do that here. The syntax rules for a class definition are the same
as for other compound statements. There is a header which begins with the keyword, class,
followed by the name of the class, and ending with a colon. Indentation levels tell us where the
class ends.
If the first line after the class header is a string, it becomes the docstring of the class, and will
be recognized by various tools. (This is also the way docstrings work in functions.)
Every class should have a method with the special name __init__. This initializer method
is automatically called whenever a new instance of Point is created. It gives the programmer the opportunity to set up the attributes required within the new instance by giving them
their initial state/values. The self parameter (we could choose any other name, but self
is the convention) is automatically set to reference the newly created object that needs to be
initialized.
So let’s use our new Point class now:
1
2
p = Point()
q = Point()
# Instantiate an object of type Point
# Make a second point
3
4
print(p.x, p.y, q.x, q.y)
˓→x and y
# Each point object has its own
This program prints:
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0 0 0 0
because during the initialization of the objects, we created two attributes called x and y for
each, and gave them both the value 0.
This should look familiar — we’ve used classes before to create more than one object:
1
from turtle import Turtle
2
3
4
tess = Turtle()
alex = Turtle()
# Instantiate objects of type Turtle
The variables p and q are assigned references to two new Point objects. A function like
Turtle or Point that creates a new object instance is called a constructor, and every class
automatically provides a constructor function which is named the same as the class.
It may be helpful to think of a class as a factory for making objects. The class itself isn’t an
instance of a point, but it contains the machinery to make point instances. Every time we call
the constructor, we’re asking the factory to make us a new object. As the object comes off the
production line, its initialization method is executed to get the object properly set up with its
factory default settings.
The combined process of “make me a new object” and “get its settings initialized to the factory
default settings” is called instantiation.
Attributes
Like real world objects, object instances have both attributes and methods.
We can modify the attributes in an instance using dot notation:
>>> p.x = 3
>>> p.y = 4
Both modules and instances create their own namespaces, and the syntax for accessing names
contained in each, called attributes, is the same. In this case the attribute we are selecting is a
data item from an instance.
The following state diagram shows the result of these assignments:
The variable p refers to a Point object, which contains two attributes. Each attribute refers to
a number.
We can access the value of an attribute using the same syntax:
>>> print(p.y)
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>>> x = p.x
>>> print(x)
3
The expression p.x means, “Go to the object p refers to and get the value of x”. In this case,
we assign that value to a variable named x. There is no conflict between the variable x (in the
global namespace here) and the attribute x (in the namespace belonging to the instance). The
purpose of dot notation is to fully qualify which variable we are referring to unambiguously.
We can use dot notation as part of any expression, so the following statements are legal:
1
2
print("(x={0}, y={1})".format(p.x, p.y))
distance_squared_from_origin = p.x * p.x + p.y * p.y
The first line outputs (x=3, y=4). The second line calculates the value 25.
Improving our initializer
To create a point at position (7, 6) currently needs three lines of code:
1
2
3
p = Point()
p.x = 7
p.y = 6
We can make our class constructor more general by placing extra parameters into the
__init__ method, as shown in this example:
1
2
class Point:
""" Point class represents and manipulates x,y coords. "
˓→""
3
def __init__(self, x=0, y=0):
""" Create a new point at x, y """
self.x = x
self.y = y
4
5
6
7
8
9
# Other statements outside the class continue below here.
The x and y parameters here are both optional. If the caller does not supply arguments, they’ll
get the default values of 0. Here is our improved class in action:
>>>
>>>
>>>
>>>
4 3
p = Point(4, 2)
q = Point(6, 3)
r = Point()
# r represents the origin (0, 0)
print(p.x, q.y, r.x)
0
Technically speaking ...
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If we are really fussy, we would argue that the __init__ method’s docstring is inaccurate.
__init__ doesn’t create the object (i.e. set aside memory for it), — it just initializes the
object to its factory-default settings after its creation.
But tools like PyScripter understand that instantiation — creation and initialization — happen together, and they choose to display the initializer’s docstring as the tooltip to guide the
programmer that calls the class constructor.
So we’re writing the docstring so that it makes the most sense when it pops up to help the
programmer who is using our Point class:
Adding other methods to our class
The key advantage of using a class like Point rather than a simple tuple (6, 7) now becomes apparent. We can add methods to the Point class that are sensible operations for points,
but which may not be appropriate for other tuples like (25, 12) which might represent, say,
a day and a month, e.g. Christmas day. So being able to calculate the distance from the origin
is sensible for points, but not for (day, month) data. For (day, month) data, we’d like different
operations, perhaps to find what day of the week it will fall on in 2020.
Creating a class like Point brings an exceptional amount of “organizational power” to our
programs, and to our thinking. We can group together the sensible operations, and the kinds of
data they apply to, and each instance of the class can have its own state.
A method behaves like a function but it is invoked on a specific instance, e.g. tess.
right(90). Like a data attribute, methods are accessed using dot notation.
Let’s add another method, distance_from_origin, to see better how methods work:
1
2
class Point:
""" Create a new Point, at coordinates x, y """
3
4
5
6
7
def __init__(self, x=0, y=0):
""" Create a new point at x, y """
self.x = x
self.y = y
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9
10
11
def distance_from_origin(self):
""" Compute my distance from the origin """
return ((self.x ** 2) + (self.y ** 2)) ** 0.5
Let’s create a few point instances, look at their attributes, and call our new method on them:
(We must run our program first, to make our Point class available to the interpreter.)
>>> p = Point(3, 4)
>>> p.x
3
>>> p.y
4
>>> p.distance_from_origin()
5.0
>>> q = Point(5, 12)
>>> q.x
5
>>> q.y
12
>>> q.distance_from_origin()
13.0
>>> r = Point()
>>> r.x
0
>>> r.y
0
>>> r.distance_from_origin()
0.0
When defining a method, the first parameter refers to the instance being manipulated. As
already noted, it is customary to name this parameter self.
Notice that the caller of distance_from_origin does not explicitly supply an argument
to match the self parameter — this is done for us, behind our back.
Instances as arguments and parameters
We can pass an object as an argument in the usual way. We’ve already seen this in some
of the turtle examples, where we passed the turtle to some function like draw_bar in the
chapter titled Conditionals, so that the function could control and use whatever turtle instance
we passed to it.
Be aware that our variable only holds a reference to an object, so passing tess into a function
creates an alias: both the caller and the called function now have a reference, but there is only
one turtle!
Here is a simple function involving our new Point objects:
1
2
336
def print_point(pt):
print("({0}, {1})".format(pt.x, pt.y))
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print_point takes a point as an argument and formats the output in whichever way we
choose. If we call print_point(p) with point p as defined previously, the output is (3,
4).
Converting an instance to a string
Most object-oriented programmers probably would not do what we’ve just done in
print_point. When we’re working with classes and objects, a preferred alternative is to
add a new method to the class. And we don’t like chatterbox methods that call print. A
better approach is to have a method so that every instance can produce a string representation
of itself. Let’s initially call it to_string:
1
2
class Point:
# ...
3
4
5
def to_string(self):
return "({0}, {1})".format(self.x, self.y)
Now we can say:
>>> p = Point(3, 4)
>>> print(p.to_string())
(3, 4)
But don’t we already have a str type converter that can turn our object into a string? Yes! And
doesn’t print automatically use this when printing things? Yes again! But these automatic
mechanisms do not yet do exactly what we want:
>>> str(p)
'<__main__.Point object at 0x01F9AA10>'
>>> print(p)
'<__main__.Point object at 0x01F9AA10>'
Python has a clever trick up its sleeve to fix this. If we call our new method __str__ instead
of to_string, the Python interpreter will use our code whenever it needs to convert a Point
to a string. Let’s re-do this again, now:
1
2
class Point:
# ...
3
4
5
def __str__(self):
# All we have done is renamed
˓→the method
return "({0}, {1})".format(self.x, self.y)
and now things are looking great!
>>> str(p)
˓→wrote.
(3, 4)
# Python now uses the __str__ method that we
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>>> print(p)
(3, 4)
Instances as return values
Functions and methods can return instances. For example, given two Point objects, find their
midpoint. First we’ll write this as a regular function:
1
2
3
4
5
def midpoint(p1, p2):
""" Return the midpoint of points p1 and p2 """
mx = (p1.x + p2.x)/2
my = (p1.y + p2.y)/2
return Point(mx, my)
The function creates and returns a new Point object:
>>> p
>>> q
>>> r
>>> r
(4.0,
= Point(3, 4)
= Point(5, 12)
= midpoint(p, q)
8.0)
Now let us do this as a method instead. Suppose we have a point object, and wish to find the
midpoint halfway between it and some other target point:
1
2
class Point:
# ...
3
4
5
6
7
8
def halfway(self, target):
""" Return the halfway point between myself and the
˓→target """
mx = (self.x + target.x)/2
my = (self.y + target.y)/2
return Point(mx, my)
This method is identical to the function, aside from some renaming. It’s usage might be like
this:
>>> p
>>> q
>>> r
>>> r
(4.0,
= Point(3, 4)
= Point(5, 12)
= p.halfway(q)
8.0)
While this example assigns each point to a variable, this need not be done. Just as function
calls are composable, method calls and object instantiation are also composable, leading to this
alternative that uses no variables:
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>>> print(Point(3, 4).halfway(Point(5, 12)))
(4.0, 8.0)
A change of perspective
The original syntax for a function call, print_time(current_time), suggests that the
function is the active agent. It says something like, “Hey, print_time! Here’s an object for you
to print.”
In object-oriented programming, the objects are considered the active agents. An invocation
like current_time.print_time() says “Hey current_time! Please print yourself!”
In our early introduction to turtles, we used an object-oriented style, so that we said tess.
forward(100), which asks the turtle to move itself forward by the given number of steps.
This change in perspective might be more polite, but it may not initially be obvious that it
is useful. But sometimes shifting responsibility from the functions onto the objects makes it
possible to write more versatile functions, and makes it easier to maintain and reuse code.
The most important advantage of the object-oriented style is that it fits our mental chunking and
real-life experience more accurately. In real life our cook method is part of our microwave
oven — we don’t have a cook function sitting in the corner of the kitchen, into which we pass
the microwave! Similarly, we use the cellphone’s own methods to send an sms, or to change its
state to silent. The functionality of real-world objects tends to be tightly bound up inside the
objects themselves. OOP allows us to accurately mirror this when we organize our programs.
Objects can have state
Objects are most useful when we also need to keep some state that is updated from time to
time. Consider a turtle object. Its state consists of things like its position, its heading, its color,
and its shape. A method like left(90) updates the turtle’s heading, forward changes its
position, and so on.
For a bank account object, a main component of the state would be the current balance, and
perhaps a log of all transactions. The methods would allow us to query the current balance,
deposit new funds, or make a payment. Making a payment would include an amount, and a
description, so that this could be added to the transaction log. We’d also want a method to show
the transaction log.
Glossary
attribute One of the named data items that makes up an instance.
class A user-defined compound type. A class can also be thought of as a template for the
objects that are instances of it. (The iPhone is a class. By December 2010, estimates are
that 50 million instances had been sold!)
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constructor Every class has a “factory”, called by the same name as the class, for making new
instances. If the class has an initializer method, this method is used to get the attributes
(i.e. the state) of the new object properly set up.
initializer method A special method in Python (called __init__) that is invoked automatically to set a newly created object’s attributes to their initial (factory-default) state.
instance An object whose type is of some class. Instance and object are used interchangeably.
instantiate To create an instance of a class, and to run its initializer.
method A function that is defined inside a class definition and is invoked on instances of that
class.
object A compound data type that is often used to model a thing or concept in the real world.
It bundles together the data and the operations that are relevant for that kind of data.
Instance and object are used interchangeably.
object-oriented programming A powerful style of programming in which data and the operations that manipulate it are organized into objects.
object-oriented language A language that provides features, such as user-defined classes and
inheritance, that facilitate object-oriented programming.
Exercises
1. Rewrite the distance function from the chapter titled Fruitful functions so that it takes
two Points as parameters instead of four numbers.
2. Add a method reflect_x to Point which returns a new Point, one which is the
reflection of the point about the x-axis. For example, Point(3, 5).reflect_x()
is (3, -5)
3. Add a method slope_from_origin which returns the slope of the line joining the
origin to the point. For example,
>>> Point(4, 10).slope_from_origin()
2.5
What cases will cause this method to fail?
4. The equation of a straight line is “y = ax + b”, (or perhaps “y = mx + c”). The coefficients
a and b completely describe the line. Write a method in the Point class so that if a point
instance is given another point, it will compute the equation of the straight line joining
the two points. It must return the two coefficients as a tuple of two values. For example,
>>> print(Point(4, 11).get_line_to(Point(6, 15)))
>>> (2, 3)
This tells us that the equation of the line joining the two points is “y = 2x + 3”. When
will this method fail?
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5. Given four points that fall on the circumference of a circle, find the midpoint of the circle.
When will this function fail?
Hint: You must know how to solve the geometry problem before you think of going
anywhere near programming. You cannot program a solution to a problem if you don’t
understand what you want the computer to do!
6. Create a new class, SMS_store. The class will instantiate SMS_store objects, similar to
an inbox or outbox on a cellphone:
my_inbox = SMS_store()
This store can hold multiple SMS messages (i.e. its internal state will just be a list of
messages). Each message will be represented as a tuple:
(has_been_viewed, from_number, time_arrived, text_of_SMS)
The inbox object should provide these methods:
my_inbox.add_new_arrival(from_number, time_arrived, text_of_SMS)
# Makes new SMS tuple, inserts it after other messages
# in the store. When creating this message, its
# has_been_viewed status is set False.
my_inbox.message_count()
# Returns the number of sms messages in my_inbox
my_inbox.get_unread_indexes()
# Returns list of indexes of all not-yet-viewed SMS messages
my_inbox.get_message(i)
# Return (from_number, time_arrived, text_of_sms) for
˓→message[i]
# Also change its state to "has been viewed".
# If there is no message at position i, return None
my_inbox.delete(i)
my_inbox.clear()
# Delete the message at index i
# Delete all messages from inbox
Write the class, create a message store object, write tests for these methods, and implement the methods.
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Classes and Objects — Digging a little deeper
Rectangles
Let’s say that we want a class to represent a rectangle which is located somewhere in the XY
plane. The question is, what information do we have to provide in order to specify such a
rectangle? To keep things simple, assume that the rectangle is oriented either vertically or
horizontally, never at an angle.
There are a few possibilities: we could specify the center of the rectangle (two coordinates)
and its size (width and height); or we could specify one of the corners and the size; or we could
specify two opposing corners. A conventional choice is to specify the upper-left corner of the
rectangle, and the size.
Again, we’ll define a new class, and provide it with an initializer and a string converter method:
1
2
class Rectangle:
""" A class to manufacture rectangle objects """
3
4
5
6
7
8
def __init__(self, posn, w, h):
""" Initialize rectangle at posn, with width w,
˓→height h """
self.corner = posn
self.width = w
self.height = h
9
10
11
12
def __str__(self):
return "({0}, {1}, {2})"
.format(self.corner, self.width, self.
˓→height)
13
14
15
16
17
box = Rectangle(Point(0, 0), 100, 200)
bomb = Rectangle(Point(100, 80), 5, 10)
˓→game
print("box: ", box)
print("bomb: ", bomb)
# In my video
To specify the upper-left corner, we have embedded a Point object (as we used it in the previous chapter) within our new Rectangle object! We create two new Rectangle objects,
and then print them producing:
box: ((0, 0), 100, 200)
bomb: ((100, 80), 5, 10)
The dot operator composes. The expression box.corner.x means, “Go to the object that
box refers to and select its attribute named corner, then go to that object and select its
attribute named x”.
The figure shows the state of this object:
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Objects are mutable
We can change the state of an object by making an assignment to one of its attributes. For
example, to grow the size of a rectangle without changing its position, we could modify the
values of width and height:
box.width += 50
box.height += 100
Of course, we’d probably like to provide a method to encapsulate this inside the class. We will
also provide another method to move the position of the rectangle elsewhere:
1
2
class Rectangle:
# ...
3
4
5
6
7
def grow(self, delta_width, delta_height):
""" Grow (or shrink) this object by the deltas """
self.width += delta_width
self.height += delta_height
8
9
10
11
12
def move(self, dx, dy):
""" Move this object by the deltas """
self.corner.x += dx
self.corner.y += dy
Let us try this:
>>> r = Rectangle(Point(10,5), 100, 50)
>>> print(r)
((10, 5), 100, 50)
>>> r.grow(25, -10)
>>> print(r)
((10, 5), 125, 40)
>>> r.move(-10, 10)
print(r)
((0, 15), 125, 40)
Sameness
The meaning of the word “same” seems perfectly clear until we give it some thought, and then
we realize there is more to it than we initially expected.
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For example, if we say, “Alice and Bob have the same car”, we mean that her car and his are
the same make and model, but that they are two different cars. If we say, “Alice and Bob have
the same mother”, we mean that her mother and his are the same person.
When we talk about objects, there is a similar ambiguity. For example, if two Points are the
same, does that mean they contain the same data (coordinates) or that they are actually the same
object?
We’ve already seen the is operator in the chapter on lists, where we talked about aliases: it
allows us to find out if two references refer to the same object:
>>> p1 = Point(3, 4)
>>> p2 = Point(3, 4)
>>> p1 is p2
False
Even though p1 and p2 contain the same coordinates, they are not the same object. If we
assign p1 to p3, then the two variables are aliases of the same object:
>>> p3 = p1
>>> p1 is p3
True
This type of equality is called shallow equality because it compares only the references, not
the contents of the objects.
To compare the contents of the objects — deep equality — we can write a function called
same_coordinates:
1
2
def same_coordinates(p1, p2):
return (p1.x == p2.x) and (p1.y == p2.y)
Now if we create two different objects that contain the same data, we can use same_point
to find out if they represent points with the same coordinates.
>>> p1 = Point(3, 4)
>>> p2 = Point(3, 4)
>>> same_coordinates(p1, p2)
True
Of course, if the two variables refer to the same object, they have both shallow and deep equality.
Beware of ==
“When I use a word,” Humpty Dumpty said, in a rather scornful tone, “it means just what I
choose it to mean — neither more nor less.” Alice in Wonderland
Python has a powerful feature that allows a designer of a class to decide what an operation like
== or < should mean. (We’ve just shown how we can control how our own objects are converted
to strings, so we’ve already made a start!) We’ll cover more detail later. But sometimes the
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implementors will attach shallow equality semantics, and sometimes deep equality, as shown
in this little experiment:
1
2
3
4
p = Point(4, 2)
s = Point(4, 2)
print("== on Points returns", p == s)
# By default, == on Point objects does a shallow equality
˓→test
5
6
7
8
9
a = [2,3]
b = [2,3]
print("== on lists returns", a == b)
# But by default, == does a deep equality test on lists
This outputs:
== on Points returns False
== on lists returns True
So we conclude that even though the two lists (or tuples, etc.) are distinct objects with different
memory addresses, for lists the == operator tests for deep equality, while in the case of points
it makes a shallow test.
Copying
Aliasing can make a program difficult to read because changes made in one place might have
unexpected effects in another place. It is hard to keep track of all the variables that might refer
to a given object.
Copying an object is often an alternative to aliasing. The copy module contains a function
called copy that can duplicate any object:
>>> import copy
>>> p1 = Point(3, 4)
>>> p2 = copy.copy(p1)
>>> p1 is p2
False
>>> same_coordinates(p1, p2)
True
Once we import the copy module, we can use the copy function to make a new Point. p1
and p2 are not the same point, but they contain the same data.
To copy a simple object like a Point, which doesn’t contain any embedded objects, copy is
sufficient. This is called shallow copying.
For something like a Rectangle, which contains a reference to a Point, copy doesn’t do
quite the right thing. It copies the reference to the Point object, so both the old Rectangle
and the new one refer to a single Point.
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If we create a box, b1, in the usual way and then make a copy, b2, using copy, the resulting
state diagram looks like this:
This is almost certainly not what we want. In this case, invoking grow on one of the
Rectangle objects would not affect the other, but invoking move on either would affect
both! This behavior is confusing and error-prone. The shallow copy has created an alias to the
Point that represents the corner.
Fortunately, the copy module contains a function named deepcopy that copies not only the
object but also any embedded objects. It won’t be surprising to learn that this operation is called
a deep copy.
>>> b2 = copy.deepcopy(b1)
Now b1 and b2 are completely separate objects.
Glossary
deep copy To copy the contents of an object as well as any embedded objects, and any objects
embedded in them, and so on; implemented by the deepcopy function in the copy
module.
deep equality Equality of values, or two references that point to objects that have the same
value.
shallow copy To copy the contents of an object, including any references to embedded objects;
implemented by the copy function in the copy module.
shallow equality Equality of references, or two references that point to the same object.
Exercises
1. Add a method area to the Rectangle class that returns the area of any instance:
r = Rectangle(Point(0, 0), 10, 5)
test(r.area() == 50)
2. Write a perimeter method in the Rectangle class so that we can find the perimeter
of any rectangle instance:
r = Rectangle(Point(0, 0), 10, 5)
test(r.perimeter() == 30)
3. Write a flip method in the Rectangle class that swaps the width and the height of
any rectangle instance:
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r = Rectangle(Point(100, 50), 10, 5)
test(r.width == 10 and r.height == 5)
r.flip()
test(r.width == 5 and r.height == 10)
4. Write a new method in the Rectangle class to test if a Point falls within the rectangle. For this exercise, assume that a rectangle at (0,0) with width 10 and height 5 has
open upper bounds on the width and height, i.e. it stretches in the x direction from [0 to
10), where 0 is included but 10 is excluded, and from [0 to 5) in the y direction. So it
does not contain the point (10, 2). These tests should pass:
r = Rectangle(Point(0, 0), 10, 5)
test(r.contains(Point(0, 0)))
test(r.contains(Point(3, 3)))
test(not r.contains(Point(3, 7)))
test(not r.contains(Point(3, 5)))
test(r.contains(Point(3, 4.99999)))
test(not r.contains(Point(-3, -3)))
5. In games, we often put a rectangular “bounding box” around our sprites. (A sprite is an
object that can move about in the game, as we will see shortly.) We can then do collision
detection between, say, bombs and spaceships, by comparing whether their rectangles
overlap anywhere.
Write a function to determine whether two rectangles collide. Hint: this might be quite a
tough exercise! Think carefully about all the cases before you code.
Even more OOP
MyTime
As another example of a user-defined type, we’ll define a class called MyTime that records the
time of day. We’ll provide an __init__ method to ensure that every instance is created with
appropriate attributes and initialization. The class definition looks like this:
1
class MyTime:
2
3
4
˓→
5
6
7
def __init__(self, hrs=0, mins=0, secs=0):
""" Create a MyTime object initialized to hrs, mins,
secs """
self.hours = hrs
self.minutes = mins
self.seconds = secs
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We can instantiate a new MyTime object:
1
tim1 = MyTime(11, 59, 30)
The state diagram for the object looks like this:
We’ll leave it as an exercise for the readers to add a __str__ method so that MyTime objects
can print themselves decently.
Pure functions
In the next few sections, we’ll write two versions of a function called add_time, which
calculates the sum of two MyTime objects. They will demonstrate two kinds of functions:
pure functions and modifiers.
The following is a rough version of add_time:
1
2
3
4
5
6
def add_time(t1, t2):
h = t1.hours + t2.hours
m = t1.minutes + t2.minutes
s = t1.seconds + t2.seconds
sum_t = MyTime(h, m, s)
return sum_t
The function creates a new MyTime object and returns a reference to the new object. This is
called a pure function because it does not modify any of the objects passed to it as parameters
and it has no side effects, such as updating global variables, displaying a value, or getting user
input.
Here is an example of how to use this function. We’ll create two MyTime objects:
current_time, which contains the current time; and bread_time, which contains the
amount of time it takes for a breadmaker to make bread. Then we’ll use add_time to figure
out when the bread will be done.
>>> current_time = MyTime(9, 14, 30)
>>> bread_time = MyTime(3, 35, 0)
>>> done_time = add_time(current_time, bread_time)
>>> print(done_time)
12:49:30
The output of this program is 12:49:30, which is correct. On the other hand, there are cases
where the result is not correct. Can you think of one?
The problem is that this function does not deal with cases where the number of seconds or
minutes adds up to more than sixty. When that happens, we have to carry the extra seconds into
the minutes column or the extra minutes into the hours column.
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Here’s a better version of the function:
1
def add_time(t1, t2):
2
3
4
5
h = t1.hours + t2.hours
m = t1.minutes + t2.minutes
s = t1.seconds + t2.seconds
6
7
8
9
if s >= 60:
s -= 60
m += 1
10
11
12
13
if m >= 60:
m -= 60
h += 1
14
15
16
sum_t = MyTime(h, m, s)
return sum_t
This function is starting to get bigger, and still doesn’t work for all possible cases. Later we
will suggest an alternative approach that yields better code.
Modifiers
There are times when it is useful for a function to modify one or more of the objects it gets as
parameters. Usually, the caller keeps a reference to the objects it passes, so any changes the
function makes are visible to the caller. Functions that work this way are called modifiers.
increment, which adds a given number of seconds to a MyTime object, would be written
most naturally as a modifier. A rough draft of the function looks like this:
1
2
def increment(t, secs):
t.seconds += secs
3
4
5
6
if t.seconds >= 60:
t.seconds -= 60
t.minutes += 1
7
8
9
10
if t.minutes >= 60:
t.minutes -= 60
t.hours += 1
The first line performs the basic operation; the remainder deals with the special cases we saw
before.
Is this function correct? What happens if the parameter seconds is much greater than sixty?
In that case, it is not enough to carry once; we have to keep doing it until seconds is less than
sixty. One solution is to replace the if statements with while statements:
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1
2
def increment(t, seconds):
t.seconds += seconds
3
4
5
6
while t.seconds >= 60:
t.seconds -= 60
t.minutes += 1
7
8
9
10
while t.minutes >= 60:
t.minutes -= 60
t.hours += 1
This function is now correct when seconds is not negative, and when hours does not exceed 23,
but it is not a particularly good solution.
Converting increment to a method
Once again, OOP programmers would prefer to put functions that work with MyTime objects
into the MyTime class, so let’s convert increment to a method. To save space, we will leave
out previously defined methods, but you should keep them in your version:
1
2
class MyTime:
# Previous method definitions here...
3
4
5
def increment(self, seconds):
self.seconds += seconds
6
7
8
9
while self.seconds >= 60:
self.seconds -= 60
self.minutes += 1
10
11
12
13
while self.minutes >= 60:
self.minutes -= 60
self.hours += 1
The transformation is purely mechanical — we move the definition into the class definition and
(optionally) change the name of the first parameter to self, to fit with Python style conventions.
Now we can invoke increment using the syntax for invoking a method.
1
current_time.increment(500)
Again, the object on which the method is invoked gets assigned to the first parameter, self.
The second parameter, seconds gets the value 500.
An “Aha!” insight
Often a high-level insight into the problem can make the programming much easier.
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In this case, the insight is that a MyTime object is really a three-digit number in base 60! The
second component is the ones column, the minute component is the sixties column, and the
hour component is the thirty-six hundreds column.
When we wrote add_time and increment, we were effectively doing addition in base 60,
which is why we had to carry from one column to the next.
This observation suggests another approach to the whole problem — we can convert a MyTime
object into a single number and take advantage of the fact that the computer knows how to do
arithmetic with numbers. The following method is added to the MyTime class to convert any
instance into a corresponding number of seconds:
1
2
class MyTime:
# ...
3
4
5
6
7
8
def to_seconds(self):
""" Return the number of seconds represented
by this instance
"""
return self.hours * 3600 + self.minutes * 60 + self.
˓→seconds
Now, all we need is a way to convert from an integer back to a MyTime object. Supposing we
have tsecs seconds, some integer division and mod operators can do this for us:
1
2
3
4
hrs = tsecs // 3600
leftoversecs = tsecs % 3600
mins = leftoversecs // 60
secs = leftoversecs % 60
You might have to think a bit to convince yourself that this technique to convert from one base
to another is correct.
In OOP we’re really trying to wrap together the data and the operations that apply to it. So
we’d like to have this logic inside the MyTime class. A good solution is to rewrite the class
initializer so that it can cope with initial values of seconds or minutes that are outside the
normalized values. (A normalized time would be something like 3 hours 12 minutes and 20
seconds. The same time, but unnormalized could be 2 hours 70 minutes and 140 seconds.)
Let’s rewrite a more powerful initializer for MyTime:
1
2
class MyTime:
# ...
3
4
5
6
7
8
def __init__(self, hrs=0, mins=0, secs=0):
""" Create a new MyTime object initialized to hrs,
˓→mins, secs.
The values of mins and secs may be outside the
˓→range 0-59,
but the resulting MyTime object will be
˓→normalized.
"""
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9
# Calculate total seconds to represent
totalsecs = hrs*3600 + mins*60 + secs
self.hours = totalsecs // 3600
# Split in h,
10
11
12
˓→
13
14
15
m, s
leftoversecs = totalsecs % 3600
self.minutes = leftoversecs // 60
self.seconds = leftoversecs % 60
Now we can rewrite add_time like this:
1
2
3
def add_time(t1, t2):
secs = t1.to_seconds() + t2.to_seconds()
return MyTime(0, 0, secs)
This version is much shorter than the original, and it is much easier to demonstrate or reason
that it is correct.
Generalization
In some ways, converting from base 60 to base 10 and back is harder than just dealing with
times. Base conversion is more abstract; our intuition for dealing with times is better.
But if we have the insight to treat times as base 60 numbers and make the investment of writing
the conversions, we get a program that is shorter, easier to read and debug, and more reliable.
It is also easier to add features later. For example, imagine subtracting two MyTime objects to
find the duration between them. The naive approach would be to implement subtraction with
borrowing. Using the conversion functions would be easier and more likely to be correct.
Ironically, sometimes making a problem harder (or more general) makes the programming
easier, because there are fewer special cases and fewer opportunities for error.
Specialization versus Generalization
Computer Scientists are generally fond of specializing their types, while mathematicians often
take the opposite approach, and generalize everything.
What do we mean by this?
If we ask a mathematician to solve a problem involving weekdays, days of the century, playing
cards, time, or dominoes, their most likely response is to observe that all these objects can
be represented by integers. Playing cards, for example, can be numbered from 0 to 51. Days
within the century can be numbered. Mathematicians will say “These things are enumerable —
the elements can be uniquely numbered (and we can reverse this numbering to get back to the
original concept). So let’s number them, and confine our thinking to integers. Luckily, we have
powerful techniques and a good understanding of integers, and so our abstractions — the way
we tackle and simplify these problems — is to try to reduce them to problems about integers.”
Computer Scientists tend to do the opposite. We will argue that there are many integer operations that are simply not meaningful for dominoes, or for days of the century. So we’ll often
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define new specialized types, like MyTime, because we can restrict, control, and specialize the
operations that are possible. Object-oriented programming is particularly popular because it
gives us a good way to bundle methods and specialized data into a new type.
Both approaches are powerful problem-solving techniques. Often it may help to try to think
about the problem from both points of view — “What would happen if I tried to reduce everything to very few primitive types?”, versus “What would happen if this thing had its own
specialized type?”
Another example
The after function should compare two times, and tell us whether the first time is strictly
after the second, e.g.
>>> t1 = MyTime(10, 55, 12)
>>> t2 = MyTime(10, 48, 22)
>>> after(t1, t2)
True
# Is t1 after t2?
This is slightly more complicated because it operates on two MyTime objects, not just one.
But we’d prefer to write it as a method anyway — in this case, a method on the first argument:
1
2
class MyTime:
# Previous method definitions here...
3
4
5
6
7
8
9
def after(self, time2):
""" Return True if I am strictly greater than time2
˓→"""
if self.hours > time2.hours:
return True
if self.hours < time2.hours:
return False
10
11
12
13
14
15
16
if self.minutes > time2.minutes:
return True
if self.minutes < time2.minutes:
return False
if self.seconds > time2.seconds:
return True
17
18
return False
We invoke this method on one object and pass the other as an argument:
1
2
if current_time.after(done_time):
print("The bread will be done before it starts!")
We can almost read the invocation like English: If the current time is after the done time, then...
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The logic of the if statements deserve special attention here. Lines 11-18 will only be reached
if the two hour fields are the same. Similarly, the test at line 16 is only executed if both times
have the same hours and the same minutes.
Could we make this easier by using our “Aha!” insight and extra work from earlier, and reducing both times to integers? Yes, with spectacular results!
1
2
class MyTime:
# Previous method definitions here...
3
4
5
6
def after(self, time2):
""" Return True if I am strictly greater than time2
˓→"""
return self.to_seconds() > time2.to_seconds()
This is a great way to code this: if we want to tell if the first time is after the second time, turn
them both into integers and compare the integers.
Operator overloading
Some languages, including Python, make it possible to have different meanings for the same
operator when applied to different types. For example, + in Python means quite different things
for integers and for strings. This feature is called operator overloading.
It is especially useful when programmers can also overload the operators for their own userdefined types.
For example, to override the addition operator +, we can provide a method named __add__:
1
2
class MyTime:
# Previously defined methods here...
3
4
5
def __add__(self, other):
return MyTime(0, 0, self.to_seconds() + other.to_
˓→seconds())
As usual, the first parameter is the object on which the method is invoked. The second parameter is conveniently named other to distinguish it from self. To add two MyTime objects,
we create and return a new MyTime object that contains their sum.
Now, when we apply the + operator to MyTime objects, Python invokes the __add__ method
that we have written:
>>> t1 = MyTime(1, 15, 42)
>>> t2 = MyTime(3, 50, 30)
>>> t3 = t1 + t2
>>> print(t3)
05:06:12
The expression t1 + t2 is equivalent to t1.__add__(t2), but obviously more elegant.
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As an exercise, add a method __sub__(self, other) that overloads the subtraction operator, and try it out.
For the next couple of exercises we’ll go back to the Point class defined in our first chapter about objects, and overload some of its operators. Firstly, adding two points adds their
respective (x, y) coordinates:
1
2
class Point:
# Previously defined methods here...
3
4
5
def __add__(self, other):
return Point(self.x + other.x,
self.y + other.y)
There are several ways to override the behavior of the multiplication operator: by defining a
method named __mul__, or __rmul__, or both.
If the left operand of * is a Point, Python invokes __mul__, which assumes that the other
operand is also a Point. It computes the dot product of the two Points, defined according to
the rules of linear algebra:
1
2
def __mul__(self, other):
return self.x * other.x + self.y * other.y
If the left operand of * is a primitive type and the right operand is a Point, Python invokes
__rmul__, which performs scalar multiplication:
1
2
def __rmul__(self, other):
return Point(other * self.x,
other * self.y)
The result is a new Point whose coordinates are a multiple of the original coordinates. If
other is a type that cannot be multiplied by a floating-point number, then __rmul__ will
yield an error.
This example demonstrates both kinds of multiplication:
>>> p1 = Point(3, 4)
>>> p2 = Point(5, 7)
>>> print(p1 * p2)
43
>>> print(2 * p2)
(10, 14)
What happens if we try to evaluate p2 * 2? Since the first parameter is a Point, Python
invokes __mul__ with 2 as the second argument. Inside __mul__, the program tries to
access the x coordinate of other, which fails because an integer has no attributes:
>>> print(p2 * 2)
AttributeError: 'int' object has no attribute 'x'
Unfortunately, the error message is a bit opaque. This example demonstrates some of the
difficulties of object-oriented programming. Sometimes it is hard enough just to figure out
what code is running.
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Polymorphism
Most of the methods we have written only work for a specific type. When we create a new
object, we write methods that operate on that type.
But there are certain operations that we will want to apply to many types, such as the arithmetic
operations in the previous sections. If many types support the same set of operations, we can
write functions that work on any of those types.
For example, the multadd operation (which is common in linear algebra) takes three parameters; it multiplies the first two and then adds the third. We can write it in Python like this:
1
2
def multadd (x, y, z):
return x * y + z
This function will work for any values of x and y that can be multiplied and for any value of z
that can be added to the product.
We can invoke it with numeric values:
>>> multadd (3, 2, 1)
7
Or with Points:
>>> p1 = Point(3,
>>> p2 = Point(5,
>>> print(multadd
(11, 15)
>>> print(multadd
44
4)
7)
(2, p1, p2))
(p1, p2, 1))
In the first case, the Point is multiplied by a scalar and then added to another Point. In
the second case, the dot product yields a numeric value, so the third parameter also has to be a
numeric value.
A function like this that can take arguments with different types is called polymorphic.
As another example, consider the function front_and_back, which prints a list twice, forward and backward:
1
2
3
4
5
def front_and_back(front):
import copy
back = copy.copy(front)
back.reverse()
print(str(front) + str(back))
Because the reverse method is a modifier, we make a copy of the list before reversing it.
That way, this function doesn’t modify the list it gets as a parameter.
Here’s an example that applies front_and_back to a list:
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>>> my_list = [1, 2, 3, 4]
>>> front_and_back(my_list)
[1, 2, 3, 4][4, 3, 2, 1]
Of course, we intended to apply this function to lists, so it is not surprising that it works. What
would be surprising is if we could apply it to a Point.
To determine whether a function can be applied to a new type, we apply Python’s fundamental
rule of polymorphism, called the duck typing rule: If all of the operations inside the function
can be applied to the type, the function can be applied to the type. The operations in the
front_and_back function include copy, reverse, and print.
Not all programming languages define polymorphism in this way. Look up duck typing, and
see if you can figure out why it has this name.
copy works on any object, and we have already written a __str__ method for Point objects, so all we need is a reverse method in the Point class:
1
2
def reverse(self):
(self.x , self.y) = (self.y, self.x)
Then we can pass Points to front_and_back:
>>> p = Point(3, 4)
>>> front_and_back(p)
(3, 4)(4, 3)
The most interesting polymorphism is the unintentional kind, where we discover that a function
we have already written can be applied to a type for which we never planned.
Glossary
dot product An operation defined in linear algebra that multiplies two Points and yields a
numeric value.
functional programming style A style of program design in which the majority of functions
are pure.
modifier A function or method that changes one or more of the objects it receives as parameters. Most modifier functions are void (do not return a value).
normalized Data is said to be normalized if it fits into some reduced range or set of rules.
We usually normalize our angles to values in the range [0..360). We normalize minutes
and seconds to be values in the range [0..60). And we’d be surprised if the local store
advertised its cold drinks at “One dollar, two hundred and fifty cents”.
operator overloading Extending built-in operators ( +, -, *, >, <, etc.) so that they do
different things for different types of arguments. We’ve seen early in the book how + is
overloaded for numbers and strings, and here we’ve shown how to further overload it for
user-defined types.
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polymorphic A function that can operate on more than one type. Notice the subtle distinction:
overloading has different functions (all with the same name) for different types, whereas
a polymorphic function is a single function that can work for a range of types.
pure function A function that does not modify any of the objects it receives as parameters.
Most pure functions are fruitful rather than void.
scalar multiplication An operation defined in linear algebra that multiplies each of the coordinates of a Point by a numeric value.
Exercises
1. Write a Boolean function between that takes two MyTime objects, t1 and t2, as
arguments, and returns True if the invoking object falls between the two times. Assume
t1 <= t2, and make the test closed at the lower bound and open at the upper bound,
i.e. return True if t1 <= obj < t2.
2. Turn the above function into a method in the MyTime class.
3. Overload the necessary operator(s) so that instead of having to write
if t1.after(t2): ...
we can use the more convenient
if t1 > t2: ...
4. Rewrite increment as a method that uses our “Aha” insight.
5. Create some test cases for the increment method. Consider specifically the case where
the number of seconds to add to the time is negative. Fix up increment so that it
handles this case if it does not do so already. (You may assume that you will never
subtract more seconds than are in the time object.)
6. Can physical time be negative, or must time always move in the forward direction? Some
serious physicists think this is not such a dumb question. See what you can find on the
Internet about this.
Collections of objects
Composition
By now, we have seen several examples of composition. One of the first examples was using a
method invocation as part of an expression. Another example is the nested structure of statements; we can put an if statement within a while loop, within another if statement, and so
on.
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Having seen this pattern, and having learned about lists and objects, we should not be surprised
to learn that we can create lists of objects. We can also create objects that contain lists (as
attributes); we can create lists that contain lists; we can create objects that contain objects; and
so on.
In this chapter and the next, we will look at some examples of these combinations, using Card
objects as an example.
Card objects
If you are not familiar with common playing cards, now would be a good time to get a deck, or
else this chapter might not make much sense. There are fifty-two cards in a deck, each of which
belongs to one of four suits and one of thirteen ranks. The suits are Spades, Hearts, Diamonds,
and Clubs (in descending order in bridge). The ranks are Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack,
Queen, and King. Depending on the game that we are playing, the rank of Ace may be higher
than King or lower than 2. The rank is sometimes called the face-value of the card.
If we want to define a new object to represent a playing card, it is obvious what the attributes
should be: rank and suit. It is not as obvious what type the attributes should be. One
possibility is to use strings containing words like "Spade" for suits and "Queen" for ranks.
One problem with this implementation is that it would not be easy to compare cards to see
which had a higher rank or suit.
An alternative is to use integers to encode the ranks and suits. By encode, we do not mean
what some people think, which is to encrypt or translate into a secret code. What a computer
scientist means by encode is to define a mapping between a sequence of numbers and the items
I want to represent. For example:
Spades
Hearts
Diamonds
Clubs
-->
-->
-->
-->
3
2
1
0
An obvious feature of this mapping is that the suits map to integers in order, so we can compare
suits by comparing integers. The mapping for ranks is fairly obvious; each of the numerical
ranks maps to the corresponding integer, and for face cards:
Jack
Queen
King
-->
-->
-->
11
12
13
The reason we are using mathematical notation for these mappings is that they are not part of
the Python program. They are part of the program design, but they never appear explicitly in
the code. The class definition for the Card type looks like this:
1
2
3
4
class Card:
def __init__(self, suit=0, rank=0):
self.suit = suit
self.rank = rank
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As usual, we provide an initialization method that takes an optional parameter for each attribute.
To create some objects, representing say the 3 of Clubs and the Jack of Diamonds, use these
commands:
1
2
three_of_clubs = Card(0, 3)
card1 = Card(1, 11)
In the first case above, for example, the first argument, 0, represents the suit Clubs.
Save this code for later use ...
In the next chapter we assume that we have save the Cards class, and the upcoming Deck
class in a file called Cards.py.
Class attributes and the __str__ method
In order to print Card objects in a way that people can easily read, we want to map the integer
codes onto words. A natural way to do that is with lists of strings. We assign these lists to class
attributes at the top of the class definition:
1
2
3
4
class Card:
suits = ["Clubs", "Diamonds", "Hearts", "Spades"]
ranks = ["narf", "Ace", "2", "3", "4", "5", "6", "7",
"8", "9", "10", "Jack", "Queen", "King"]
5
6
7
8
def __init__(self, suit=0, rank=0):
self.suit = suit
self.rank = rank
9
10
11
def __str__(self):
return (self.ranks[self.rank] + " of " + self.
˓→suits[self.suit])
A class attribute is defined outside of any method, and it can be accessed from any of the
methods in the class.
Inside __str__, we can use suits and ranks to map the numerical values of suit and
rank to strings. For example, the expression self.suits[self.suit] means use the
attribute suit from the object self as an index into the class attribute named suits, and
select the appropriate string.
The reason for the "narf" in the first element in ranks is to act as a place keeper for the zeroeth element of the list, which will never be used. The only valid ranks are 1 to 13. This wasted
item is not entirely necessary. We could have started at 0, as usual, but it is less confusing to
encode the rank 2 as integer 2, 3 as 3, and so on.
With the methods we have so far, we can create and print cards:
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>>> card1 = Card(1, 11)
>>> print(card1)
Jack of Diamonds
Class attributes like suits are shared by all Card objects. The advantage of this is that we
can use any Card object to access the class attributes:
>>> card2 = Card(1, 3)
>>> print(card2)
3 of Diamonds
>>> print(card2.suits[1])
Diamonds
Because every Card instance references the same class attribute, we have an aliasing situation.
The disadvantage is that if we modify a class attribute, it affects every instance of the class. For
example, if we decide that Jack of Diamonds should really be called Jack of Swirly Whales,
we could do this:
>>> card1.suits[1] = "Swirly Whales"
>>> print(card1)
Jack of Swirly Whales
The problem is that all of the Diamonds just became Swirly Whales:
>>> print(card2)
3 of Swirly Whales
It is usually not a good idea to modify class attributes.
Comparing cards
For primitive types, there are six relational operators ( <, >, ==, etc.) that compare values
and determine when one is greater than, less than, or equal to another. If we want our own
types to be comparable using the syntax of these relational operators, we need to define six
corresponding special methods in our class.
We’d like to start with a single method named cmp that houses the logic of ordering. By
convention, a comparison method takes two parameters, self and other, and returns 1 if the
first object is greater, -1 if the second object is greater, and 0 if they are equal to each other.
Some types are completely ordered, which means that we can compare any two elements and
tell which is bigger. For example, the integers and the floating-point numbers are completely
ordered. Some types are unordered, which means that there is no meaningful way to say that
one element is bigger than another. For example, the fruits are unordered, which is why we
cannot compare apples and oranges, and we cannot meaningfully order a collection of images,
or a collection of cellphones.
Playing cards are partially ordered, which means that sometimes we can compare cards and
sometimes not. For example, we know that the 3 of Clubs is higher than the 2 of Clubs, and
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the 3 of Diamonds is higher than the 3 of Clubs. But which is better, the 3 of Clubs or the 2 of
Diamonds? One has a higher rank, but the other has a higher suit.
In order to make cards comparable, we have to decide which is more important, rank or suit.
To be honest, the choice is arbitrary. For the sake of choosing, we will say that suit is more
important, because a new deck of cards comes sorted with all the Clubs together, followed by
all the Diamonds, and so on.
With that decided, we can write cmp:
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2
3
4
5
6
7
8
9
def cmp(self, other):
# Check the suits
if self.suit > other.suit: return 1
if self.suit < other.suit: return -1
# Suits are the same... check ranks
if self.rank > other.rank: return 1
if self.rank < other.rank: return -1
# Ranks are the same... it's a tie
return 0
In this ordering, Aces appear lower than Deuces (2s).
Now, we can define the six special methods that do the overloading of each of the relational
operators for us:
1
2
def __eq__(self, other):
return self.cmp(other) == 0
3
4
5
def __le__(self, other):
return self.cmp(other) <= 0
6
7
8
def __ge__(self, other):
return self.cmp(other) >= 0
9
10
11
def __gt__(self, other):
return self.cmp(other) > 0
12
13
14
def __lt__(self, other):
return self.cmp(other) < 0
15
16
17
def __ne__(self, other):
return self.cmp(other) != 0
With this machinery in place, the relational operators now work as we’d like them to:
>>> card1
>>> card2
>>> card3
>>> card1
False
>>> card1
True
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=
=
=
<
Card(1, 11)
Card(1, 3)
Card(1, 11)
card2
== card3
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Decks
Now that we have objects to represent Cards, the next logical step is to define a class to
represent a Deck. Of course, a deck is made up of cards, so each Deck object will contain a
list of cards as an attribute. Many card games will need at least two different decks — a red
deck and a blue deck.
The following is a class definition for the Deck class. The initialization method creates the
attribute cards and generates the standard pack of fifty-two cards:
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2
3
4
5
6
class Deck:
def __init__(self):
self.cards = []
for suit in range(4):
for rank in range(1, 14):
self.cards.append(Card(suit, rank))
The easiest way to populate the deck is with a nested loop. The outer loop enumerates the suits
from 0 to 3. The inner loop enumerates the ranks from 1 to 13. Since the outer loop iterates
four times, and the inner loop iterates thirteen times, the total number of times the body is
executed is fifty-two (thirteen times four). Each iteration creates a new instance of Card with
the current suit and rank, and appends that card to the cards list.
With this in place, we can instantiate some decks:
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2
red_deck = Deck()
blue_deck = Deck()
Printing the deck
As usual, when we define a new type we want a method that prints the contents of an instance.
To print a Deck, we traverse the list and print each Card:
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2
3
4
5
class Deck:
...
def print_deck(self):
for card in self.cards:
print(card)
Here, and from now on, the ellipsis (...) indicates that we have omitted the other methods in
the class.
As an alternative to print_deck, we could write a __str__ method for the Deck class.
The advantage of __str__ is that it is more flexible. Rather than just printing the contents of
the object, it generates a string representation that other parts of the program can manipulate
before printing, or store for later use.
Here is a version of __str__ that returns a string representation of a Deck. To add a bit of
pizzazz, it arranges the cards in a cascade where each card is indented one space more than the
previous card:
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1
2
3
4
5
6
7
class Deck:
...
def __str__(self):
s = ""
for i in range(len(self.cards)):
s = s + " " * i + str(self.cards[i]) + "\n"
return s
This example demonstrates several features. First, instead of traversing self.cards and
assigning each card to a variable, we are using i as a loop variable and an index into the list of
cards.
Second, we are using the string multiplication operator to indent each card by one more space
than the last. The expression " " * i yields a number of spaces equal to the current value of
i.
Third, instead of using the print command to print the cards, we use the str function.
Passing an object as an argument to str is equivalent to invoking the __str__ method on
the object.
Finally, we are using the variable s as an accumulator. Initially, s is the empty string. Each
time through the loop, a new string is generated and concatenated with the old value of s to get
the new value. When the loop ends, s contains the complete string representation of the Deck,
which looks like this:
>>> red_deck = Deck()
>>> print(red_deck)
Ace of Clubs
2 of Clubs
3 of Clubs
4 of Clubs
5 of Clubs
6 of Clubs
7 of Clubs
8 of Clubs
9 of Clubs
10 of Clubs
Jack of Clubs
Queen of Clubs
King of Clubs
Ace of Diamonds
2 of Diamonds
...
And so on. Even though the result appears on 52 lines, it is one long string that contains
newlines.
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Shuffling the deck
If a deck is perfectly shuffled, then any card is equally likely to appear anywhere in the deck,
and any location in the deck is equally likely to contain any card.
To shuffle the deck, we will use the randrange function from the random module. With
two integer arguments, a and b, randrange chooses a random integer in the range a <= x
< b. Since the upper bound is strictly less than b, we can use the length of a list as the second
parameter, and we are guaranteed to get a legal index. For example, if rng has already been
instantiated as a random number source, this expression chooses the index of a random card in
a deck:
1
rng.randrange(0, len(self.cards))
An easy way to shuffle the deck is by traversing the cards and swapping each card with a
randomly chosen one. It is possible that the card will be swapped with itself, but that is fine. In
fact, if we precluded that possibility, the order of the cards would be less than entirely random:
1
2
3
4
5
6
7
8
9
class Deck:
...
def shuffle(self):
import random
rng = random.Random()
# Create a random
˓→generator
num_cards = len(self.cards)
for i in range(num_cards):
j = rng.randrange(i, num_cards)
(self.cards[i], self.cards[j]) = (self.cards[j],
˓→ self.cards[i])
Rather than assume that there are fifty-two cards in the deck, we get the actual length of the list
and store it in num_cards.
For each card in the deck, we choose a random card from among the cards that haven’t been
shuffled yet. Then we swap the current card (i) with the selected card (j). To swap the cards
we use a tuple assignment:
1
(self.cards[i], self.cards[j]) = (self.cards[j], self.
˓→cards[i])
While this is a good shuffling method, a random number generator object also has a shuffle
method that can shuffle elements in a list, in place. So we could rewrite this function to use the
one provided for us:
1
2
3
4
5
6
class Deck:
...
def shuffle(self):
import random
rng = random.Random()
˓→generator
rng.shuffle(self.cards)
˓→method
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# Create a random
# uUse its shuffle
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Removing and dealing cards
Another method that would be useful for the Deck class is remove, which takes a card as a
parameter, removes it, and returns True if the card was in the deck and False otherwise:
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2
3
4
5
6
7
8
class Deck:
...
def remove(self, card):
if card in self.cards:
self.cards.remove(card)
return True
else:
return False
The in operator returns True if the first operand is in the second. If the first operand is
an object, Python uses the object’s __eq__ method to determine equality with items in the
list. Since the __eq__ we provided in the Card class checks for deep equality, the remove
method checks for deep equality.
To deal cards, we want to remove and return the top card. The list method pop provides a
convenient way to do that:
1
2
3
4
class Deck:
...
def pop(self):
return self.cards.pop()
Actually, pop removes the last card in the list, so we are in effect dealing from the bottom of
the deck.
One more operation that we are likely to want is the Boolean function is_empty, which
returns True if the deck contains no cards:
1
2
3
4
class Deck:
...
def is_empty(self):
return self.cards == []
Glossary
encode To represent one type of value using another type of value by constructing a mapping
between them.
class attribute A variable that is defined inside a class definition but outside any method.
Class attributes are accessible from any method in the class and are shared by all instances
of the class.
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accumulator A variable used in a loop to accumulate a series of values, such as by concatenating them onto a string or adding them to a running sum.
Exercises
1. Modify cmp so that Aces are ranked higher than Kings.
Inheritance
Inheritance
The language feature most often associated with object-oriented programming is inheritance.
Inheritance is the ability to define a new class that is a modified version of an existing class.
The primary advantage of this feature is that you can add new methods to a class without
modifying the existing class. It is called inheritance because the new class inherits all of the
methods of the existing class. Extending this metaphor, the existing class is sometimes called
the parent class. The new class may be called the child class or sometimes subclass.
Inheritance is a powerful feature. Some programs that would be complicated without inheritance can be written concisely and simply with it. Also, inheritance can facilitate code reuse,
since you can customize the behavior of parent classes without having to modify them. In some
cases, the inheritance structure reflects the natural structure of the problem, which makes the
program easier to understand.
On the other hand, inheritance can make programs difficult to read. When a method is invoked,
it is sometimes not clear where to find its definition. The relevant code may be scattered among
several modules. Also, many of the things that can be done using inheritance can be done as
elegantly (or more so) without it. If the natural structure of the problem does not lend itself to
inheritance, this style of programming can do more harm than good.
In this chapter we will demonstrate the use of inheritance as part of a program that plays the
card game Old Maid. One of our goals is to write code that could be reused to implement other
card games.
A hand of cards
For almost any card game, we need to represent a hand of cards. A hand is similar to a deck,
of course. Both are made up of a set of cards, and both require operations like adding and
removing cards. Also, we might like the ability to shuffle both decks and hands.
A hand is also different from a deck. Depending on the game being played, we might want
to perform some operations on hands that don’t make sense for a deck. For example, in poker
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we might classify a hand (straight, flush, etc.) or compare it with another hand. In bridge, we
might want to compute a score for a hand in order to make a bid.
This situation suggests the use of inheritance. If Hand is a subclass of Deck, it will have all
the methods of Deck, and new methods can be added.
We add the code in this chapter to our Cards.py file from the previous chapter. In the class
definition, the name of the parent class appears in parentheses:
1
2
class Hand(Deck):
pass
This statement indicates that the new Hand class inherits from the existing Deck class.
The Hand constructor initializes the attributes for the hand, which are name and cards. The
string name identifies this hand, probably by the name of the player that holds it. The name is
an optional parameter with the empty string as a default value. cards is the list of cards in the
hand, initialized to the empty list:
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2
3
4
class Hand(Deck):
def __init__(self, name=""):
self.cards = []
self.name = name
For just about any card game, it is necessary to add and remove cards from the deck. Removing
cards is already taken care of, since Hand inherits remove from Deck. But we have to write
add:
1
2
3
4
class Hand(Deck):
...
def add(self, card):
self.cards.append(card)
Again, the ellipsis indicates that we have omitted other methods. The list append method
adds the new card to the end of the list of cards.
Dealing cards
Now that we have a Hand class, we want to deal cards from the Deck into hands. It is not
immediately obvious whether this method should go in the Hand class or in the Deck class,
but since it operates on a single deck and (possibly) several hands, it is more natural to put it in
Deck.
deal should be fairly general, since different games will have different requirements. We may
want to deal out the entire deck at once or add one card to each hand.
deal takes two parameters, a list (or tuple) of hands and the total number of cards to deal. If
there are not enough cards in the deck, the method deals out all of the cards and stops:
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2
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4
5
6
7
8
9
10
def deal(self, hands, num_cards=999):
num_hands = len(hands)
for i in range(num_cards):
if self.is_empty():
break
#
˓→cards
card = self.pop()
#
˓→card
hand = hands[i % num_hands] #
˓→next?
hand.add(card)
#
˓→the hand
Break if out of
Take the top
Whose turn is
Add the card to
The second parameter, num_cards, is optional; the default is a large number, which effectively means that all of the cards in the deck will get dealt.
The loop variable i goes from 0 to num_cards-1. Each time through the loop, a card is
removed from the deck using the list method pop, which removes and returns the last item in
the list.
The modulus operator (%) allows us to deal cards in a round robin (one card at a time to each
hand). When i is equal to the number of hands in the list, the expression i % num_hands
wraps around to the beginning of the list (index 0).
Printing a Hand
To print the contents of a hand, we can take advantage of the __str__ method inherited from
Deck. For example:
>>> deck = Deck()
>>> deck.shuffle()
>>> hand = Hand("frank")
>>> deck.deal([hand], 5)
>>> print(hand)
Hand frank contains
2 of Spades
3 of Spades
4 of Spades
Ace of Hearts
9 of Clubs
It’s not a great hand, but it has the makings of a straight flush.
Although it is convenient to inherit the existing methods, there is additional information in
a Hand object we might want to include when we print one. To do that, we can provide a
__str__ method in the Hand class that overrides the one in the Deck class:
1
2
3
class Hand(Deck)
...
def __str__(self):
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4
5
6
7
8
9
s = "Hand " + self.name
if self.is_empty():
s += " is empty\n"
else:
s += " contains\n"
return s + Deck.__str__(self)
Initially, s is a string that identifies the hand. If the hand is empty, the program appends the
words is empty and returns s.
Otherwise, the program appends the word contains and the string representation of the
Deck, computed by invoking the __str__ method in the Deck class on self.
It may seem odd to send self, which refers to the current Hand, to a Deck method, until you
remember that a Hand is a kind of Deck. Hand objects can do everything Deck objects can,
so it is legal to send a Hand to a Deck method.
In general, it is always legal to use an instance of a subclass in place of an instance of a parent
class.
The CardGame class
The CardGame class takes care of some basic chores common to all games, such as creating
the deck and shuffling it:
1
2
3
4
class CardGame:
def __init__(self):
self.deck = Deck()
self.deck.shuffle()
This is the first case we have seen where the initialization method performs a significant computation, beyond initializing attributes.
To implement specific games, we can inherit from CardGame and add features for the new
game. As an example, we’ll write a simulation of Old Maid.
The object of Old Maid is to get rid of cards in your hand. You do this by matching cards by
rank and color. For example, the 4 of Clubs matches the 4 of Spades since both suits are black.
The Jack of Hearts matches the Jack of Diamonds since both are red.
To begin the game, the Queen of Clubs is removed from the deck so that the Queen of Spades
has no match. The fifty-one remaining cards are dealt to the players in a round robin. After the
deal, all players match and discard as many cards as possible.
When no more matches can be made, play begins. In turn, each player picks a card (without
looking) from the closest neighbor to the left who still has cards. If the chosen card matches a
card in the player’s hand, the pair is removed. Otherwise, the card is added to the player’s hand.
Eventually all possible matches are made, leaving only the Queen of Spades in the loser’s hand.
In our computer simulation of the game, the computer plays all hands. Unfortunately, some
nuances of the real game are lost. In a real game, the player with the Old Maid goes to some
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effort to get their neighbor to pick that card, by displaying it a little more prominently, or
perhaps failing to display it more prominently, or even failing to fail to display that card more
prominently. The computer simply picks a neighbor’s card at random.
OldMaidHand class
A hand for playing Old Maid requires some abilities beyond the general abilities of a Hand. We
will define a new class, OldMaidHand, that inherits from Hand and provides an additional
method called remove_matches:
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2
3
4
5
6
7
8
9
10
11
12
13
class OldMaidHand(Hand):
def remove_matches(self):
count = 0
original_cards = self.cards[:]
for card in original_cards:
match = Card(3 - card.suit, card.rank)
if match in self.cards:
self.cards.remove(card)
self.cards.remove(match)
print("Hand {0}: {1} matches {2}"
.format(self.name, card, match))
count += 1
return count
We start by making a copy of the list of cards, so that we can traverse the copy while removing
cards from the original. Since self.cards is modified in the loop, we don’t want to use it
to control the traversal. Python can get quite confused if it is traversing a list that is changing!
For each card in the hand, we figure out what the matching card is and go looking for it. The
match card has the same rank and the other suit of the same color. The expression 3 - card.
suit turns a Club (suit 0) into a Spade (suit 3) and a Diamond (suit 1) into a Heart (suit 2).
You should satisfy yourself that the opposite operations also work. If the match card is also in
the hand, both cards are removed.
The following example demonstrates how to use remove_matches:
>>> game = CardGame()
>>> hand = OldMaidHand("frank")
>>> game.deck.deal([hand], 13)
>>> print(hand)
Hand frank contains
Ace of Spades
2 of Diamonds
7 of Spades
8 of Clubs
6 of Hearts
8 of Spades
7 of Clubs
Queen of Clubs
7 of Diamonds
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5 of Clubs
Jack of Diamonds
10 of Diamonds
10 of Hearts
>>> hand.remove_matches()
Hand frank: 7 of Spades matches 7 of Clubs
Hand frank: 8 of Spades matches 8 of Clubs
Hand frank: 10 of Diamonds matches 10 of Hearts
>>> print(hand)
Hand frank contains
Ace of Spades
2 of Diamonds
6 of Hearts
Queen of Clubs
7 of Diamonds
5 of Clubs
Jack of Diamonds
Notice that there is no __init__ method for the OldMaidHand class. We inherit it from
Hand.
OldMaidGame class
Now we can turn our attention to the game itself. OldMaidGame is a subclass of CardGame
with a new method called play that takes a list of players as a parameter.
Since __init__ is inherited from CardGame, a new OldMaidGame object contains a new
shuffled deck:
1
2
3
4
class OldMaidGame(CardGame):
def play(self, names):
# Remove Queen of Clubs
self.deck.remove(Card(0,12))
5
6
7
8
9
# Make a hand for each player
self.hands = []
for name in names:
self.hands.append(OldMaidHand(name))
10
11
12
13
14
# Deal the cards
self.deck.deal(self.hands)
print("---------- Cards have been dealt")
self.print_hands()
15
16
17
18
19
# Remove initial matches
matches = self.remove_all_matches()
print("---------- Matches discarded, play begins")
self.print_hands()
20
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21
22
23
24
25
26
# Play until all 50 cards are matched
turn = 0
num_hands = len(self.hands)
while matches < 25:
matches += self.play_one_turn(turn)
turn = (turn + 1) % num_hands
27
28
29
print("---------- Game is Over")
self.print_hands()
The writing of print_hands has been left as an exercise.
Some of the steps of the game have been separated into methods. remove_all_matches
traverses the list of hands and invokes remove_matches on each:
1
2
3
4
5
6
7
class OldMaidGame(CardGame):
...
def remove_all_matches(self):
count = 0
for hand in self.hands:
count += hand.remove_matches()
return count
count is an accumulator that adds up the number of matches in each hand. When we’ve gone
through every hand, the total is returned (count).
When the total number of matches reaches twenty-five, fifty cards have been removed from the
hands, which means that only one card is left and the game is over.
The variable turn keeps track of which player’s turn it is. It starts at 0 and increases by one
each time; when it reaches num_hands, the modulus operator wraps it back around to 0.
The method play_one_turn takes a parameter that indicates whose turn it is. The return
value is the number of matches made during this turn:
1
2
3
4
5
6
7
8
9
10
11
12
class OldMaidGame(CardGame):
...
def play_one_turn(self, i):
if self.hands[i].is_empty():
return 0
neighbor = self.find_neighbor(i)
picked_card = self.hands[neighbor].pop()
self.hands[i].add(picked_card)
print("Hand", self.hands[i].name, "picked", picked_
˓→card)
count = self.hands[i].remove_matches()
self.hands[i].shuffle()
return count
If a player’s hand is empty, that player is out of the game, so he or she does nothing and returns
0.
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Otherwise, a turn consists of finding the first player on the left that has cards, taking one card
from the neighbor, and checking for matches. Before returning, the cards in the hand are
shuffled so that the next player’s choice is random.
The method find_neighbor starts with the player to the immediate left and continues
around the circle until it finds a player that still has cards:
1
2
3
4
5
6
7
8
class OldMaidGame(CardGame):
...
def find_neighbor(self, i):
num_hands = len(self.hands)
for next in range(1,num_hands):
neighbor = (i + next) % num_hands
if not self.hands[neighbor].is_empty():
return neighbor
If find_neighbor ever went all the way around the circle without finding cards, it would
return None and cause an error elsewhere in the program. Fortunately, we can prove that that
will never happen (as long as the end of the game is detected correctly).
We have omitted the print_hands method. You can write that one yourself.
The following output is from a truncated form of the game where only the top fifteen cards (tens
and higher) were dealt to three players. With this small deck, play stops after seven matches
instead of twenty-five.
>>> import cards
>>> game = cards.OldMaidGame()
>>> game.play(["Allen","Jeff","Chris"])
---------- Cards have been dealt
Hand Allen contains
King of Hearts
Jack of Clubs
Queen of Spades
King of Spades
10 of Diamonds
Hand Jeff contains
Queen of Hearts
Jack of Spades
Jack of Hearts
King of Diamonds
Queen of Diamonds
Hand Chris contains
Jack of Diamonds
King of Clubs
10 of Spades
10 of Hearts
10 of Clubs
Hand Jeff: Queen of Hearts matches Queen of Diamonds
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Hand Chris: 10 of Spades matches 10 of Clubs
---------- Matches discarded, play begins
Hand Allen contains
King of Hearts
Jack of Clubs
Queen of Spades
King of Spades
10 of Diamonds
Hand Jeff contains
Jack of Spades
Jack of Hearts
King of Diamonds
Hand Chris contains
Jack of Diamonds
King of Clubs
10 of Hearts
Hand Allen picked King of Diamonds
Hand Allen: King of Hearts matches King of Diamonds
Hand Jeff picked 10 of Hearts
Hand Chris picked Jack of Clubs
Hand Allen picked Jack of Hearts
Hand Jeff picked Jack of Diamonds
Hand Chris picked Queen of Spades
Hand Allen picked Jack of Diamonds
Hand Allen: Jack of Hearts matches Jack of Diamonds
Hand Jeff picked King of Clubs
Hand Chris picked King of Spades
Hand Allen picked 10 of Hearts
Hand Allen: 10 of Diamonds matches 10 of Hearts
Hand Jeff picked Queen of Spades
Hand Chris picked Jack of Spades
Hand Chris: Jack of Clubs matches Jack of Spades
Hand Jeff picked King of Spades
Hand Jeff: King of Clubs matches King of Spades
---------- Game is Over
Hand Allen is empty
Hand Jeff contains
Queen of Spades
Hand Chris is empty
So Jeff loses.
D.5. Inheritance
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Glossary
inheritance The ability to define a new class that is a modified version of a previously defined
class.
parent class The class from which a child class inherits.
child class A new class created by inheriting from an existing class; also called a subclass.
Exercises
1. Add a method, print_hands, to the OldMaidGame class which traverses self.
hands and prints each hand.
2. Define a new kind of Turtle, TurtleGTX, that comes with some extra features: it can
jump forward a given distance, and it has an odometer that keeps track of how far the
turtle has travelled since it came off the production line. (The parent class has a number
of synonyms like fd, forward, back, backward, and bk: for this exercise, just focus
on putting this functionality into the forward method.) Think carefully about how to
count the distance if the turtle is asked to move forward by a negative amount. (We
would not want to buy a second-hand turtle whose odometer reading was faked because
its previous owner drove it backwards around the block too often. Try this in a car near
you, and see if the car’s odometer counts up or down when you reverse.)
3. After travelling some random distance, your turtle should break down with a flat tyre.
After this happens, raise an exception whenever forward is called. Also provide a
change_tyre method that can fix the flat.
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APPENDIX E
Exceptions
Catching exceptions
Whenever a runtime error occurs, it creates an exception object. The program stops running
at this point and Python prints out the traceback, which ends with a message describing the
exception that occurred.
For example, dividing by zero creates an exception:
>>> print(55/0)
Traceback (most recent call last):
File "<interactive input>", line 1, in <module>
ZeroDivisionError: integer division or modulo by zero
So does accessing a non-existent list item:
>>> a = []
>>> print(a[5])
Traceback (most recent call last):
File "<interactive input>", line 1, in <module>
IndexError: list index out of range
Or trying to make an item assignment on a tuple:
>>> tup = ("a", "b", "d", "d")
>>> tup[2] = "c"
Traceback (most recent call last):
File "<interactive input>", line 1, in <module>
TypeError: 'tuple' object does not support item assignment
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In each case, the error message on the last line has two parts: the type of error before the colon,
and specifics about the error after the colon.
Sometimes we want to execute an operation that might cause an exception, but we don’t want
the program to stop. We can handle the exception using the try statement to “wrap” a region
of code.
For example, we might prompt the user for the name of a file and then try to open it. If the file
doesn’t exist, we don’t want the program to crash; we want to handle the exception:
1
2
3
4
5
filename = input("Enter a file name: ")
try:
f = open(filename, "r")
except FileNotFoundError:
print("There is no file named", filename)
The try statement has four separate clauses—or parts—introduced by the keywords try,
except, else, and finally. All clauses but the try can be omitted.
The interpretor executes the block under the try statement, and monitors for exceptions. If
one occurs, the interpretor moves to the except statement; it executes the expect block if
the exception raised match the exception requested in the except statement. If no exception
occurs, the interpretor skips the block under the except clause. A else block is executed
after the try one, if no exception occurred. A finally block is executed in any case. With
all the statements, a try clause looks like:
1
2
3
4
5
6
7
8
9
10
11
12
13
14
user_input = input('Type a number:')
try:
# Try do do something that could fail.
user_input_as_number = float(user_input)
except ValueError:
# This will be executed if a ``ValueError`` is raised.
print('You did not enter a number.')
else:
# This will be executed if not exception got raised in
˓→the
# ``try`` statement.
print('The square of your number is ', user_input_as_
˓→number**2)
finally:
# This will be executed whether or not an exception is
˓→raised.
print('Thank you')
When using a try clause, you should have as little as possible in the try block. If too many
things happen in that block, you risk handling an unexpected exception.
If the try block can fail if various way, you can handle different exceptions in the same try
clause:
It is also possible not to specify a particular exception in the except statement. In this case,
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any exception will be handled. Such bare except statement should be avoided, though, as
they can easily mask bugs.
Raising our own exceptions
Can our program deliberately cause its own exceptions? If our program detects an error condition, we can raise an exception. Here is an example that gets input from the user and checks
that the number is non-negative:
1
2
3
4
5
6
7
def get_age():
age = int(input("Please enter your age: "))
if age < 0:
# Create a new instance of an exception
my_error = ValueError("{0} is not a valid age".
˓→format(age))
raise my_error
return age
Line 5 creates an exception object, in this case, a ValueError object, which encapsulates
specific information about the error. Assume that in this case function A called B which called
C which called D which called get_age. The raise statement on line 6 carries this object
out as a kind of “return value”, and immediately exits from get_age() to its caller D. Then
D again exits to its caller C, and C exits to B and so on, each returning the exception object to
their caller, until it encounters a try ... except that can handle the exception. We call
this “unwinding the call stack”.
ValueError is one of the built-in exception types which most closely matches the kind of
error we want to raise. The complete listing of built-in exceptions can be found at the Built-in
Exceptions section of the Python Library Reference , again by Python’s creator, Guido van
Rossum.
If the function that called get_age (or its caller, or their caller, ...) handles the error, then the
program can carry on running; otherwise, Python prints the traceback and exits:
>>> get_age()
Please enter your age: 42
42
>>> get_age()
Please enter your age: -2
Traceback (most recent call last):
File "<interactive input>", line 1, in <module>
File "learn_exceptions.py", line 4, in get_age
raise ValueError("{0} is not a valid age".format(age))
ValueError: -2 is not a valid age
The error message includes the exception type and the additional information that was provided
when the exception object was first created.
It is often the case that lines 5 and 6 (creating the exception object, then raising the exception)
are combined into a single statement, but there are really two different and independent things
E.2. Raising our own exceptions
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happening, so perhaps it makes sense to keep the two steps separate when we first learn to work
with exceptions. Here we show it all in a single statement:
1
raise ValueError("{0} is not a valid age".format(age))
Revisiting an earlier example
Using exception handling, we can now modify our recursion_depth example from the
previous chapter so that it stops when it reaches the maximum recursion depth allowed:
1
2
3
4
5
6
def recursion_depth(number):
print("Recursion depth number", number)
try:
recursion_depth(number + 1)
except:
print("I cannot go any deeper into this wormhole.")
7
8
recursion_depth(0)
Run this version and observe the results.
The finally clause of the try statement
A common programming pattern is to grab a resource of some kind — e.g. we create a window
for turtles to draw on, or we dial up a connection to our internet service provider, or we may
open a file for writing. Then we perform some computation which may raise an exception, or
may work without any problems.
Whatever happens, we want to “clean up” the resources we grabbed — e.g. close the window,
disconnect our dial-up connection, or close the file. The finally clause of the try statement
is the way to do just this. Consider this (somewhat contrived) example:
1
2
import turtle
import time
3
4
5
6
7
def show_poly():
try:
win = turtle.Screen()
˓→g. a window
tess = turtle.Turtle()
# Grab/create a resource, e.
8
9
10
11
12
380
# This dialog could be cancelled,
#
or the conversion to int might fail, or n might
˓→be zero.
n = int(input("How many sides do you want in your
˓→polygon?"))
angle = 360 / n
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13
14
15
16
17
18
for i in range(n):
tess.forward(10)
tess.left(angle)
time.sleep(3)
˓→seconds
finally:
win.bye()
# Draw the polygon
# Make program wait a few
# Close the turtle's window
19
20
21
22
show_poly()
show_poly()
show_poly()
In lines 20–22, show_poly is called three times. Each one creates a new window for its
turtle, and draws a polygon with the number of sides input by the user. But what if the user
enters a string that cannot be converted to an int? What if they close the dialog? We’ll
get an exception, but even though we’ve had an exception, we still want to close the turtle’s
window. Lines 17–18 does this for us. Whether we complete the statements in the try clause
successfully or not, the finally block will always be executed.
Notice that the exception is still unhandled — only an except clause can handle an exception,
so our program will still crash. But at least its turtle window will be closed before it crashes!
Glossary
exception An error that occurs at runtime.
handle an exception To prevent an exception from causing our program to crash, by wrapping
the block of code in a try ... except construct.
raise To create a deliberate exception by using the raise statement.
Exercises
1. Write a function named readposint that uses the input dialog to prompt the user
for a positive integer and then checks the input to confirm that it meets the requirements.
It should be able to handle inputs that cannot be converted to int, as well as negative
ints, and edge cases (e.g. when the user closes the dialog, or does not enter anything at
all.)
E.5. Glossary
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APPENDIX F
Fitting
Suppose we want to determine the gravitational acceleration. To this end, we could drop an
object from the building and measure how long it takes for the object to reach the ground with
a stopwatch. Newton’s laws predict the following model:
1
ℎ = 2
2
where ℎ is the height from which we dropped the object, and  the time it takes to hit the ground.
So from our one measurement we could now calculate the gravitational acceleration .
We can measure the height very accurately, but since we use a stopwatch to measure time,
that value is a lot less reliable because you might have started and stopped your stopwatch at
the wrong moments. Therefore, the result will not be very accurate. To make the value more
accurate we should repeat the same measurement  times to obtain an average  and use that
instead. We’ll get back to this later.
For now we are more interested in the question: is this model correct? To test this question,
we drop our object from different heights, doing multiple measurements for each height to get
reliable values. The data, obtained by simulation for health and safety reasons, are given in the
following table:
y
10
20
30
40
50
t
1.4
2.1
2.6
3.0
3.3
n
5
3
8
15
30
Since the model predicts a parabola, we want to fit the data to this model to see how good it
works. It might be a bit confusing, but ℎ is our x axis, and  is the y axis.
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We use the symfit package to do our fitting. You can find the installation instructions here.
To fit the data to the model we run the following code:
import numpy as np
from symfit.api import Variable, Parameter, Fit, sqrt
t_data = np.array([1.4, 2.1, 2.6, 3.0, 3.3])
h_data = np.array([10, 20, 30, 40, 50])
# We now define our model
h = Variable()
g = Parameter()
t_model = sqrt(2 * h / g)
fit = Fit(t_model, h_data, t_data)
fit_result = fit.execute()
print(fit_result)
Looking at these results, we see that  = 9.09 ± 0.15 for this dataset. In order to plot this result
alongside the data, we need to calculate values for the model. In the same script, we can do:
# Make an array from 0 to 50 in 1000 steps
h_range = np.linspace(0, 50, 1000)
fit_data = t_model(h=h_range, g=fit_result.params.g)
This gives the model evaluated at all the points in h_range. Making the actual plot is left to
you as an exercise. We see that we can reach the value of g by calling fit_result.params.g, this
returns 9.09.
Let’s think for a second about the implications. The value of  is  = 9.81 in the Netherlands.
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Based on our result, the textbooks should be rewritten because that value is extremely unlikely
to be true given the small standard deviation in our data. It is at this point that we remember
that our data point were not infinitely precise: we took many measurements and averaged
them. This means there is an uncertainty in each of our data points. We will now account for
this additional uncertainty and see what this does to our conclusion. To do this we first have to
describe how the fitting actually works.
How does it work?
In fitting we want to find values for the parameters such that the differences between the model
and the data are as small as possible. The differences (residuals) are easy to calculate:
 ( , ⃗) − 
Here we have written the parameters as a vector ⃗, to indicate that we can have multiple parameters.  and  are the x and y coordinates of the i’th datapoint. However, if we were to
minimize the sum over all these differences we would have a problem, because these differences can be either possitive or negative. This means there’s many ways to add these values
and get zero out of the sum. We therefore take the sum over the residuals squared:
2
 =

∑︁
( ( , ⃗) −  )2
=1
Now if we minimize 2 , we get the best possible values for our parameters. The fitting algorithm actually just takes some values for the parameters, calculates 2 , then changes the values
slightly by adding or subtracting a number, and checks if this new value is smaller than the old
one. If this is true it keeps going in the same direction until the value of 2 starts to increase.
That’s when you know you’ve hit a minimum. Of cource the trick is to do this smartly, and a
lot of algorithms have been developed in order to do this.
Propagating Uncertanties
In the example above we the fitting process assumed that every measurement was equally reliable. But this is not true. By repeating a measurement and averaging the result, we can improve
the accuracy. So in our example, we dropped our object from every height a couple of times
and took the average. Therefore, we want to assign a weight depending on how accurate the
average value for that height is. Statistically the weight  to use is  = 12 , where  is the

standard deviation for each point.
Our sum to minimize now changes to:
F.1. How does it work?
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2
 =

∑︁
2
 ( ( , ⃗) −  ) =

∑︁
( ( , ⃗) −  )2
=1
=1
2
But how do we know the standard deviation in the mean value we calculate for every height?
Suppose the standard deviation of our stopwatch is ℎ = 0.2. If we do  measurements
from the same height, the average time is found by calculating

1 ∑︁

¯ =
 =1
It can be shown that the standard deviation of the mean is now:
¯ =
ℎ
√

So we see that by increasing the amount of measurements, we can decrease the uncertainty in
¯. Our simulated data now changes to:
y
10
20
30
40
50
t
1.4
2.1
2.6
3.0
3.3
n
5
3
8
15
30

0.089
0.115
0.071
0.052
0.037
The values of  have been calculated by using the above formula. Let’s fit to this new data set
using symfit. Notice that there are some small differences to the code:
import numpy as np
from symfit.api import Variable, Parameter, Fit, sqrt
t_data = np.array([1.4, 2.1, 2.6, 3.0, 3.3])
h_data = np.array([10, 20, 30, 40, 50])
n = np.array([5, 3, 8, 15, 30])
sigma = 0.2
sigma_t = sigma / np.sqrt(n)
# We now define our model
h = Variable()
t = Variable()
g = Parameter()
t_model = {t: sqrt(2 * h / g)}
fit = Fit(t_model, h=h_data, t=t_data, sigma_t=sigma_t)
fit_result = fit.execute()
print(fit_result)
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Note: Named Models
Looking at the definition of t_model, we see it is now a dict. This has been done so
we can tell symfit which of our variables are uncertain by the name of the variable,
in this case t has an uncertainty sigma_t.
Including these uncertainties in the fit yields  = 9.10 ± 0.16. The accepted value of  = 9.81
is well outside the uncertainty in this data. Therefore the textbooks must be rewriten!
This example shows the importance of propagating your errors consistently. (And of the importance of performing the actual measurement as the author of a chapter on error propagation
so you don’t end up claiming the textbooks have to rewritten.)
More on symfit
There are a lot more features in symfit to help you on your quest to fitting the universe. You can
find the tutorial there.
It is recommended you read this as well before starting to fit your own data.
F.3. More on symfit
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APPENDIX G
PyGame
PyGame is a package that is not part of the standard Python distribution, so if you do not already
have it installed (i.e. import pygame fails), download and install a suitable version from
http://pygame.org/download.shtml. These notes are based on PyGame 1.9.1, the most recent
version at the time of writing.
PyGame comes with a substantial set of tutorials, examples, and help, so there is ample opportunity to stretch yourself on the code. You may need to look around a bit to find these resources,
though: if you’ve installed PyGame on a Windows machine, for example, they’ll end up in a
folder like C:\Python31\Lib\site-packages\pygame\ where you will find directories for docs and
examples.
The game loop
The structure of the games we’ll consider always follows this fixed pattern:
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In every game, in the setup section we’ll create a window, load and prepare some content, and
then enter the game loop. The game loop continuously does four main things:
• it polls for events — i.e. asks the system whether events have occurred — and responds
appropriately,
• it updates whatever internal data structures or objects need changing,
• it draws the current state of the game into a (non-visible) surface,
• it puts the just-drawn surface on display.
1
import pygame
2
3
4
5
6
def main():
""" Set up the game and run the main game loop """
pygame.init()
# Prepare the pygame module for use
surface_size = 480
# Desired physical surface size,
˓→in pixels.
7
8
9
# Create surface of (width, height), and its window.
main_surface = pygame.display.set_mode((surface_size,
˓→surface_size))
10
11
# Set up some data to describe a small rectangle and
its color
small_rect = (300, 200, 150, 90)
some_color = (255, 0, 0)
# A color is a mix of
˓→(Red, Green, Blue)
˓→
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13
14
15
16
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while True:
event = pygame.event.poll()
# Look for any event
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17
˓→
18
if event.type == pygame.QUIT:
button clicked?
break
#
# Window close
... leave game loop
19
# Update your game objects and data structures here.
20
˓→
..
21
# We draw everything from scratch on each frame.
# So first fill everything with the background
22
23
˓→
color
main_surface.fill((0, 200, 255))
24
25
# Overpaint a smaller rectangle on the main surface
main_surface.fill(some_color, small_rect)
26
27
28
# Now the surface is ready, tell pygame to display
29
˓→
it!
pygame.display.flip()
30
31
32
pygame.quit()
˓→window.
# Once we leave the loop, close the
33
34
main()
This program pops up a window which stays there until we close it:
PyGame does all its drawing onto rectangular surfaces. After initializing PyGame at line 5, we
create a window holding our main surface. The main loop of the game extends from line 15 to
30, with the following key bits of logic:
• First (line 16) we poll to fetch the next event that might be ready for us. This step will
always be followed by some conditional statements that will determine whether any event
that we’re interested in has happened. Polling for the event consumes it, as far as PyGame
is concerned, so we only get one chance to fetch and use each event. On line 17 we test
whether the type of the event is the predefined constant called pygame.QUIT. This is the
G.1. The game loop
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event that we’ll see when the user clicks the close button on the PyGame window. In
response to this event, we leave the loop.
• Once we’ve left the loop, the code at line 32 closes window, and we’ll return from function main. Your program could go on to do other things, or reinitialize pygame and
create another window, but it will usually just end too.
• There are different kinds of events — key presses, mouse motion, mouse clicks, joystick
movement, and so on. It is usual that we test and handle all these cases with new code
squeezed in before line 19. The general idea is “handle events first, then worry about the
other stuff”.
• At line 20 we’d update objects or data — for example, if we wanted to vary the color,
position, or size of the rectangle we’re about to draw, we’d re-assign some_color, and
small_rect here.
• A modern way to write games (now that we have fast computers and fast graphics cards)
is to redraw everything from scratch on every iteration of the game loop. So the first thing
we do at line 24 is fill the entire surface with a background color. The fill method of a
surface takes two arguments — the color to use for filling, and the rectangle to be filled.
But the second argument is optional, and if it is left out the entire surface is filled.
• In line 27 we fill a second rectangle, this time using some_color. The placement and
size of the rectangle are given by the tuple small_rect, a 4-element tuple (x, y,
width, height).
• It is important to understand that the origin of the PyGame’s surface is at the top left
corner (unlike the turtle module that puts its origin in the middle of the screen). So,
if you wanted the rectangle closer to the top of the window, you need to make its y
coordinate smaller.
• If your graphics display hardware tries to read from memory at the same time as the
program is writing to that memory, they will interfere with each other, causing video
noise and flicker. To get around this, PyGame keeps two buffers in the main surface —
the back buffer that the program draws to, while the front buffer is being shown to the
user. Each time the program has fully prepared its back buffer, it flips the back/front role
of the two buffers. So the drawing on lines 24 and 27 does does not change what is seen
on the screen until we flip the buffers, on line 30.
Displaying images and text
To draw an image on the main surface, we load the image, say a beach ball, into its own new
surface. The main surface has a blit method that copies pixels from the beach ball surface
into its own surface. When we call blit, we can specify where the beach ball should be placed
on the main surface. The term blit is widely used in computer graphics, and means to make a
fast copy of pixels from one area of memory to another.
So in the setup section, before we enter the game loop, we’d load the image, like this:
1
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ball = pygame.image.load("ball.png")
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and after line 28 in the program above, we’d add this code to display our image at position
(100,120):
1
main_surface.blit(ball, (100, 120))
To display text, we need do do three things. Before we enter the game loop, we instantiate a
font object:
1
2
# Instantiate 16 point Courier font to draw text.
my_font = pygame.font.SysFont("Courier", 16)
and after line 28, again, we use the font’s render method to create a new surface containing
the pixels of the drawn text, and then, as in the case for images, we blit our new surface onto the
main surface. Notice that render takes two extra parameters — the second tells it whether to
carefully smooth edges of the text while drawing (this process is called anti-aliasing), and the
second is the color that we want the text text be. Here we’ve used (0,0,0) which is black:
1
2
the_text = my_font.render("Hello, world!", True, (0,0,0))
main_surface.blit(the_text, (10, 10))
We’ll demonstrate these two new features by counting the frames — the iterations of the game
loop — and keeping some timing information. On each frame, we’ll display the frame count,
and the frame rate. We will only update the frame rate after every 500 frames, when we’ll look
at the timing interval and can do the calculations.
1
2
import pygame
import time
3
4
def main():
5
pygame.init()
# Prepare the PyGame module for use
main_surface = pygame.display.set_mode((480, 240))
6
7
8
# Load an image to draw. Substitute your own.
# PyGame handles gif, jpg, png, etc. image types.
ball = pygame.image.load("ball.png")
9
10
11
12
# Create a font for rendering text
my_font = pygame.font.SysFont("Courier", 16)
13
14
15
frame_count = 0
frame_rate = 0
t0 = time.clock()
16
17
18
19
while True:
20
21
# Look for an event from keyboard, mouse, joystick,
22
˓→
23
24
etc.
ev = pygame.event.poll()
if ev.type == pygame.QUIT:
˓→clicked?
G.2. Displaying images and text
# Window close button
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break
25
# Leave game loop
26
# Do other bits of logic for the game here
frame_count += 1
if frame_count % 500 == 0:
t1 = time.clock()
frame_rate = 500 / (t1-t0)
t0 = t1
27
28
29
30
31
32
33
34
˓→
35
# Completely redraw the surface, starting with
background
main_surface.fill((0, 200, 255))
36
# Put a red rectangle somewhere on the surface
main_surface.fill((255,0,0), (300, 100, 150, 90))
37
38
39
# Copy our image to the surface, at this (x,y) posn
main_surface.blit(ball, (100, 120))
40
41
42
43
44
45
46
47
# Make a new surface with an image of the text
the_text = my_font.render("Frame = {0}, rate = {1:.
˓→2f} fps"
.format(frame_count, frame_rate), True,
˓→(0,0,0))
# Copy the text surface to the main surface
main_surface.blit(the_text, (10, 10))
48
# Now that everything is drawn, put it on display!
pygame.display.flip()
49
50
51
52
pygame.quit()
53
54
55
main()
The frame rate is close to ridiculous — a lot faster than one’s eye can process frames. (Commercial video games usually plan their action for 60 frames per second (fps).) Of course, our
rate will drop once we start doing something a little more strenuous inside our game loop.
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Drawing a board for the N queens puzzle
We previously solved our N queens puzzle. For the 8x8 board, one of the solutions was the
list [6,4,2,0,5,7,1,3]. Let’s use that solution as testdata, and now use PyGame to draw
that chessboard with its queens.
We’ll create a new module for the drawing code, called draw_queens.py. When we have
our test case(s) working, we can go back to our solver, import this new module, and add a call
to our new function to draw a board each time a solution is discovered.
We begin with a background of black and red squares for the board. Perhaps we could create an
image that we could load and draw, but that approach would need different background images
for different size boards. Just drawing our own red and black rectangles of the appropriate size
sounds like much more fun!
1
2
def draw_board(the_board):
""" Draw a chess board with queens, from the_board. """
3
4
5
pygame.init()
colors = [(255,0,0), (0,0,0)]
˓→black]
# Set up colors [red,
6
7
8
9
10
n = len(the_board)
# This is an NxN chess board.
surface_size = 480
# Proposed physical
˓→surface size.
square_size = surface_size // n
# sq_sz is length of
˓→a square.
surface_size = n * square_size
# Adjust to exactly
˓→fit n squares.
11
12
13
# Create the surface of (width, height), and its window.
surface = pygame.display.set_mode((surface_size,
˓→surface_size))
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Here we precompute square_size, the integer size that each square will be, so that we can
fit the squares nicely into the available window. So if we’d like the board to be 480x480, and
we’re drawing an 8x8 chessboard, then each square will need to have a size of 60 units. But we
notice that a 7x7 board cannot fit nicely into 480 — we’re going to get some ugly border that
our squares don’t fill exactly. So we recompute the surface size to exactly fit our squares before
we create the window.
Now let’s draw the squares, in the game loop. We’ll need a nested loop: the outer loop will run
over the rows of the chessboard, the inner loop over the columns:
1
2
3
4
5
6
7
8
# Draw a fresh background (a blank chess board)
for row in range(n):
# Draw each row of the board.
color_index = row % 2
# Change starting color
˓→on each row
for col in range(n):
# Run through cols drawing
˓→squares
the_square = (col*square_size, row*square_size,
˓→square_size, square_size)
surface.fill(colors[color_index], the_square)
# now flip the color index for the next square
c_index = (c_index + 1) % 2
There are two important ideas in this code: firstly, we compute the rectangle to be filled from the
row and col loop variables, multiplying them by the size of the square to get their position.
And, of course, each square is a fixed width and height. So the_square represents the
rectangle to be filled on the current iteration of the loop. The second idea is that we have to
alternate colors on every square. In the earlier setup code we created a list containing two
colors, here we manipulate color_index (which will always either have the value 0 or 1) to
start each row on a color that is different from the previous row’s starting color, and to switch
colors each time a square is filled.
This (together with the other fragments not shown to flip the surface onto the display) leads to
the pleasing backgrounds like this, for different size boards:
Now, on to drawing the queens! Recall that our solution [6,4,2,0,5,7,1,3] means that
in column 0 of the board we want a queen at row 6, at column 1 we want a queen at row 4, and
so on. So we need a loop running over each queen:
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1
2
for (col, row) in enumerate(the_board):
# draw a queen at col, row...
In this chapter we already have a beach ball image, so we’ll use that for our queens. In the
setup code before our game loop, we load the ball image (as we did before), and in the body of
the loop, we add the line:
1
surface.blit(ball, (col * square_size, row * square_size))
We’re getting there, but those queens need to be centred in their squares! Our problem arises
from the fact that both the ball and the rectangle have their upper left corner as their reference
points. If we’re going to centre this ball in the square, we need to give it an extra offset in
both the x and y direction. (Since the ball is round and the square is square, the offset in the
two directions will be the same, so we’ll just compute a single offset value, and use it in both
directions.)
The offset we need is half the (size of the square less the size of the ball). So we’ll precompute
this in the game’s setup section, after we’ve loaded the ball and determined the square size:
1
ball_offset = (square_size - ball.get_width()) // 2
Now we touch up the drawing code for the ball and we’re done:
1
surface.blit(ball, (col * square_size + ball_offset, row *
˓→square_size + ball_offset))
We might just want to think about what would happen if the ball was bigger than the square. In
that case, ball_offset would become negative. So it would still be centered in the square
- it would just spill over the boundaries, or perhaps obscure the square entirely!
Here is the complete program:
1
import pygame
2
3
4
def draw_board(the_board):
""" Draw a chess board with queens, as determined by
˓→the the_board. """
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5
6
7
pygame.init()
colors = [(255,0,0), (0,0,0)]
˓→black]
# Set up colors [red,
8
9
10
11
12
n = len(the_board)
# This is an NxN chess board.
surface_size = 480
# Proposed physical
˓→surface size.
square_size = surface_size // n
# sq_sz is length of
˓→a square.
surface_size = n * square_size
# Adjust to exactly
˓→fit n squares.
13
14
15
# Create the surface of (width, height), and its window.
surface = pygame.display.set_mode((surface_size,
˓→surface_size))
16
17
ball = pygame.image.load("ball.png")
18
19
20
21
22
# Use an extra offset to centre the ball in its square.
# If the square is too small, offset becomes negative,
#
but it will still be centered :-)
ball_offset = (square_size-ball.get_width()) // 2
23
24
while True:
25
26
27
28
29
# Look for an event from keyboard, mouse, etc.
event = pygame.event.poll()
if event.type == pygame.QUIT:
break;
30
31
32
33
34
35
36
37
38
# Draw a fresh background (a blank chess board)
for row in range(n):
# Draw each row of
˓→the board.
color_index = row % 2
# Alternate
˓→starting color
for col in range(n):
# Run through cols
˓→drawing squares
the_square = (col*square_size, row*square_
˓→size, square_size, square_size)
surface.fill(colors[color_index], the_
˓→square)
# Now flip the color index for the next
˓→square
color_index = (color_index + 1) % 2
39
40
41
42
43
398
# Now that squares are drawn, draw the queens.
for (col, row) in enumerate(the_board):
surface.blit(ball,
(col*square_size+ball_offset,row*square_
˓→size+ball_offset))
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44
pygame.display.flip()
45
46
47
48
pygame.quit()
49
50
51
52
53
54
if __name__ == "__main__":
draw_board([0, 5, 3, 1, 6, 4, 2])
# 7 x 7 to test
˓→window size
draw_board([6, 4, 2, 0, 5, 7, 1, 3])
draw_board([9, 6, 0, 3, 10, 7, 2, 4, 12, 8, 11, 5, 1])
˓→# 13 x 13
draw_board([11, 4, 8, 12, 2, 7, 3, 15, 0, 14, 10, 6, 13,
˓→ 1, 5, 9])
There is one more thing worth reviewing here. The conditional statement on line 50 tests
whether the name of the currently executing program is __main__. This allows us to distinguish whether this module is being run as a main program, or whether it has been imported
elsewhere, and used as a module. If we run this module in Python, the test cases in lines 51-54
will be executed. However, if we import this module into another program (i.e. our N queens
solver from earlier) the condition at line 50 will be false, and the statements on lines 51-54
won’t run.
Previously, our main program looked like this:
1
def main():
2
3
4
5
6
7
8
9
10
11
12
board = list(range(8))
# Generate the initial
˓→permutation
num_found = 0
tries = 0
while num_found < 10:
random.shuffle(bd)
tries += 1
if not has_clashes(bd):
print("Found solution {0} in {1} tries.".
˓→format(board, tries))
tries = 0
num_found += 1
13
14
main()
Now we just need two changes. At the top of that program, we import the module that we’ve
been working on here (assume we called it draw_queens). (You’ll have to ensure that the
two modules are saved in the same folder.) Then after line 10 here we add a call to draw the
solution that we’ve just discovered:
draw_queens.draw_board(bd)
And that gives a very satisfying combination of program that can search for solutions to the N
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queens problem, and when it finds each, it pops up the board showing the solution.
Sprites
A sprite is an object that can move about in a game, and has internal behaviour and state of its
own. For example, a spaceship would be a sprite, the player would be a sprite, and bullets and
bombs would all be sprites.
Object oriented programming (OOP) is ideally suited to a situation like this: each object can
have its own attributes and internal state, and a couple of methods. Let’s have some fun with
our N queens board. Instead of placing the queen in her final position, we’d like to drop her in
from the top of the board, and let her fall into position, perhaps bouncing along the way.
The first encapsulation we need is to turn each of our queens into an object. We’ll keep a list of
all the active sprites (i.e. a list of queen objects), and arrange two new things in our game loop:
• After handling events, but before drawing, call an update method on every sprite. This
will give each sprite a chance to modify its internal state in some way — perhaps change
its image, or change its position, or rotate itself, or make itself grow a bit bigger or a bit
smaller.
• Once all the sprites have updated themselves, the game loop can begin drawing - first the
background, and then call a draw method on each sprite in turn, and delegate (hand off)
the task of drawing to the object itself. This is in line with the OOP idea that we don’t
say “Hey, draw, show this queen!”, but we prefer to say “Hey, queen, draw yourself!”.
We start with a simple object, no movement or animation yet, just scaffolding, to see how to fit
all the pieces together:
1
class QueenSprite:
2
3
4
5
6
7
8
9
def __init__(self, img, target_posn):
""" Create and initialize a queen for this
target position on the board
"""
self.image = img
self.target_posn = target_positionn
self.position = target_position
10
11
12
def update(self):
return
# Do nothing for the moment.
13
14
15
def draw(self, target_surface):
target_surface.blit(self.image, self.position)
We’ve given the sprite three attributes: an image to be drawn, a target position, and a current
position. If we’re going to move the spite about, the current position may need to be different
from the target, which is where we want the queen finally to end up. In this code at this time
we’ve done nothing in the update method, and our draw method (which can probably remain
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this simple in future) simply draws itself at its current position on the surface that is provided
by the caller.
With its class definition in place, we now instantiate our N queens, put them into a list of sprites,
and arrange for the game loop to call the update and draw methods on each frame. The new
bits of code, and the revised game loop look like this:
1
˓→
all_sprites = []
the game
# Keep a list of all sprites in
2
3
# Create a sprite object for each queen, and populate
our list.
for (col, row) in enumerate(the_board):
a_queen = QueenSprite(ball,
(col*square_size+ball_offset, row*square_
˓→size+ball_offset))
all_sprites.append(a_queen)
˓→
4
5
6
7
8
9
10
11
12
13
while True:
# Look for an event from keyboard, mouse, etc.
event = pygame.event.poll()
if event.type == pygame.QUIT:
break;
14
15
16
17
# Ask every sprite to update itself.
for sprite in all_sprites:
sprite.update()
18
19
20
# Draw a fresh background (a blank chess board)
# ... same as before ...
21
22
23
24
# Ask every sprite to draw itself.
for sprite in all_sprites:
sprite.draw(surface)
25
26
pygame.display.flip()
This works just like it did before, but our extra work in making objects for the queens has
prepared the way for some more ambitious extensions.
Let us begin with a falling queen object. At any instant, it will have a velocity i.e. a speed,
in a certain direction. (We are only working with movement in the y direction, but use your
imagination!) So in the object’s update method, we want to change its current position by its
velocity. If our N queens board is floating in space, velocity would stay constant, but hey, here
on Earth we have gravity! Gravity changes the velocity on each time interval, so we’ll want
a ball that speeds up as it falls further. Gravity will be constant for all queens, so we won’t
keep it in the instances — we’ll just make it a variable in our module. We’ll make one other
change too: we will start every queen at the top of the board, so that it can fall towards its target
position. With these changes, we now get the following:
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gravity = 0.0001
2
3
class QueenSprite:
4
5
6
7
8
9
10
def __init__(self, img, target_posn):
self.image = img
self.target_position = target_position
(x, y) = target_position
self.position = (x, 0)
# Start ball at top of
˓→its column
self.y_velocity = 0
#
with zero initial
˓→velocity
11
12
13
14
15
16
def update(self):
self.y_velocity += gravity
˓→velocity
(x, y) = self.position
new_y_pos = y + self.y_velocity
˓→the ball
self.position = (x, new_y_pos)
˓→new position.
# Gravity changes
# Velocity moves
#
to this
17
18
19
def draw(self, target_surface):
# Same as before.
target_surface.blit(self.image, self.position)
Making these changes gives us a new chessboard in which each queen starts at the top of its
column, and speeds up, until it drops off the bottom of the board and disappears forever. A
good start — we have movement!
The next step is to get the ball to bounce when it reaches its own target position. It is pretty easy
to bounce something — you just change the sign of its velocity, and it will move at the same
speed in the opposite direction. Of course, if it is travelling up towards the top of the board it
will be slowed down by gravity. (Gravity always sucks down!) And you’ll find it bounces all
the way up to where it began from, reaches zero velocity, and starts falling all over again. So
we’ll have bouncing balls that never settle.
A realistic way to settle the object is to lose some energy (probably to friction) each time it
bounces — so instead of simply reversing the sign of the velocity, we multiply it by some
fractional factor — say -0.65. This means the ball only retains 65% of its energy on each
bounce, so it will, as in real life, stop bouncing after a short while, and settle on its “ground”.
The only changes are in the update method, which now looks like this:
1
2
3
4
5
6
402
def update(self):
self.y_velocity += gravity
(x, y) = self.postion
new_y_pos = y + self.y_velocity
(target_x, target_y) = self.target_posn
˓→position
dist_to_go = target_y - new_y_pos
˓→our floor?
# Unpack the
# How far to
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8
9
10
if dist_to_go < 0:
# Are we
˓→under floor?
self.y_velocity = -0.65 * self.y_velocity
#
˓→Bounce
new_y_pos = target_y + dist_to_go
# Move back
˓→above floor
11
12
self.position = (x, new_y_pos)
˓→new position.
# Set our
Heh, heh, heh! We’re not going to show animated screenshots, so copy the code into your
Python environment and see for yourself.
Events
The only kind of event we’re handled so far has been the QUIT event. But we can also detect
keydown and keyup events, mouse motion, and mousebutton down or up events. Consult the
PyGame documentation and follow the link to Event.
When your program polls for and receives an event object from PyGame, its event type will
determine what secondary information is available. Each event object carries a dictionary
(which you may only cover in due course in these notes). The dictionary holds certain keys and
values that make sense for the type of event.
For example, if the type of event is MOUSEMOTION, we’ll be able to find the mouse position
and information about the state of the mouse buttons in the dictionary attached to the event.
Similarly, if the event is KEYDOWN, we can learn from the dictionary which key went down,
and whether any modifier keys (shift, control, alt, etc.) are also down. You also get events when
the game window becomes active (i.e. gets focus) or loses focus.
The event object with type NOEVENT is returned if there are no events waiting. Events can be
printed, allowing you to experiment and play around. So dropping these lines of code into the
game loop directly after polling for any event is quite informative:
1
2
if event.type != pygame.NOEVENT:
˓→interesting!
print(event)
# Only print if it is
With this is place, hit the space bar and the escape key, and watch the events you get. Click
your three mouse buttons. Move your mouse over the window. (This causes a vast cascade of
events, so you may also need to filter those out of the printing.) You’ll get output that looks
something like this:
<Event(17-VideoExpose {})>
<Event(1-ActiveEvent {'state': 1, 'gain': 0})>
<Event(2-KeyDown {'scancode': 57, 'key': 32, 'unicode': ' ', 'mod':
˓→0})>
<Event(3-KeyUp {'scancode': 57, 'key': 32, 'mod': 0})>
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<Event(2-KeyDown {'scancode': 1, 'key': 27, 'unicode': '\x1b',
˓→': 0})>
<Event(3-KeyUp {'scancode': 1, 'key': 27, 'mod': 0})>
...
<Event(4-MouseMotion {'buttons': (0, 0, 0), 'pos': (323, 194),
˓→': (-3, -1)})>
<Event(4-MouseMotion {'buttons': (0, 0, 0), 'pos': (322, 193),
˓→': (-1, -1)})>
<Event(4-MouseMotion {'buttons': (0, 0, 0), 'pos': (321, 192),
˓→': (-1, -1)})>
<Event(4-MouseMotion {'buttons': (0, 0, 0), 'pos': (319, 192),
˓→': (-2, 0)})>
<Event(5-MouseButtonDown {'button': 1, 'pos': (319, 192)})>
<Event(6-MouseButtonUp {'button': 1, 'pos': (319, 192)})>
<Event(4-MouseMotion {'buttons': (0, 0, 0), 'pos': (319, 191),
˓→': (0, -1)})>
<Event(5-MouseButtonDown {'button': 2, 'pos': (319, 191)})>
<Event(5-MouseButtonDown {'button': 5, 'pos': (319, 191)})>
<Event(6-MouseButtonUp {'button': 5, 'pos': (319, 191)})>
<Event(6-MouseButtonUp {'button': 2, 'pos': (319, 191)})>
<Event(5-MouseButtonDown {'button': 3, 'pos': (319, 191)})>
<Event(6-MouseButtonUp {'button': 3, 'pos': (319, 191)})>
...
<Event(1-ActiveEvent {'state': 1, 'gain': 0})>
<Event(12-Quit {})>
'mod
'rel
'rel
'rel
'rel
'rel
So let us now make these changes to the code near the top of our game loop:
1
while True:
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
# Look for an event from keyboard,
ev = pygame.event.poll()
if event.type == pygame.QUIT:
break;
if event.type == pygame.KEYDOWN:
key = ev.dict["key"]
if key == 27:
break
˓→loop.
if key == ord("r"):
colors[0] = (255, 0, 0)
˓→black.
elif key == ord("g"):
colors[0] = (0, 255, 0)
˓→black.
elif key == ord("b"):
colors[0] = (0, 0, 255)
˓→black.
mouse, etc.
# On Escape key ...
#
leave the game
# Change to red +
# Change to green +
# Change to blue +
17
18
˓→
404
if event.type == pygame.MOUSEBUTTONDOWN: # Mouse gone
down?
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19
˓→
20
posn_of_click = event.dict["pos"]
# Get the
coordinates.
print(posn_of_click)
# Just print them.
Lines 7-16 show typical processing for a KEYDOWN event — if a key has gone down, we
test which key it is, and take some action. With this in place, we have another way to quit our
queens program — by hitting the escape key. Also, we can use keys to change the color of the
board that is drawn.
Finally, at line 20, we respond (pretty lamely) to the mouse button going down.
As a final exercise in this section, we’ll write a better response handler to mouse clicks. What
we will do is figure out if the user has clicked the mouse on one of our sprites. If there is a sprite
under the mouse when the click occurs, we’ll send the click to the sprite and let it respond in
some sensible way.
We’ll begin with some code that finds out which sprite is under the clicked position, perhaps
none! We add a method to the class, contains_point, which returns True if the point is
within the rectangle of the sprite:
1
2
3
4
5
6
7
8
def contains_point(self, point):
""" Return True if my sprite rectangle contains point
˓→pt """
(my_x, my_y) = self.position
my_width = self.image.get_width()
my_height = self.image.get_height()
(x, y) = point
return ( x >= my_x and x < my_x + my_width and
y >= my_y and y < my_y + my_height)
Now in the game loop, once we’ve seen the mouse event, we determine which queen, if any,
should be told to respond to the event:
1
2
3
4
5
6
if ev.type == pygame.MOUSEBUTTONDOWN:
posn_of_click = event.dict["pos"]
for sprite in all_sprites:
if sprite.contains_point(posn_of_click):
sprite.handle_click()
break
And the final thing is to write a new method called handle_click in the QueenSprite
class. When a sprite is clicked, we’ll just add some velocity in the up direction, i.e. kick it back
into the air.
1
2
def handle_click(self):
self.y_velocity += -0.3
# Kick it up
With these changes we have a playable game! See if you can keep all the balls on the move,
not allowing any one to settle!
G.5. Events
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A wave of animation
Many games have sprites that are animated: they crouch, jump and shoot. How do they do that?
Consider this sequence of 10 images: if we display them in quick succession, Duke will wave
at us. (Duke is a friendly visitor from the kingdom of Javaland.)
A compound image containing smaller patches which are intended for animation is called a
sprite sheet. Download this sprite sheet by right-clicking in your browser and saving it in your
working directory with the name duke_spritesheet.png.
The sprite sheet has been quite carefully prepared: each of the 10 patches are spaced exactly
50 pixels apart. So, assuming we want to draw patch number 4 (numbering from 0), we want
to draw only the rectangle that starts at x position 200, and is 50 pixels wide, within the sprite
sheet. Here we’ve shown the patches and highlighted the patch we want to draw.
The blit method we’ve been using — for copying pixels from one surface to another — can
copy a sub-rectangle of the source surface. So the grand idea here is that each time we draw
Duke, we won’t blit the whole sprite sheet. Instead we’ll provide an extra rectangle argument
that determines which portion of the sprite sheet will be blitted.
We’re going to add new code in this section to our existing N queens drawing game. What we
want is to put some instances of Duke on the chessboard somewhere. If the user clicks on one
of them, we’ll get him to respond by waving back, for one cycle of his animation.
But before we do that, we need another change. Up until now, our game loop has been running
at really fast frame rates that are unpredictable. So we’ve chosen some magic numbers for
gravity and for bouncing and kicking the ball on the basis of trial-and-error. If we’re going to
start animating more sprites, we need to tame our game loop to operate at a fixed, known frame
rate. This will allow us to plan our animation better.
PyGame gives us the tools to do this in just two lines of code. In the setup section of the game,
we instantiate a new Clock object:
1
my_clock = pygame.time.Clock()
and right at the bottom of the game loop, we call a method on this object that limits the frame
rate to whatever we specify. So let’s plan our game and animation for 60 frames per second, by
adding this line at the bottom of our game loop:
1
406
my_clock.tick(60)
˓→60 fps
# Waste time so that frame rate becomes
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You’ll find that you have to go back and adjust the numbers for gravity and kicking the ball
now, to match this much slower frame rate. When we plan an animation so that it only works
sensibly at a fixed frame rate, we say that we’ve baked the animation. In this case we’re baking
our animations for 60 frames per second.
To fit into the existing framework that we already have for our queens board, we want to create
a DukeSprite class that has all the same methods as the QueenSprite class. Then we can
add one or more Duke instances onto our list of all_sprites, and our existing game loop
will then call methods of the Duke instance. Let us start with skeleton scaffolding for the new
class:
1
class DukeSprite:
2
3
4
5
def __init__(self, img, target_position):
self.image = img
self.position = target_position
6
7
8
def update(self):
return
9
10
11
def draw(self, target_surface):
return
12
13
14
def handle_click(self):
return
15
16
17
18
def contains_point(self, pt):
# Use code from QueenSprite here
return
The only changes we’ll need to the existing game are all in the setup section. We load up the
new sprite sheet and instantiate a couple of instances of Duke, at the positions we want on the
chessboard. So before entering the game loop, we add this code:
1
2
# Load the sprite sheet
duke_sprite_sheet = pygame.image.load("duke_spritesheet.png
˓→")
3
4
5
6
# Instantiate two duke instances, put them on the
˓→chessboard
duke1 = DukeSprite(duke_sprite_sheet,(square_size*2, 0))
duke2 = DukeSprite(duke_sprite_sheet,(square_size*5, sq_sz))
7
8
9
10
# Add them to the list of sprites which our game loop
˓→manages
all_sprites.append(duke1)
all_sprites.append(duke2)
Now the game loop will test if each instance has been clicked, will call the click handler for
that instance. It will also call update and draw for all sprites. All the remaining changes we
need to make will be made in the methods of the DukeSprite class.
G.6. A wave of animation
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Let’s begin with drawing one of the patches.
We’ll introduce a new attribute
curr_patch_num into the class. It holds a value between 0 and 9, and determines which
patch to draw. So the job of the draw method is to compute the sub-rectangle of the patch to
be drawn, and to blit only that portion of the spritesheet:
1
2
3
4
def draw(self, target_surface):
patch_rect = (self.curr_patch_num * 50, 0,
50, self.image.get_height())
target_surface.blit(self.image, self.posn, patch_rect)
Now on to getting the animation to work. We need to arrange logic in update so that if we’re
busy animating, we change the curr_patch_num every so often, and we also decide when
to bring Duke back to his rest position, and stop the animation. An important issue is that the
game loop frame rate — in our case 60 fps — is not the same as the animation rate — the rate
at which we want to change Duke’s animation patches. So we’ll plan Duke wave’s animation
cycle for a duration of 1 second. In other words, we want to play out Duke’s 10 animation
patches over 60 calls to update. (This is how the baking of the animation takes place!) So
we’ll keep another animation frame counter in the class, which will be zero when we’re not
animating, and each call to update will increment the counter up to 59, and then back to
0. We can then divide that animation counter by 6, to set the curr_patch_num variable to
select the patch we want to show.
1
2
3
4
def update(self):
if self.anim_frame_count > 0:
self.anim_frame_count = (self.anim_frame_count + 1 )
˓→% 60
self.curr_patch_num = self.anim_frame_count // 6
Notice that if anim_frame_count is zero, i.e. Duke is at rest, nothing happens here.
But if we start the counter running, it will count up to 59 before settling back to zero. Notice also, that because anim_frame_count can only be a value between 0 and 59, the
curr_patch_num will always stay between 0 and 9. Just what we require!
Now how do we trigger the animation, and start it running? On the mouse click.
1
2
3
def handle_click(self):
if self.anim_frame_count == 0:
self.anim_frame_count = 5
Two things of interest here. We only start the animation if Duke is at rest. Clicks on Duke while
he is already waving get ignored. And when we do start the animation, we set the counter to
5 — this means that on the very next call to update the counter becomes 6, and the image
changes. If we had set the counter to 1, we would have needed to wait for 5 more calls to
update before anything happened — a slight lag, but enough to make things feel sluggish.
The final touch-up is to initialize our two new attributes when we instantiate the class. Here is
the code for the whole class now:
1
class DukeSprite:
2
3
408
def __init__(self, img, target_posn):
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4
5
6
7
self.image = img
self.position = target_posn
self.anim_frame_count = 0
self.curr_patch_num = 0
8
9
10
11
12
def update(self):
if self.anim_frame_count > 0:
self.anim_frame_count = (self.anim_frame_count +
˓→1 ) % 60
self.curr_patch_num = self.anim_frame_count // 6
13
14
15
16
17
def draw(self, target_surface):
patch_rect = (self.curr_patch_num * 50, 0,
50, self.image.get_height())
target_surface.blit(self.image, self.posn, patch_
˓→rect)
18
19
20
21
22
23
24
25
26
def contains_point(self, pt):
""" Return True if my sprite rectangle contains
˓→pt """
(my_x, my_y) = self.posn
my_width = self.image.get_width()
my_height = self.image.get_height()
(x, y) = pt
return ( x >= my_x and x < my_x + my_width and
y >= my_y and y < my_y + my_height)
27
28
29
30
def handle_click(self):
if self.anim_frame_count == 0:
self.anim_frame_count = 5
Now we have two extra Duke instances on our chessboard, and clicking on either causes that
instance to wave.
G.6. A wave of animation
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Aliens - a case study
Find the example games with the PyGame package, (On a windows system, something like
C:\Python3\Lib\site-packages\pygame\examples) and play the Aliens game. Then read the
code, in an editor or Python environment that shows line numbers.
It does a number of much more advanced things that we do, and relies on the PyGame framework for more of its logic. Here are some of the points to notice:
• The frame rate is deliberately constrained near the bottom of the game loop at line 311.
If we change that number we can make the game very slow or unplayably fast!
• There are different kinds of sprites: Explosions, Shots, Bombs, Aliens and a Player.
Some of these have more than one image — by swapping the images, we get animation
of the sprites, i.e. the Alien spacecraft lights change, and this is done at line 112.
• Different kinds of objects are referenced in different groups of sprites, and PyGame helps
maintain these. This lets the program check for collisions between, say, the list of shots
fired by the player, and the list of spaceships that are attacking. PyGame does a lot of the
hard work for us.
• Unlike our game, objects in the Aliens game have a limited lifetime, and have to get
killed. For example, if we shoot, a Shot object is created — if it reaches the top of the
screen without expoding against anything, it has to be removed from the game. Lines
141-142 do this. Similarly, when a falling bomb gets close to the ground (line 156), it
instantiates a new Explosion sprite, and the bomb kills itself.
• There are random timings that add to the fun — when to spawn the next Alien, when an
Alien drops the next bomb, etc.
• The game plays sounds too: a less-than-relaxing loop sound, plus sounds for the shots
and explosions.
Reflections
Object oriented programming is a good organizational tool for software. In the examples in this
chapter, we’ve started to use (and hopefully appreciate) these benefits. Here we had N queens
each with its own state, falling to its own floor level, bouncing, getting kicked, etc. We might
have managed without the organizational power of objects — perhaps we could have kept lists
of velocities for each queen, and lists of target positions, and so on — our code would likely
have been much more complicated, ugly, and a lot poorer!
Glossary
animation rate The rate at which we play back successive patches to create the illusion of
movement. In the sample we considered in this chapter, we played Duke’s 10 patches
over the duration of one second. Not the same as the frame rate.
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baked animation An animation that is designed to look good at a predetermined fixed frame
rate. This reduces the amount of computation that needs to be done when the game is
running. High-end commercial games usually bake their animations.
blit A verb used in computer graphics, meaning to make a fast copy of an image or pixels
from a sub-rectangle of one image or surface to another surface or image.
frame rate The rate at which the game loop executes and updates the display.
game loop A loop that drives the logic of a game. It will usually poll for events, then update
each of the objects in the game, then get everything drawn, and then put the newly drawn
frame on display.
pixel A single picture element, or dot, from which images are made.
poll To ask whether something like a keypress or mouse movement has happened. Game loops
usually poll to discover what events have occurred. This is different from event-driven
programs like the ones seen in the chapter titled “Events”. In those cases, the button click
or keypress event triggers the call of a handler function in your program, but this happens
behind your back.
sprite An active agent or element in a game, with its own state, position and behaviour.
surface This is PyGame’s term for what the Turtle module calls a canvas. A surface is a
rectangle of pixels used for displaying shapes and images.
Exercises
1. Have fun with Python, and with PyGame.
2. We deliberately left a bug in the code for animating Duke. If you click on one of the
chessboard squares to the right of Duke, he waves anyway. Why? Find a one-line fix for
the bug.
3. Use your preferred search engine to search their image library for “sprite sheet playing
cards”. Create a list [0..51] to represent an encoding of the 52 cards in a deck. Shuffle
the cards, slice off the top five as your hand in a poker deal. Display the hand you have
been dealt.
4. So the Aliens game is in outer space, without gravity. Shots fly away forever, and bombs
don’t speed up when they fall. Add some gravity to the game. Decide if you’re going to
allow your own shots to fall back on your head and kill you.
5. Those pesky Aliens seem to pass right through each other! Change the game so that they
collide, and destroy each other in a mighty explosion.
G.10. Exercises
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APPENDIX H
Plotting data with matplotlib
Introduction
There are many scientific plotting packages. In this chapter we focus on matplotlib, chosen
because it is the de facto plotting library and integrates very well with Python.
This is just a short introduction to the matplotlib plotting package. Its capabilities and
customizations are described at length in the project’s webpage, the Beginner’s Guide, the
matplotlib.pyplot tutorial, and the matplotlib.pyplot documentation. (Check in
particular the specific documentation of pyplot.plot).
Basic Usage – pyplot.plot
Simple use of matplotlib is straightforward:
>>> from matplotlib import pyplot as plt
>>> plt.plot([1,2,3,4])
[<matplotlib.lines.Line2D at 0x7faa8d9ba400>]
>>> plt.show()
If you run this code in the interactive Python interpreter, you should get a plot like this:
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Two things to note from this plot:
• pyplot.plot assumed our single data list to be the y-values;
• in the absence of an x-values list, [0, 1, 2, 3] was used instead.
Note: pyplot is commonly used abbreviated as plt, just as numpy is commonly abbreviated as np. The remainder of this chapter uses the abbreviated
form.
Note: Enhanced interactive python interpreters such as IPython can automate
some of the plotting calls for you. For instance, you can run %matplotlib
in IPython, after which you no longer need to run plt.show everytime when
calling plt.plot. For simplicity, plt.show will also be left out of the
remainder of these examples.
If you pass two lists to plt.plot you then explicitly set the x values:
>>> plt.plot([0.1, 0.2, 0.3, 0.4], [1, 2, 3, 4])
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Understandably, if you provide two lists their lengths must match:
>>> plt.plot([0.1, 0.2, 0.3, 0.4], [1, 2, 3, 4, 5])
ValueError: x and y must have same first dimension
To plot multiple curves simply call plt.plot with as many x–y list pairs as needed:
>>> plt.plot([0.1, 0.2, 0.3, 0.4], [1, 2, 3, 4],
[0.1, 0.2, 0.3, 0.4], [1, 4, 9, 16])
Alternaltively, more plots may be added by repeatedly calling plt.plot. The following code
snippet produces the same plot as the previous code example:
>>> plt.plot([0.1, 0.2, 0.3, 0.4], [1, 2, 3, 4])
>>> plt.plot([0.1, 0.2, 0.3, 0.4], [1, 4, 9, 16])
Adding information to the plot axes is straightforward to do:
H.2. Basic Usage – pyplot.plot
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>>>
>>>
>>>
>>>
plt.plot([0.1, 0.2, 0.3, 0.4], [1, 2, 3, 4])
plt.plot([0.1, 0.2, 0.3, 0.4], [1, 4, 9, 16])
plt.xlabel("Time (s)")
plt.ylabel("Scale (Bananas)")
Also, adding an legend is rather simple:
>>> plt.plot([0.1, 0.2, 0.3, 0.4], [1, 2, 3, 4], label=
˓→'first plot')
>>> plt.plot([0.1, 0.2, 0.3, 0.4], [1, 4, 9, 16], label=
˓→'second plot')
>>> plt.legend()
And adjusting axis ranges can be done by calling plt.xlim and plt.ylim with the lower
and higher limits for the respective axes.
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>>>
>>>
>>>
>>>
>>>
>>>
plt.plot([0.1, 0.2, 0.3, 0.4], [1, 2, 3, 4])
plt.plot([0.1, 0.2, 0.3, 0.4], [1, 4, 9, 16])
plt.xlabel("Time (s)")
plt.ylabel("Scale (Bananas)")
plt.xlim(0, 1)
plt.ylim(-5, 20)
In addition to x and y data lists, plt.plot can also take strings that define the plotting style:
>>>
>>>
>>>
>>>
plt.plot([0.1, 0.2, 0.3, 0.4], [1, 2, 3, 4], 'rx')
plt.plot([0.1, 0.2, 0.3, 0.4], [1, 4, 9, 16], 'b-.')
plt.xlabel("Time (s)")
plt.ylabel("Scale (Bananas)")
The style strings, one per x–y pair, specify color and shape: ‘rx’ stands for red crosses, and
‘b-.’ stands for blue dash-point line. Check the documentation of pyplot.plot for the list
H.2. Basic Usage – pyplot.plot
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of colors and shapes.
Finally, plt.plot can also, conveniently, take numpy arrays as its arguments.
More plots
While plt.plot can satisfy basic plotting needs, matplotlib provides many more plotting functions. Below we try out the plt.bar function, for plotting bar charts. The full list of
plotting functions can be found in the the matplotlib.pyplot documentation.
Bar charts can be plotted using plt.bar, in a similar fashion to plt.plot:
>>> plt.bar(range(7), [1, 2, 3, 4, 3, 2, 1])
Note, however, that contrary to plt.plot you must always specify x and y (which correspond, in bar chart terms to the left bin edges and the bar heights). Also note that you can
only plot one chart per call. For multiple, overlapping charts you’ll need to call plt.bar
repeatedly.
One of the optional arguments to plt.bar is width, which lets you specify the width of the
bars. Its default of 0.8 might not be the most suited for all cases, especially when the x values
are small:
>>> plt.bar(numpy.arange(0., 1.4, .2), [1, 2, 3, 4, 3, 2,
˓→1])
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Specifying narrower bars gives us a much better result:
>>> plt.bar(numpy.arange(0., 1.4, .2), [1, 2, 3, 4, 3, 2,
˓→1], width=0.2)
Sometimes you will want to compare a function to your measured data; for example when you
just fitted a function. Of course this is possible with matplotlib. Let’s say we fitted an quadratic
function to the first 10 prime numbers, and want to check how good our fit matches our data.
1
import matplotlib.pyplot as plt
2
3
4
def found_fit(x):
return 0.388 * x**2
# Found with symfit.
5
6
7
x_data = list(range(10))
y_data = [2, 3, 5, 7, 11, 13, 17, 19, 23, 29]
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x_func = np.linspace(0, 10, 50)
# numpy will do the right thing and evaluate found_fit for
˓→all elements
y_func = found_fit(x_func)
9
10
11
12
# From here the plotting starts
13
14
plt.scatter(x_data, y_data, c='r', label='data')
plt.plot(x_func, y_func, label='$f(x) = 0.388 x^2$')
plt.xlabel('x')
plt.ylabel('y')
plt.title('Fitting primes')
plt.legend()
plt.show()
15
16
17
18
19
20
21
We made the scatter plot red by passing it the keyword argument c='r'; c stands for colour,
r for red. In addition, the label we gave to the plot statement is in LaTeX format, making it
very pretty indeed. It’s not a great fit, but that’s besides the point here.
Interactivity and saving to file
If you tried out the previous examples using a Python/IPython console you probably got for
each plot an interactive window. Through the four rightmost buttons in this window you can
do a number of actions:
• Pan around the plot area;
• Zoom in and out;
• Access interactive plot size control;
• Save to file.
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The three leftmost buttons will allow you to navigate between different plot views, after zooming/panning.
As explained above, saving to file can be easily done from the interactive plot window. However, the need might arise to have your script write a plot directly as an image, and not bring up
any interactive window. This is easily done by calling plt.savefig:
>>>
>>>
>>>
>>>
>>>
plt.plot([0.1, 0.2, 0.3, 0.4], [1, 2, 3, 4], 'rx')
plt.plot([0.1, 0.2, 0.3, 0.4], [1, 4, 9, 16], 'b-.')
plt.xlabel("Time (s)")
plt.ylabel("Scale (Bananas)")
plt.savefig('the_best_plot.pdf')
Note: When saving a plot, you’ll want to choose a vector format (either pdf,
ps, eps, or svg). These are resolution-independent formats and will yield the best
quality, even if printed at very large sizes. Saving as png should be avoided, and
saving as jpg should be avoided even more.
Multiple figures
With this groundwork out of the way, we can move on to some more advanced matplotlib use.
It is also possible to use it in an object-oriented manner, which allows for more separation
between several plots and figures. Let’s say we have two sets of data we want to plot next to
eachother, rather than in the same figure. Matplotlib has several layers of organisation: first,
there’s an Figure object, which basically is the window your plot is drawn in. On top of that,
there are Axes objects, which are your separate graphs. It is perfectly possible to have multiple
(or no) Axes in one Figure. We’ll explain the add_subplot method a bit later. For now, it
just creates an Axis instance.
1
import matplotlib.pyplot as plt
2
3
4
x_data = [0.1, 0.2, 0.3, 0.4]
y_data = [1, 2, 3, 4]
5
6
fig = plt.figure()
H.5. Multiple figures
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8
9
10
11
ax = fig.add_subplot(1, 1, 1)
ax.plot([0.1, 0.2, 0.3, 0.4], [1, 2, 3, 4])
ax.plot([0.1, 0.2, 0.3, 0.4], [1, 4, 9, 16])
ax.set_xlabel('Time (s)')
ax.set_ylabel('Scale (Bananas)')
12
13
plt.show()
This example also neatly highlights one of Matplotlib’s shortcomings: the API is highly inconsistent. Where we could do xlabel() before, we now need to do set_xlabel(). In
addition, we can’t show the figures one by one (i.e. fig.show()); instead we can only show
them all at the same time with plt.show().
Now, we want to make multiple plots next to each other. We do that by calling plot on two
different axes:
1
2
x_data1 = [0.1, 0.2, 0.3, 0.4]
y_data1 = [1, 2, 3, 4]
3
4
5
x_data2 = [0.1, 0.2, 0.3, 0.4]
y_data2 = [1, 4, 9, 16]
6
7
8
9
10
11
12
13
14
15
16
17
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fig = plt.figure()
ax1 = fig.add_subplot(1, 2, 1)
ax2 = fig.add_subplot(1, 2, 2)
ax1.plot(x_data1, y_data1, label='data 1')
ax2.plot(x_data2, y_data2, label='data 2')
ax1.set_xlabel('Time (s)')
ax1.set_ylabel('Scale (Bananas)')
ax1.set_title('first data set')
ax1.legend()
ax2.set_xlabel('Time (s)')
ax2.set_ylabel('Scale (Bananas)')
Appendix H. Plotting data with matplotlib
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19
ax2.set_title('second data set')
ax2.legend()
20
21
plt.show()
The add_subplot method returns an Axis instance and takes three arguments: the first is
the number of rows to create; the second is the number of columns; and the last is which plot
number we add right now. So in common usage you will need to call add_subplot once for
every axis you want to make with the same first two arguments. What would happen if you first
ask for one row and two columns, and for two rows and one column in the next call?
Exercises
1. Plot a dashed line.
2. Search the matplotlib documentation, and plot a line with plotmarkers on all it’s datapoints. You can do this with just one call to plt.plot.
H.6. Exercises
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Appendix H. Plotting data with matplotlib
Index
A
chained conditional, 44, 70
character, 109
abecedarian series, 112
chatterbox function, 97
accumulator, 245, 367
child class, 254, 376
algorithm, 4, 9, 64, 70
chunking, 35, 85
deterministic, 176, 298
class, 13, 217, 339
aliases, 139, 148, 155
class attribute, 244, 366
alternative execution, 43
clone, 140, 148
ambiguity, 7
Collatz 3n + 1 sequence, 52
animation rate, 288, 410
collection, 133
argument, 82, 86
comment, 8, 9
assignment, 15, 49
comparison of strings, 114
tuple, 130
comparison operator, 39, 70
assignment statement, 15, 24, 49
compile, 3
assignment token, 24
attribute, 29, 69, 184, 185, 211, 217, 306, 307, composition, 24, 82, 95
composition (of functions), 97
333, 339
composition of functions, 22
B
compound data type, 109, 124, 209, 331
compound statement, 41, 87
baked animation, 289, 411
body, 41
base case, 197, 204, 319, 326
header, 41
blit, 289, 411
computation pattern, 116
block, 41, 69
concatenate, 24
body, 41, 69, 70, 86
concatenation, 22, 112
Boolean algebra, 69
condition, 51, 70
Boolean expression, 39, 69
conditional
Boolean function, 95, 97
chained, 44
Boolean value, 39, 69
conditional branching, 41
branch, 43, 69
conditional execution, 41
break statement, 58
conditional statement, 41, 70
bug, 5, 9
conditionals
builtin scope, 182, 304
nested, 45
bump, 70
constructor, 218, 340
C
continue statement, 61, 70
control flow, 34, 65, 69, 103
call graph, 156
copy,
223, 345
canvas, 29, 69
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deep, 223, 345
shallow, 223, 345
counter, 70
counting pattern, 117
cursor, 70
D
data structure, 132, 196, 318
recursive, 196, 318
data type, 13, 25
dead code, 90, 97
debugging, 5, 9, 94
decrement, 70
deep copy, 224, 346
deep equality, 221, 224, 343, 346
default value, 117, 124
definite iteration, 70
definition
function, 29, 77
recursive, 196, 318
del statement, 138, 153
delimiter, 148, 172
deterministic algorithm, 176, 298
dictionary, 151, 156
dir function, 118
directory, 170, 172
docstring, 87, 118, 124
dot notation, 118, 124
dot operator, 184, 185, 306, 307
dot product, 235, 357
Doyle, Arthur Conan, 6
E
element, 133, 148
elif, 41
else, 41
encode, 244, 366
enumerate, 140
equality, 221, 343
deep, 221, 343
shallow, 221, 343
escape sequence, 56, 70
eureka traversal, 116
evaluate, 25
exception, 5, 9, 255, 259, 377, 381
handling, 255, 377
expression, 18, 25
Boolean, 39
426
F
fibonacci numbers, 198, 320
field width, 120
file, 167, 172
file handle, 167
file system, 172
float, 13, 20, 25
floor division, 19, 25
flow of execution, 34, 81, 87
for loop, 33, 50, 69, 112, 140
for loop traversal (for), 124
formal language, 6, 9
formatting
strings, 120
fractal
Cesaro torn square, 204, 326
Sierpinski triangle, 205, 327
frame, 87
frame rate, 289, 411
fruitful function, 87, 97
fully qualified name, 185, 307
function, 29, 61, 77, 87
argument, 82
composition, 82
len, 112
parameter, 82
pure, 226, 348
function call, 87
function composition, 22, 87, 95
function definition, 29, 77, 87
function tips, 102
function type, 118
functional programming style, 235, 357
G
game loop, 265, 289, 387, 411
generalization, 230, 352
global scope, 182, 304
H
hand trace, 54
handle, 167, 172
handle an exception, 259, 381
handling an exception, 255, 377
header line, 87
help, 55
high-level language, 3, 9
Holmes, Sherlock, 6
Index
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I
lifetime, 85, 87
Linux, 6
if, 41
list, 133, 149
if statement, 41
append, 142
immediate mode, 9
nested, 133, 146
immutable, 115, 129, 136
list
index,
133
immutable data value, 124, 132, 148, 156
import statement, 82, 87, 180, 184, 185, 302, list traversal, 133, 149
literalness, 7
306, 307
local scope, 182, 304
in and not in operator (in, not in), 124
local variable, 85, 87
in operator, 115
logarithm, 56
increment, 70
logical operator, 39, 40, 70
incremental development, 92, 97
loop, 51, 71
indefinite iteration, 70
loop body, 51, 69
index, 109, 124, 133, 149
loop variable, 69, 71
negative, 112
low-level language, 3, 9
indexing ([]), 124
infinite loop, 51, 70
M
infinite recursion, 197, 204, 319, 326
Make Way for Ducklings, 112
inheritance, 254, 376
mapping type, 151, 156
initialization (of a variable), 70
matrix, 148
initializer method, 218, 340
McCloskey, Robert, 112
input, 22
memo, 156
input dialog, 22
meta-notation, 55, 71
instance, 31, 69, 218, 340
method, 29, 69, 185, 218, 307, 340
instantiate, 218, 340
middle-test loop, 71
int, 13, 20, 25
mode, 172
integer, 13
modifier, 143, 149, 227, 235, 349, 357
Intel, 56
module,
29, 69, 118, 186, 308
interpret, 3
modulus operator, 23, 25
interpreter, 9
mutable, 115, 129, 136
invoke, 29, 69
mutable data value, 124, 132, 149, 156
is operator, 138
item, 133, 149
N
item assignment, 136
namespace, 180, 186, 302, 308
iteration, 49, 51, 71
naming collision, 186, 308
natural language, 6, 9
J
negative index, 112
join, 144
nested conditionals, 45
justification, 120
nested list, 133, 146, 149
nested loop, 71
K
nesting, 70
key, 151, 156
newline, 56, 71
key:value pair, 151, 156
Newton’s method, 63
keyword, 16, 25
non-volatile memory, 172
None, 90, 97, 102
L
normalized, 235, 357
len function, 112
length function (len), 124
Index
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O
R
raise, 259, 381
object, 29, 69, 149, 218, 340
random numbers, 175, 297
object code, 9
range, 69
object-oriented language, 218, 340
object-oriented programming, 209, 218, 331, range function, 35, 145
rectangle, 220, 342
340
recursion, 197, 204, 319, 326
objects and values, 138
infinite, 197, 319
operand, 19, 25
recursive call, 197, 204, 319, 326
operations on strings, 120
recursive data structure, 196, 318
operator, 19, 25
recursive definition, 196, 204, 318, 326
comparison, 39
redundancy, 7
in, 115
refactor, 87
logical, 39, 40
refactoring code, 85
modulus, 23
return, 102
operator overloading, 235, 357
return a tuple, 131
optional parameter, 117, 125
return statement, 47, 90
order of operations, 21
return value, 90, 97
P
rules of precedence, 21, 25
runtime error, 5, 10, 112, 115
parameter, 82, 87, 141
optional, 117
S
parent class, 254, 376
safe language, 5
parse, 6, 9
scaffolding, 92, 97
pass statement, 41
scalar multiplication, 236, 358
path, 172
scope, 182, 304
pattern, 149
builtin, 182, 304
pattern of computation, 116
global, 182, 304
Pentium, 56
local, 182, 304
pixel, 289, 411
script,
10
poetry, 7
semantic error, 5, 10
poll, 265, 289, 387, 411
semantics, 5, 10
polymorphic, 236, 358
sequence, 133, 149
portability, 9
shallow copy, 224, 346
portable, 3
shallow equality, 221, 224, 343, 346
post-test loop, 71
short-circuit evaluation, 116, 125
pre-test loop, 71
shuffle, 175, 297
print function, 9
side effect, 143, 149
problem solving, 10
slice, 113, 125, 136
program, 4, 8, 10
slicing ([:]), 124
program tracing, 54
source code, 10
programming language, 3
split, 144
promise, 145, 149, 175, 297
sprite, 289, 411
prompt, 70
stack diagram, 88
prose, 8
state, 29
pure function, 149, 236, 358
state snapshot, 15, 25
PyGame, 265, 387
statement, 18, 25
PyScripter, 3
assignment, 49
Python shell, 10
428
Index
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Documentation, Release 3rd Edition
continue, 61
del, 138, 153
if, 41
import, 82, 180, 302
pass, 41
return, 47
statement block, 41
statement: break, 58
step size, 149
str, 20, 25
string, 13, 106
string comparison, 114
string comparison (>, <, >=, <=, ==,
=), 124
string formatting, 120
string module, 118
string operations, 22
string slice, 113
strings and lists, 144
style, 96
sublist, 113, 136
subscript operator, 109
substring, 113
surface, 265, 289, 387, 411
syntax, 5, 10
syntax error, 5, 10
T
tuple assignment, 132
turtle module, 29
two-dimensional table, 57
type, 13
type conversion, 70
type converter functions, 20
U
underscore character, 16
unit tests, 176, 298
unreachable code, 90
V
value, 13, 25, 90, 151
Boolean, 39
variable, 15, 25
local, 85
temporary, 90
variable name, 25
variables local, 105
void function, 88
volatile memory, 172
W
while loop, 51
while statement, 51
whitespace, 125
wrapping code in a function, 43
tab, 56, 71
table, 56
temporary variable, 90, 97
terminating condition, 69
text file, 172
token, 6, 10
trace, 71
traceback, 88
tracing a program, 54
traversal, 112, 116
traverse, 125
trichotomy, 71
triple quoted string, 13
truncation, 20
truth table, 70
try ... except, 255, 377
try ... except ... finally, 258, 380
tuple, 129, 132
assignment, 130
return value, 131
Index
429

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