Topic 1 Numbers

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NCV 2
MATHEMATICS
WORKBOOK
This material was written by
Gayle Staegermann
And edited by
Jackie Scheiber
RADMASTE Centre
University of the Witwatersrand
Telephone (011) 717-6070
Fax (011) 339-1054
Copyright of the material remains with the authors.
However, this material may be photocopied and used for educational purposes.
ii
CONTENTS
Page
TOPIC 1: NUMBERS
Chapter 1
Natural Numbers, Whole Numbers and Integers ………………...
1
Chapter 2
Fractions and Decimals …………………………………………..
20
Chapter 3
Rational Numbers and Real Numbers ……………………………
39
TOPIC 2: FUNCTIONS
Chapter 4
Working with Algebraic Expressions ……………………………
58
Chapter 5
Linear Equations and Linear Functions ………………………….
75
Chapter 6
Quadratic Equations and Functions, Hyperbolic functions and
Exponential Equations and Functions ……………………………
100
TOPIC 3: SPACE. SHAPE AND ORIENTATION
Chapter 7
Properties of Shapes ……………………………………………...
128
Chapter 8
Analytical and Transformation Geometry ………………………..
144
Chapter 9
Trigonometry ……………………………………………………..
153
TOPIC 4: STATISTICAL AND PROBABILITY MODELS
Chapter 10
Data Handling ……………………………………………………
170
TOPIC 5: FINANCIAL MATHEMATICS
Chapter 11
Financial Mathematics …………………………………………...
iii
190
NCV2 – Chapter 1: Natural numbers, whole numbers and integers
CHAPTER 1
Natural numbers, whole
numbers and integers
In this chapter you will:
• Use the order of operations to do calculations
• Calculate factors of a number
• Classify numbers as either prime or composite
• Calculate prime factors
• Calculate the Highest Common Factor (HCF)
• Calculate multiples of a number
• Calculate the Lowest Common Multiple (LCM)
• Use laws of exponents to simplify expressions
• Calculate square roots and cube roots
• Distinguish between sets of natural numbers and whole numbers
• Add, subtract, multiply and divide using a calculator
• Find squares, cubes and higher powers on a calculator
• Find square roots, cube roots and higher roots on a calculator
• Estimate answers
• Add and subtract integers using a number line
• Add and subtract integers without using a number line
• Multiply and divide integers
• Distinguish between integers, whole numbers and natural numbers.
This chapter covers material from Topic 1: Numbers
SUBJECT OUTCOME 1.1:
Use Computational Tools and Strategies and Make Estimates and Approximations
Use a scientific calculator competently and efficiently
Learning Outcome 1:
Execute algorithms appropriately in calculations
Learning Outcome 2:
1
NCV2 – Chapter 1: Natural numbers, whole numbers and integers
§ 1.1
1.1
ORDER OF OPERATIONS
If a calculation has more than one operation, we do them in the following fixed order:
1. ( ) Brackets
2. Exponents
3. × ÷ Multiplication and Division, working from left to right
4. + – Addition and Subtraction, working from left to right
BEMDAS will help you to remember the order in which you must work. Unless you follow this
order, you will get wrong answers.
EXAMPLES:
Calculate the answers without
using a calculator:
1)
5+2–7
2)
24 × 2 ÷ 8 ÷ 3
3)
9–4 × 2
4)
7 + 4 × (3 + 2)
5)
22 + 4 × 3
6)
8–2 × 6+5
7)
6 + 32 ÷ 3
8)
(6 + 9) ÷ 3
9)
2(3 + 1) – 4 ÷ 2
SOLUTIONS
5 + 2 – 7 = 7 – 7 = 0 ….. Work from left to right
24 × 2 ÷ 8 ÷ 3 = 48 ÷ 8 ÷ 3 = 6 ÷ 3 = 2
….. Work from left to right
9 – 4 × 2 = 9 – (4 × 2) = 9 – 8 = 1
….. Multiply before subtracting
7 + 4 × (3 + 2) = 7 + (4 × 5) = 7 + 20 = 27
…..Simplify brackets, multiply, then add
22 + 4 × 3 = 4 + (4 × 3) = 4 + 12 = 16
..… Simplify the exponent, multiply, then add
8 – 2 × 6 + 5 = 8 – (2 × 6 ) + 5 = 8 – 12 + 5 = 1
..… Multiply, subtract, then add and subtract
6 + 32 ÷ 3 = 6 + (9 ÷ 3) = 6 + 3 = 9
..…Simplify the exponent, divide, then add
(6 + 9) ÷ 3 = 15 ÷ 3 = 5
..… Simplify brackets, then divide
2(3 + 1) – 4 ÷ 2 = 2(4) – (4 ÷ 2) = 8 – 2 = 6
..… Simplify brackets, divide, then subtract
Exercise 1.1
Calculate the answers without using a calculator:
1)
9 – 8 + 1 = ……………………………...
2)
13 – 10 – 3 = …………………………….
3)
8 + 5 – 2 + 3 = ………………………….
4)
12 + 5 – 2 = ……………………………...
5)
12 ÷ 6 ÷ 2 = …………………………….
6)
4 × 6 – 3 = ……………………………….
7)
15 – 3 × 4 = …………………………….
8)
(15 – 3) × 4 = …………………………….
9)
12 ÷ 4 + 2 = ……………………………
10)
12 ÷ (4 + 2) = …………………………….
11) 3 × (2 + 1) – 3 ÷ 3 = ……………………………………………………………………………….
12) 7 – 2 × 3 + 4 ÷ 4 = …………………………………………………………………………………
13) 19 – 4(5 – 3) × 2 = …………………………………………………………………………………
14) 9 × 105 × 28 × (3 – 3) = ………………………………………………………………………….
15) 23 – 15 + 4 – 12 + 9 – 3 = ………………………………………………………………………….
16)
6 + 2 × 8 – 8 – 5 = …………………………………………………………………………………
2
NCV2 – Chapter 1: Natural numbers, whole numbers and integers
§ 1.2
1.2 FACTORS
A factor of a number divides into that number exactly, without a remainder.
3 is a factor of 12, since 12 ÷ 3 = 4. 3 divides exactly into 12.
3 is not a factor of 16, since 16 ÷ 3 = 5 remainder 1.
There are other factors of 12. To find them, divide by 1, 2, 3, ….. in this order. Those that divide
into 12 exactly are factors.
Note:
• F stands for factor and F12 means "the set of factors of 12".
EXAMPLE
SOLUTION
1)
Is 7 a factor of 42? Why?
7 is a factor of 42 since 42 ÷ 7 = 6
2)
3)
Is 7 a factor of 41? Why?
List all the factors of 12.
4)
List all the factors of 30.
7 is not a factor of 41, since 41 ÷ 7 = 5 remainder 6
12 ÷ 1 = 12, 12 ÷ 2 = 6, 12 ÷ 3 = 4, 12 ÷ 4 = 3,
12 ÷ 6 = 2, 12 ÷ 12 = 1. So F12 = {1, 2, 3, 4, 6, 12}.
There are 6 factors of 12. 1 is the least factor and 12
is the greatest factor.
F30 = {1, 2, 3, 5, 6, 10, 15, 30}
Note:
• The least factor of any number is always 1, since every number can be divided by 1.
• The greatest factor of any number is the number itself.
Exercise 1.2
1)
Is 6 a factor of 54? Why? …………………………………………………………………………
2)
Is 6 a factor of 46? Why? …………………………………………………………………………
3)
a)
List all the factors of 18. F18 = {…………………………………………………………….
b)
How many factors does 18 have? ……………………………………………………………
c)
What is the least factor of 18? ………………………………………………………………
d)
What is the greatest factor of 18? ……………………………………………………………
4)
List all the factors of 24. F24 = { …………………………………………………………………
5)
List all the factors of 36. F36 = { …………………………………………………………………
6)
List all the factors of 45. ………………………………………………………………………….
7) What are the greatest and the least factors of 179?
Greatest factor = ……......……………………………………………
Least factor = …………………………………………………………
3
NCV2 – Chapter 1: Natural numbers, whole numbers and integers
§ 1.3
1.3 PRIME
PRIME AND COMPOSITE NUMBERS
A prime number has only 2 factors, itself and 1.
A composite number has more than 2 factors.
NOTE:
• 7 is prime, since its only factors are 1 and 7. It has only 2 factors.
• 6 is not prime, since its factors are 1, 2, 3, and 6. 6 is a composite number since it has more
than 2 factors.
• 1 is not prime. The only factor of 1 is 1, it does not have 2 factors. So 1 is neither prime nor
composite.
EXAMPLE
1) Is 1 740 prime or composite?
Give a reason for your answer.
2) Is 365 prime?
Why?
SOLUTION
1, 10, 174 and 1 740 are some of the factors of 1 740.
It is composite since it has more than 2 factors.
No, 365 is NOT prime.
1, 5, and 365 are factors of 365, so it has more than 2
factors.
NOTE:
• Numbers that end in 0 have a factor of 10
• Numbers that end in 5 have a factor of 5
• The set of even numbers = {2; 4; 6; 8; 10; ………………..}.
In other words, numbers that end in an even number have a factor of 2.
Exercise 1.3
1) Are the following numbers prime or composite? Give a reason for your answer.
a) 8 ………………………………………………………………………………………..............
b) 11 ………………………………………………………………………………………………
c) 14 ………………………………………………………………………………………………
d) 19 ………………………………………………………………………………………………
e) 1 ………………………………………………………………………………………..............
f) 23 ……………………………………………………………………………………………….
g) 445 ……………………………………………………………………………………...............
h) 986 ……………………………………………………………………………………...............
2) List all the prime numbers up to and including 20
. ……………………………………………..
………………………………………………………………………………………………………
………………………………………………………………………………………………………
3) List all the composite numbers up to and including 20 …………………………………………….
………………………………………………………………………………………………………
……………………………………………………………………………………………………….
4
NCV2 – Chapter 1: Natural numbers, whole numbers and integers
§ 1.4
1.4 PRIME FACTORS
A prime factor is a factor that is a prime number.
EXAMPLE
1) We said that F12 = {1, 2, 3, 4, 6, 12}
List the prime factors of 12
2) Which factors of 12 are composite?
3)
4)
SOLUTION
The prime factors are 2 and 3
The composite factors are 4; 6; 12
Why is 1 not a prime factor of 12?
Factorise 36 into prime factors
1 has only one factor, so it is not prime.
Prime
Other
factors factors
2 36
After dividing, we can say
36 = 2 × 2 × 3 × 3
= 2 2 × 32
2 18
3
9
3
3
Note:
We do not include 1
because it is not a prime
number
1
Exercise 1.4
1) a) List all the factors of 15:
F15 = {
;
;
;
}
b) List the prime factors of 15 …………………………………………………………………….
c) Which factors of 15 are composite? ……………………………………………………………
4)
a)
Factorise 60, 72,
and 90 into prime
factors.
Prime
factors
Prime
factors
Other
factors
60
Other
factors
72
Prime
factors
Other
factors
75
b)
List the prime factors of 60 ……………………………………………………………………
c)
List the prime factors of 72 ……………………………………………………………………
d)
List the prime factors of 75 ……………………………………………………………………
e)
Is 1 a factor of 60? Give a reason for your answer
………………………………………………………………………………………………….
f)
Is 1 a prime factor of 60? Give a reason for your answer
………………………………………………………………………………………………….
5
NCV2 – Chapter 1: Natural numbers, whole numbers and integers
§ 1.5
1.5 HIGHEST COMMON FACTOR
HCF stands for Highest Common Factor. It is the greatest factor common to two or more
numbers.
EXAMPLE
1) Find the HCF of 36 and 24 by
listing all their factors.
2)
SOLUTION
F24 = {1, 2, 3, 4, 6, 8, 12, 24}
F36 = {1, 2, 3, 4, 6, 9, 12, 18, 36}
The HCF of 36 and 24 = 12, since this is the highest factor
common to both.
Find the HCF of 36 and 24 by
breaking each number into its
prime factors.
Prime
Other
Prime
Other
factors factors
factors factors
2 24
2 36
2 12
2 18
2 6
3 9
3 3
3 3
1
1
Look for the factors that are common to both numbers
24 = 2 × 2 × 2 × 3
36 = 2 × 2 × 3 × 3
The HCF of 36 and 24 = 2 × 2 × 3 = 12.
Exercise 1.5
1) Find the HCF of 120 and 90 by breaking each number into its prime factors.
Prime
factors
Other
factors
120
Prime
factors
Other
factors
90
120 = ……………………………………………………………………….
90 = …………………………………………………………………………
HCF = ……………………………………………………………………….
2) Find the HCF of 125 and 75 by listing all their factors.
F125 = {
;
F75 = {
;
;
;
;
;
}
;
;
}
HCF = ……………………………………………………………………….
3) Find the HCF of 180 and 150 by listing all their factors.
F180 = {……………………………………………………………………….
F150 = {……………………………………………………………………….
HCF = ……………………………………………………………………....
6
NCV2 – Chapter 1: Natural numbers, whole numbers and integers
§ 1.6
1.6 MULTIPLES
A multiple of a whole number is found by multiplying it by any whole number.
Examples:
The multiples of 5 are 5 × 1, 5 × 2, 5 × 3, 5 × 4, 5 × 5, …… = 5, 10, 15, 20, 25, ………..
The multiples of 1 are 1, 2, 3, 4, 5, …………
Note that M stands for multiple and M24 means "the set of multiples of 24"
We cannot list all the multiples of a number since there are an infinite number of them, so we use
dots to show that they are ongoing
EXAMPLE
1) a) List the factors of 12
b) List the multiples of 12
2) a) Write down the factors of 18
b) Write down the multiples of 18.
3) Which of the numbers 11, 22, 41, 77 are
multiples of 11?
SOLUTION
a) F12 = {1; 2; 3; 4; 6; 12}
b) Multiples of 12 are 12 × 1; 12 × 2; 12 × 3; ….
M12 = {12; 24; 36; 48; 60; …………..}
a) F18 = {1; 2; 3; 6; 9; 18}
b) The multiples of 18 are 18 × 1 = 18,
18 × 2 = 36, 18 × 3 = 54 etc.
So M18 = {18; 36; 54; ……}
11 × 1 = 11, 11 × 2 = 22 and 11 × 7 = 77,
so 11, 22 and 77 are multiples of 11
Note:
• factors of a number are the number itself and smaller
• multiples of a number are the number itself and greater
Exercise 1.6
1) a)
List the factors of 15 …………………………………………………………………………..
b)
List the multiples of 15 ………………………………………………………………………..
c)
Which number is both a factor and a multiple of 15? …………………………………………
2) a)
List the multiples of 20 ………………………………………………………………………..
b)
List the factors of 20 …………………………………………………………………………..
c)
Which multiple of 20 is also a factor of 20? …………………………………………………..
3) Which of the following numbers are multiples of 9: 9, 18, 36, 109? Why?
……………………………………………………………………………………………………….
………………………………………………………………………………………………………
4) Write down the first twelve multiples of 8
……………………………………………………………………………………………………….
……………………………………………………………………………………………………….
7
NCV2 – Chapter 1: Natural numbers, whole numbers and integers
§ 1.7
1.7 LCM
LCM stands for Lowest Common Multiple. It is the lowest number into which two or more
numbers divide.
EXAMPLE
1) Find the Lowest Common Multiple
(LCM) of 36 and 24 by listing the
multiples of each.
SOLUTION
M36 = {36, 72, 108, …}
M24 = {24, 48, 72, …}
The LCM = 72. This is the lowest multiple which is
common to both 36 and 24.
2)
Use prime factors to find the HCF
and LCM of 504 and 180.
Prime
factors
2
2
2
3
3
7
Other
factors
504
252
126
63
21
7
1
Prime
Other
factors factors
2 180
2 90
3 45
3 15
5
5
1
504 = 2 × 2 × 2 × 3 × 3 × 7
180 = 2 × 2 × 3 × 3 × 5
HCF = 2 × 2 × 3 × 3 = 36
To find the LCM, multiply the HCF by those factors
that are not common.
LCM = 36 × 2 × 5 × 7 = 2 520
Exercise 1.7
1) Find the LCM of 15 and 20 by listing the multiples of each.
………………………………………………………………………………………………………
………………………………………………………………………………………………………
2) Find the HCF of 105 and 140 by listing the factors of each.
………………………………………………………………………………………………………
………………………………………………………………………………………………………
……………………………………………………………………………………………………..
3) Find the LCM of 105 and 140 by listing the multiples of each.
………………………………………………………………………………………………………
………………………………………………………………………………………………………
………………………………………………………………………………………………………
8
NCV2 – Chapter 1: Natural numbers, whole numbers and integers
§ 1.8
1.8 EXPONENTS, SQUARES AND CUBES
In the expression 37 , 3 is called the base and 7 is called the exponent, index or power.
The exponent gives the number of times the base is multiplied by itself.
37 expanded means 3 × 3 × 3 × 3 × 3 × 3 × 3
37 is said to be in index form or exponent form or written as a power of the base.
EXAMPLE
1) Expand the following:
a) 24
b) x 3
c) (a + b)2
2) Write the following in exponential form:
a) 6 × 6 × 6
b) x × x
c) p × p × p × p × p
d) (d + g)(d + g)
SOLUTION
24 = 2 × 2 × 2 × 2
x3 = x × x × x
(a + b)2 = (a + b)(a + b)
6 × 6 × 6 = 63
x × x = x2
p × p × p × p × p = p5
(d + g)(d + g) = (d + g)2
Exercise 1.8
1) Expand the following:
a) 54 = ……………………. b) x3 = ……………………. c) ( x + y ) 2 = …………………….
2) Write the following in index form:
a) 8 × 8 × 8 × 8 = ………...
b) b × b × b = ………… c) (p + q)(p + q)(p + q) = ……………
The square of any number is the number multiplied by itself.
The cube of any number is the number multiplied by itself and by itself again.
EXAMPLE
Evaluate:
1) 72
2) 23
3) (4 + 2)2
4) 42 + 22
SOLUTION
72 = 7 × 7 = 49
23 = 2 × 2 × 2 = 8
(4 + 2)2 = 62 = 36
42 + 22 = (4 × 4) + (2 × 2) = 16 + 4 = 20
Note:
• 72 is read "seven squared" and 23 is read "two cubed"
• 49 = 72 is called a "perfect square" and 8 = 23 is called a "perfect cube".
3) Calculate:
a) 52 + 12 = …………..…….
b) (5 + 1)2 = …………….…… c) 14 + 41 = ……………..….
d) (4 – 3)2 = ……………….
e) 43 – 32 = ………………….
g) (1 + 2)3 = …………...…..
h) (2 + 5 – 3)2 = ..........................................................................
4) a) 36 is the square of ……....
f) 25 – 2 = ………………….
b) 64 is the cube of .…….. and the square of ……...........
c) Twelve squared = ……....
d) Ten squared = …….……..
f) …..….. is the square of 11
g) The square of 9 = ................... h) Twenty squared = .............
9
e) Five cubed = ……........….
NCV2 – Chapter 1: Natural numbers, whole numbers and integers
§ 1.9
1.9
SQUARE ROOTS AND CUBE ROOTS
Note:
is the symbol for square root and means
3
is the symbol for cube root
16 is read "the square root of 16"
3
2
where the 2 is assumed
27 is read "the cube root of 27"
The square root of a number multiplied by itself gives the number.
3 × 3 = 3× 3 = 3
Take one from every pair of identical numbers under the square root sign as the answer.
5 × 5 = 3 × 5 = 15
3× 3 ×
3× 5× 3× 5 =
So
The cube root of a number multiplied by itself and by itself again gives the number.
3
2 × 3 2 × 3 2 = 3 2× 2× 2 = 2
Take one from every three identical numbers under the cube root sign as the answer.
So
3
EXAMPLE
Evaluate
1)
16
2)
3
3)
4)
5)
27
7)
8)
1
59 × 59 × 59
1
10)
1 =
1× 1× 1 = 1
25 =
5×5 = 5
78 × 78 = 78
3
59 × 59 × 59 = 59
9 + 16 =
7
9
1
8 × 18
Exercise 1.9
Calculate
1) 81 = ……….
0×0 = 0
3
9 + 16 =
9 + 16
9)
3× 3× 3 = 2 × 3 = 6
27 = 3 3 × 3 × 3 = 3
0 =
3
78 × 78
3
3
16 = 4 × 4 = 4
3
9 + 16
6)
2× 2× 2 ×
SOLUTION
0
3
3
2 × 3× 3× 2 × 3× 2 =
7
=
9
16
=
9
8 × 18 =
2)
3× 3 +
4× 4
4
=
3
3× 3
(2 × 2 × 2) × (2 × 3 × 3) = 2 × 2 × 3 = 12
144 = …..…..
4 = ………………..……
4×4 = 3 + 4 = 7
3)
3
27 = ……....
4)
3
8 = …..…..
5)
16 –
7)
169 – 16 = ……………………
8)
9)
25 − 16 = ……………………..
10)
36 + 3 125 = ………………………
12)
3
16
= …………..
25
11)
27
= ……………
125
15) 3 19 × 19 × 19 = …………….
13)
3
6)
169 − 25 = ..…………
3
1 + 81 = ……..…………………
100
14)
16)
10
64
5
3
= …………………..
4
= ………………………….…..
9
9 × 4 × 6 = ………………………………….
NCV2 – Chapter 1: Natural numbers, whole numbers and integers
§ 1.10
1.10 THE SET OF NATURAL NUMBERS AND WHOLE
NUMBERS
Numbers belong in number sets which have names. Some numbers belong to more than one
number set.
So far, all the numbers we have worked with have been either natural numbers or whole numbers.
Here are the first two sets. Remember, the dots in the set brackets mean that the numbers continue
in the same way without end.
The set of Natural numbers (also called counting numbers): N = {1; 2; 3; 4; …………..}
The set of Whole numbers: W or N0 = {0; 1; 2; 3; …………….}
These numbers can be shown on a number line. The arrow on the end means that the numbers
continue in the same way without end.
1
2
3
4
5
6
0
1
2
3
4
5
N = {1; 2; 3; 4; …….….….}
6
W = {0; 1; 2; 3; ……….}
EXAMPLE:
1) 0 is a Whole number (W)
2) 1 is a Natural numbers (N) and a Whole numbers (W)
3) 245 is a Natural numbers (N) and a Whole numbers (W)
Exercise 1.10
1) Write down the letter that stands for:
a) the set of whole numbers ………………………………………………………………
b) the set of natural numbers ……………………………………………………………...
2) What is the smallest natural number? ……………………………………………………….
3) What is the greatest natural number? ……………………………….……………………….
4) What is the smallest whole number? ……………………………….………………………..
5) Write down a number that is a whole number but not a natural number. ..……………….…
6) Are the following numbers whole numbers and/or natural numbers?
a) 4 …………………………………………………………………………………………
b) 56 ………………………………………………………………………………………..
c) 0 ………………………………………………………………………………………….
d) 249 ……………………………………………………………………………………….
11
NCV2 – Chapter 1: Natural numbers, whole numbers and integers
§ 1.11
1.11 ADDITION, SUBTRACTION, MULTIPLICATION AND
DIVISION USING A CALCULATOR
Remember BIMDAS, the correct order of operations. Your scientific calculator follows this
order automatically.
EXAMPLE
Calculate
1)
8− 2
3
2)
(8 – 2) ÷ 3
3)
8− 2÷3
4)
2
8–
3
SOLUTION
Key sequence
Display
÷ 3 =
8
–
2
=
(
8
–
2
8
–
2
÷ 3 =
)
2
÷ 3 =
2
7.33…
Answer
8− 2
=2
3
(8 – 2) ÷ 3 = 2
8− 2÷3 = 8 –
2
= 7,33…
3
Note:
For 1), you have to enter = after 8 – 2 before dividing by 4. This is because the entire numerator
is divided by 3.
For 2), you use brackets instead of = . Questions 1) and 2) are identical.
For 3) and 4) the answer is not 2. The calculator first divides 2 by 3 before subtracting the answer
from 8.
Exercise 1.11
Use your calculator to calculate the following:
1)
100 – 84 ÷ 2 = ……….
2)
(100 – 84) ÷ 2 = ……….
3)
310 × 54 – 22 = ……….
4)
310 × (54 – 22) = ……….
5)
840 ÷ 21 + 3 = ……….
6)
840 ÷ (21 + 3) = ……….
7)
321 + 12 ÷ 3 = ……….
8)
(321 + 12) ÷ 3 = ……….
9)
102 – 34 × 5 + 96 = ……….
10) 102 – 34 × (5 + 96) = ……….
11) (102 – 34) × 5 + 96 = ……….
12) (102 – 34) × (5 + 96) = ……….
13) 126 + 18 ÷ 9 = ……….
14) (126 + 18) ÷ 9 = ……….
15) 145 × 23 ÷ 5 = ……….
16) 114 ÷ 19 ÷ 2 = ……….
17) 459 ÷ 3 × 9 = ……….
18) 459 ÷ (3 × 9) = ……….
19) 154 – 6 ÷ 2 = ……….
20) (154 – 6) ÷ 2 = ……….
21) 119 ÷ 17 – 7 = ……….
22) 119 ÷ (17 – 7) = ……….
12
NCV2 – Chapter 1: Natural numbers, whole numbers and integers
§ 1.12
1.12 SQUARES, CUBES AND HIGHER POWERS ON A
CALCULATOR
On a CASIO fx-82ES scientific calculator,
x 2 is used for squaring a number
x3
is used for cubing a number
x
is used for raising a base to any exponent
Note:
On some calculators the key
x
is marked
EXAMPLE
Use your calculator
to calculate:
^
or
xy
or
yx
SOLUTION
Key Sequence
1)
1982
198
x2
2)
183
18
x3
3)
145
14
x
Answer
1982 = 39 204
=
183 = 5 832
=
5
145 = 537 824
=
4) 25 × 163
25 × 16
x3
5) 2 3792 − 453
2 379
x2
–
6) (23 + 19)3
(
23
+ 19
7) (172 – 132)4
(
17
x2
25 × 163 = 102 400
=
45
x3
=
1232 - 453 = 5 568 516
)
x3
=
(23 + 19)3 = 74 088
– 13
x2
x
)
Note:
•
On some calculators you may not have to enter the
4
=
=
(172 – 132)4 = 207 360 000
key for squares and cubes.
Exercise 1.12
Use your calculator to evaluate:
1)
542 = ……….
2)
433 = ……….
3)
184 = ……….
4)
153 – 143 = ……….
5)
172 – 162 + 52 = ……….
6)
(222 – 192)3 = ……….
7)
182 – 82 = ……….
8)
(18 – 8)2 = ……….
9)
(43 – 19)4 = ……….
10) 434 – 194 = ……….
11) 153 – 142 + 162 = ……….
12) 144 – (132 + 123) = ……….
13) 1232 – 133 + 124 = ……….
14) (252 – 202 – 52)2 = ……….
15) (233 + 124)2 = ……….
16) (56 – 23 + 45)4 = ……….
17) 12 × 152 × 132 = ……….
18) (32 + 47)3 = ……….
13
NCV2 – Chapter 1: Natural numbers, whole numbers and integers
§ 1.13
1.13 SQUARE ROOTS, CUBE ROOTS AND HIGHER ROOTS
ON A CALCULATOR
On your CASIO fx-82ES scientific calculator, the key:
is used for finding the square root
3
is used for finding the cube root
is used for finding other roots greater than 3
Note:
1
On some calculators the key
EXAMPLES
Use your calculator to
calculate:
1)
5
3)
841
5
9 + 16
(
3
9 261 –
5)
4
2 825 761
3
361
x
or
Key sequence
1889 568
4)
yx
or
or
x
y
SOLUTION
841
2)
1
xy
is marked
Answer
841 = 29
=
1 889 568
9
9 261
4
+
16
=
)
5
9 + 16 = 5
=
–
361
2 825 761
1889 568 = 18
=
=
3
9 261 –
361 = 2
4
2 825 761 = 41
Note:
• In 3) the brackets around the '9 + 16' are essential to get the right answer
If you forget the brackets in 3), your calculator will calculate
•
9 + 16 = 19
Exercise 1.13
Use your calculator to calculate:
1)
5
537 824 = ……….
2)
145 = ……….
3)
3
12 167 = ……….
4)
233 = ……….
14 400 = ……….
6)
729 +
3
3 375 = ……….
62 748 517 = ……….
8)
961 –
4
28 561 = ……….
252 − 242 = ……….
10) 323 –
5)
7)
7
9)
11)
( 4 130 321 +
13)
3
15)
289 )2 = ……….
361 – 3 1 331 = ……….
12) ( 256 –
225 )2 = ……….
35 937 − 256 = ……….
14)
4
130 000 + 321 = ……….
100 + 100 + 100 + 100 = ……………….
16)
3
19 × 19 × 19 = ……………………
14
NCV2 – Chapter 1: Natural numbers, whole numbers and integers
§ 1.14
1.14 ESTIMATING AN ANSWER
Rounding off or correcting to 1 significant figure:
As you read a number from left to right, the first figure you come to that is not zero is called the
first significant figure.
Numbers greater than 1
1st significant figure
325
1 456
2,789
The same is true for numbers less than 1
1st significant figure
0,0345
0,0092
0,000 549
An easy way to approximate a number is to round it off to the first significant figure.
Procedure for rounding to 1 significant figure:
i) Underline the first significant figure
ii) Look at the number to the right:
• If it is less than 5, the first significant figure remains the same
• If it is 5 or more, the first significant figure is increased by 1
iii) If the number is more than 1, fill in zeros to keep the place value correct
EXAMPLE
1) Round the following numbers off to 1
SOLUTION
significant figure:
a) 634 810
b) 38 126
c) 0,0346
d) 0,006 825 13
2) a) Calculate 12 012 × 45 ÷ 462 using a
calculator
b) As a check, estimate the answer to a) by
rounding each number off to 1 significant
figure
634 810 ≈ 600 000
38 126 ≈ 40 000
0,0346 ≈ 0,03
0,006 825 13 ≈ 0,007
Calculator: 12 012 × 45 ÷ 462 = 1 170
Estimate: 10 000 × 50 ÷ 500
= 500 000 ÷ 500 = 1 000
The answers to a) and b) are nearly the same, so
the answer to a) is probably correct
Exercise 1.14
1) Round each of the following numbers to 1 significant figure:
a) 5 499 ≈ …………………………………………………
b) 491 ≈ ……………………………………………………
c) 35 ≈ ……………………………………………………..
d) 0,005 53 ≈ ……………………………………………….
2) a) Use a calculator to calculate: 924 × 28 ÷ 87 ………………………………………………..
b) Check your answer, by rounding each number to 1 significant figure
…………………………………………………………………………………………………
3) Round each number to 1 significant figure, to estimate the answer to 5059 × 99 × 17
……………………………………………………………………………………………………...
15
NCV2 – Chapter 1: Natural numbers, whole numbers and integers
§ 1.15
1.15 ADDITION AND SUBTRACTION OF INTEGERS USING
A NUMBER LINE
The set of integers is the set of numbers {….; –5; –4; –3; –2; –1; 0; 1; 2; 3; 4; 5; …….}
We can show these integers on a number line like this:
●
●
●
●
●
●
●
●
●
●
-4
-3
-2
-1
0
1
2
3
4
5
The set of integers consists of the set of whole numbers as well as the negatives of the
whole numbers.
When we add integers, we move a number of digits to the right.
When we subtract integers , we move a number of digits to the left.
Note:
• If there is no sign in front of a digit, it is assumed to be +
• The sign before an integer belongs to that integer
Use the number line for the following:
–4
EXAMPLE
1. Add 2 and 3
2.
3.
4.
5.
–3
–2
–1
0
1
2
3
4
5
SOLUTION
Start at 2 and move 3 digits to the right.
Your answer is 2 + 3 = 5.
Add −3 and 4
Start at −3 and move 4 digits to the right.
Your answer is −3 + 4 = 1.
Add –1 and −3
A tricky one! Start at −1 and move –3 digits to the right, which
is the same as moving 3 digits to the left.
Your answer is –1 + (–3) = –4.
Subtract 4 from 3
Start at 3 and move 4 digits to the left.
Your answer is 3 – 4 = –1.
Subtract −3 from −2 Another tricky one! Start at –2 and move –3 to the left, which is
the same as moving +3 to the right.
Your answer is –2 – (–3) = –2 + 3 = 1.
Exercise 1.15
1) Use the number line above to help you to calculate the answers to each question.
a) –2 + 7 = ……
b) 5 – 9 = ………
c) 1 – (–2) = ……… d) 1 – 5 = ………
e) –3 – (–2) = ……...
f) –1 – 3 = ……..
g) –2 + (–1) = ……..
h) –2 – (–3) = ……..
2) Use the centimetre marks along a full length ruler if the number line above is too small.
a) –4 + 2 = ……..
b) –1 – 2 = ……..
c) –3 + (–1) = …….. d) –5 – (–2) = ……..
e) 2 – 3 = ……..
f) 2 – (–2) = ……..
g) –2 – 1 = ……..
h) –7 + 12 = ……..
i) –3 – 6 = ……..
j) –1 + (–2) = ……..
k) 4 + (–4) = ……..
l) –11 – 1 = ……..
m) 4 – 6 = ……..
n) –3 + 5 = ……..
o) 1 – (–2) = ……..
p) –10 – (–1) = …….
16
NCV2 – Chapter 1: Natural numbers, whole numbers and integers
§ 1.1
1.16 ADDITION AND SUBTRACTION OF INTEGERS
WITHOUT USING A NUMBER LINE
As a short cut, remember:
• + × +=+
• + × –=–
• – × +=–
• – × –=+
When multiplying:
• if the signs are the same, the answer is +
• if the signs are different, the answer is –
EXAMPLE
Evaluate
1) +5 + (+4)
2) –3 + (–5)
3) –8 – (+3) = –11
4) –2 – (–4)
5) –2 – (+6) + (+1) + (–3) – (–5)
6) 12 + (–6) – (–3) – (+2) – (–1)
7) –15 + (–18) + (+6) – (–2) + (–5) – (+7)
8) –11 – (–6) + (–3) + 4 – 2
SOLUTION
+5 + (+4) = 5 + 4 = 9
–3 + (–5) = – 3 – 8 = – 8
–8 – (+3) = – 8 – 3 = –11
–2 – (–4) = –2 + 4 = 2
–2 – (+6) + (+1) + (–3) – (–5)
= –2 – 6 + 1 – 3 + 5 = – 5
12 + (–6) – (–3) – (+2) – (–1)
= 12 – 6 + 3 – 2 + 1 = 8
–15 + (–18) + (+6) – (–2) + (–5) – (+7)
= –15 – 18 + 6 + 2 – 5 – 7 = – 37
–11 – (–6) + (–3) + 4 – 2
= – 11 + 6 – 3 + 4 – 2 = – 6
Exercise 1.16
Evaluate without the use of a calculator.
1) – (+4) = ……………………………………
2) – (–5) = ………………………………….
3) –7 – (+15) = ………………………………
4) 13 – (–9) = ………………………………
5) +14 + (+6) = ………………………………
6) –9 + (–5) = ………………………………
7) –9 + 0 = ……………………………………
8) 19 – (–7) = ……………………………….
9) –13 + (–5) = ………………………………
10) –13 – 0 = …………………………………
11) 23 – (–4) = ………………………………...
12) –25 – (+12) = ……………………………
13) 0 – (+13) = ………………………………... 14) 0 – (–23) = ……………………………….
15) –3 – 4 + 2 + (–7) + 6 = …………………………………………………………………………..
16) 14 – 17 + 4 + 3 = …………………………………………………………………………………
17) –4 – (–3 + –5) = ………………………………………………………………………………….
18) 11 + 9 + (–6) – 2 = ……………………………………………………………………………….
19) 24 – 7 – 5 – (+3) + 1 = …………………………………………………………………………..
20) –19 +(–21) – (–32) + 5 = ………………………………………………………………………..
21) –23 – 12 – 34 – 15 = …………………………………………………………………………….
17
NCV2 – Chapter 1: Natural numbers, whole numbers and integers
§ 1.1
1.17
MULTIPLICATION AND DIVISION OF INTEGERS
INTEGERS
Remember:
• + ÷ +=+
• + ÷ –=–
• – ÷ +=–
• – ÷ –=+
EXAMPLE
1) Evaluate
a) 7 × 11
b) 5 × –3
c) –4 × 7
d) –6 × –2
e) 18 ÷ 6
When dividing, as for multiplying:
• if the signs are the same, the answer is +
• if the signs are different, the answer is –
SOLUTION
7 × 11 = 77
5 × -3 = –15
–4 × 7 = –28
–6 × –2 = 12
18 ÷ 6 = 3
EXAMPLE
2) Evaluate
a) 24 ÷ –3
b) –15 ÷ 3
c) –49 ÷ –7
d) –2 × 5 × –6 × 4
e) (–2 + –3)(9 – 11)
SOLUTION
24 ÷ –3 = –8
–15 ÷ 3 = –5
–49 ÷ –7 = 7
–2 × 5 × –6 × 4 = 240
(–2 + –3)(9 – 11)
= –5 × –2 = 10
Note:
An integer multiplied by zero gives an answer of zero, for example –6 × 0 = –6
Zero divided by any integer other than zero, is zero, for example 0 ÷ 9 = 0
Multiplying an integer by 1 does not change the integer, for example –8 × 1 = –8
Division by zero is undefined, for example 3 ÷ 0 = undefined
Exercise 1.17
Calculate:
1) 0 × 24 = ………………
2) –6 × 1 = ……………
3) 1 × –6 = …………......
4) –2 × 7 = ………………
5) 5 × –4 = …………….
6) –4 × –1 = ……………
7) –5 × 0 = ………………
8) 15 × 1 = …………….
9)
0 ÷ 5 = ……………..
10) –24 × –10 = …………..
11) –9 ÷ 0 = …………….
12)
–1 × 4 = ……………
13) –12 ÷ –4 = ……..
14) –8 ÷ –2 = …………...
15) –320 ÷ 10 = …………
16) –2 × –3 × –1 × 5
17) –3 × 4 × –2 × 5
18) 4 × –5 × 7 × – 2
= ……………………….
19) (–2 + –1)(–4 +–5)
= ……………………….
22) 56 ÷ –8 –4
= ………………………
25) (–5 × 2) × 3 × –2
= ………………………
= ……………………
20) 28 ÷ –7 × –2
= …………………….
23) –24 ÷ (7 – 4)
= …………………….
26) 7 × (–3 – 2)
= ……………………..
18
= …………………….
21) –4(–1 – 6)
= …………………….
24) 12 ÷ –4 + 2
= …………………….
27) 24 ÷ –3 × 4 ÷ –2
= …………………….
NCV2 – Chapter 1: Natural numbers, whole numbers and integers
§ 1.1
1.18
THE SET OF INTEGERS
So far we have worked with three sets of numbers.
The set of integers:
Z = {……; -3; -2; -1; 0; 1; 2; 3; ………}
The set of whole numbers: W = {0; 1; 2; 3; …………….}
The set of natural numbers: N = {1; 2; 3; 4; ……………..}
Exercise 1.18
1) Answer the following:
a) In which set/s of numbers do negative numbers occur? ………………………………
b) Is there a smallest integer? ………………………………………………………….....
c) Is there a greatest integer? ………………………………………………………………
2) In which set/s of number do you find each of the following numbers?
a) 3 ……………………………………………………………………
b) 1 ……………………………………………………………………
c) –9 …………………………………………………………………..
d) 0 ……………………………………………………………………
e) 27 …………………………………………………………………..
f) 14 …………………………………………………………………..
g) –41 …………………………………………………………………
h) 102 …………………………………………………………………
i) –15 …………………………………………………………………
j) 200 …………………………………………………………………
3) Answer True or False:
a) The set of natural numbers includes zero ………………………………………………
b) There are negative numbers in N0 ………………………………………………………
c) Zero is an integer but not a natural number ……………………………………………
d) Zero is a whole number but not an integer …………………………………………….
e) All integers are whole numbers ………………………………………………………..
f) All natural numbers are whole numbers ……………………………………………….
g) All natural numbers are integers ……………………………………………………….
h) All whole numbers are integers ………………………………………………………..
19
NCV2 – Chapter2: Fractions and Decimals
CHAPTER 2
Fractions and decimals
In this chapter you will:
• Define fractions
• Find equivalent fractions
• Compare fractions
• Convert between mixed numbers and improper fractions
• Add and subtract fractions with the same denominators
• Add and subtract fractions with different denominators
• Multiply fraction
• Find the reciprocal of a number
• Divide fractions
• Convert fractions to decimals
• Compare the sizes of decimals
• Multiply decimals
• Divide decimals
• Round off to the nearest whole number and to 1 or 2 decimal places
• Round off and estimate answers
• Work with fractions and decimals on the scientific calculator
• Add and subtract fractions on a scientific calculator
• Use a scientific calculator to convert from recurring decimals to fractions
This chapter covers material from Topic 1: Numbers
SUBJECT OUTCOME 1.1:
Use Computational Tools and Strategies and Make Estimates and Approximations
Use a scientific calculator competently and efficiently
Learning Outcome 1:
Execute algorithms appropriately in calculations
Learning Outcome 2:
SUBJECT OUTCOME 1.2:
Demonstrate understanding of numbers and relationships among numbers and number systems and
represent numbers in different ways
Identify rational numbers and convert between terminating or recurring decimals like
Learning Outcome 1:
a
; a; b ∈ Z; b ≠ 0
b
20
NCV2 – Chapter2: Fractions and Decimals
§ 2.1
2.1 FRACTIONS
1
4
1
1
, -1 , -3,6 and 4,5.
2
4
Examples of fractions and decimals are 3 ,
The position of these numbers can be shown on a number line:
–4
–3
–2
-1
0
2
3
1
2
-1 1
-3,6
1
4
4
5
31
4,5
4
Note:
• the arrows on the ends of the number line show that it extends in both directions
• there is an infinite number of positive and negative numbers along the number line
This new set of numbers is called the set of rational numbers.
Between any two integers there are an infinite number of rational numbers, so it is
impossible to list them all.
Note:
• In the fraction
2
, 2 is called the numerator and 3 is called the denominator.
3
4
is a proper fraction, since the numerator is less than the denominator
5
12
•
is an improper fraction, since the numerator is greater than the denominator
5
3
• 2 is a mixed number, since it is made up of a whole number and a proper fraction
4
•
1
5
1
5
1
5
1
5
1
5
o The rectangle has 5 equal parts.
o Each piece is 1 part of the 5 equal parts.
o Each piece is
1
1
of the rectangle: 5 × = 1
5
5
o The shaded area is 3 parts of the 5 equal parts, so it is
written as
3
of the rectangle.
5
Exercise 2.1
For each of the figures below, answer the following questions:
A
B
C
A
1) How many equal parts does the figure have?
2) How many of these parts are shaded?
3) What fraction is the shaded area of the whole figure?
21
B
C
NCV2 – Chapter2: Fractions and Decimals
§ 2.2 EQUIVALENT FRACTIONS
FRACTIONS
We call
1 2 3
4
1
2
3
4
, , and
equivalent fractions. This is because
= = =
.
12
12
3 6 9
3
6
9
We can find equivalent fractions by:
• multiplying the numerator and the denominator by the same number
• dividing the numerator and the denominator by the same number
Examples:
1)
2)
3)
2
4
= , by multiplying both the numerator and the denominator by 2
3
6
2
6
= , by multiplying both the numerator and denominator by 3
3
9
12
6
=
, dividing both the numerator and the denominator by 2
24
12
4)
6
3
= , dividing both the numerator and the denominator by 2
12
6
5)
3
1
= , dividing both the numerator and the denominator by 3
6
2
2
4
6
= =
3
6
9
12
6
3
=
=
=
24
12
6
Exercise 2.2
1) Fill in the missing numbers to make the fractions equivalent:
b) 2
c) 2 10
a) 1
=
=
=
e)
i)
m)
q)
2
8
4
=
2
8
3
21
=
7
3
9
12
2
=
18
12
=
4
16
f)
j)
n)
10
=
3
15
21
7
=
8
36
3
=
48
24
6
=
7
r)
g)
k)
d)
5
3
9
=
4
3
=
5
45
o)
81
=
3
=
4
8
2
6
=
5
5
=
8
16
h)
l)
p)
5
9
4
=
2
9
2) Write down the first five equivalent fractions starting with the number given:
a)
c)
e)
g)
1
5
2
3
5
8
3
4
=
=
=
=
b)
=
=
=
=
d)
=
=
=
=
f)
=
=
=
=
h)
22
4
=
7
4
=
9
11
=
12
2
=
5
=
=
=
=
=
=
=
=
=
=
=
=
NCV2 – Chapter2: Fractions and Decimals
§ 2.3 COMPARING FRACTIONS
1) When the denominators are the same:
Is
4
3
greater than or less than ?
5
5
4
1
5
5
3
5
1
5
From the diagrams we can see that
1
5
1
5
1
5
1
5
1
5
1
5
1
5
1
5
4
3
4
3
is greater than . We write > .
5
5
5
5
2) When numerators are the same:
Which is the smaller,
1
8
1
6
5
5
or ?
8
6
1
1
8
8
1
8
1
6
1
8
1
6
From the diagram we can see that
1
8
1
6
1
8
1
8
1
6
1
6
5
8
5
6
5
5
5
5
is less than . So
< .
8
6
8
6
3) When numerators and denominators are different:
Compare
4
5
with . Which is greater?
5
6
Choose the lowest multiple (LCM) of 5 and 6, which is 30.
Then write each fraction as an equivalent fraction having a denominator of 30.
4
4
6
24
=
×
=
5
5
6
30
5
5
5
25
=
× =
6
6
5
30
But
25
24
5
4
>
, so > .
30
30
6
5
Exercise 2.3
1) Which fraction is greater? Show all calculations.
a)
b)
c)
d)
4
7
5
7
7
9
2
3
6
……………………………………………………………………………….....
7
5
or ………………………………………………………………………………….
9
20
or
……………………………………………………………………………..….
27
5
or …………………………………………………………………………..……...
8
or
………………………………………………………………………………….
2) Which fraction is smallest,
4
3
5
or or ? Show all calculations.
4
5
6
………………………………………………………………………………………….……
…………………..…………………………………………………………………………...
23
NCV2 – Chapter2: Fractions and Decimals
§ 2.4 CONVERTING FRACTIONS
1. Write fractions in simplest form:
A fraction is in simplest form when its numerator and denominator are as small as
possible. They cannot be divided by any further number and still remain whole numbers.
Examples:
24
15
, b)
in simplest form (simplify)
30
45
24
24 ÷ 6
4
15
15 ÷ 5
3
3÷3
1
Solution: a)
=
=
b)
=
= =
=
30
30 ÷ 6
5
45
45 ÷ 5
9
9÷3
3
Write a)
2. Convert improper fractions to mixed numbers:
Examples:
6
2
2
2
7
3
3
1
1
1
= + + =1+1+1=3
2) = + + = 1 + 1 + = 2
2
2
2
2
3
3
3
3
3
3
9
5
4
4
9
4
3) = + = 1
As a shortcut: = 1 remainder 4 = 1
5
5
5
5
5
5
1)
3. Convert mixed numbers into improper fractions:
Examples:
1
1
7
1
8
3
3
4
4
3
11
=1+ = + =
2) 2 = 1 + 1 + = + + =
4
4
4
4
4
4
7
7
7
7
7
3
2× 4 + 3
11
As a shortcut: 2 =
=
4
4
4
1) 1
Exercise 2.4
1) Write each fraction in simplest form / simplify:
4
= …………………...
10
12
d)
= …………………...
27
a)
14
= ……………………
28
16
e)
= ……………………
28
b)
25
= ………………
40
33
f)
= ………………
55
c)
2) Convert these improper fractions to mixed numbers in simplest form:
12
48
= ………………………………………… b)
= …………………..…………….
5
7
64
38
c)
= ………………………………………… d)
= …..…………………………..….
4
9
75
44
e)
= …….………………………………….. f)
= …………………………….……
9
8
a)
3) Convert these mixed numbers to improper fractions:
1
3
= ……………………………………….... b) 2 = …………………………………
4
5
5
2
c) 3 = ………………………………………… d) 4 = …………………………………
8
7
a) 1
24
NCV2 – Chapter2: Fractions and Decimals
§ 2.5 ADDING AND SUBTRACTING FRACTIONS WITH THE
SAME DENOMINATORS
To add or subtract fractions:
Make sure their denominators are the same – use equivalent fractions if they are not
Add or subtract the numerators
Simplify the answer
EXAMPLES
1)
1
8
2)
3)
1
8
1
8
1
8
1
8
1
8
7
3
4
4÷4
1
– = =
=
2
8
8
8
8÷4
1
8
1
16
1
15
7
2 – = 2( ) – =
–
=
=1
8
8
8
8
8
8
8
1
8
1
8
Note: 1 =
5
2
7
+
=
8
8
8
2
3
4
5
= = =
etc.
2
4
3
5
Exercise 2.5
Calculate and simplify your answers:
2
= ……………………………………………………………………………………
7
3
2)
+
= ……………………………………………………………………………………
5
4
3)
– = …………………………………………………………………………………….
9
2
4)
– = ……………………………………………………………………………………
6
1
5) 1 – = ……………………………………………………………………………………..
5
3
6) 1 – = ………………………..……………………………………………………………
4
1
3
2
7)
+ + = ………………………………………………………………………………
8
8
8
2
1
8) 1 + – = ………………………………………………………………………………..
9
9
2
1
9) 1 – – = ………………………………………………………………………………..
7
7
14
11
2
10)
–
+
= ……………………………………………………………………………
15
15
15
4
1
11) 2 – + =..........................................................................................................................
5
5
3
1
12) 1 + = …………………………………………………………………………………..
8
8
7
3
1
13)
–
+
= …………………………………………………………………………...
20
20
20
4
14) 3 – = ……………………………………………………………………………………..
5
1)
3
7
1
5
7
9
5
6
+
25
NCV2 – Chapter2: Fractions and Decimals
§ 2.6 ADDING AND SUBTRACTING FRACTIONS WITH
DIFFERENT DENOMINATORS
To add or subtract fractions like
1
1
and
they must each be written as equivalent
2
3
fractions with the same denominator.
EXAMPLES
Calculate: 1)
1
1
+
2
3
2)
5
1
–
12
3
3)
SOLUTION
1) The LCM of 2 and 3 is 6.
3)
1
1× 3
3
1
1× 2
2
=
=
and
=
=
2
2×3
6
3
3× 2
6
1
1
3
2
5
= + =
So +
2
3
6
6
6
3
2
+
The LCM of 4 and 5 is 20.
4
5
3× 5
2× 4
=
+
4×5
5× 4
15
8
=
+
20
20
23
3
=
=1
20
20
3
2
+
4
5
4) 3
1
3
–1
4
8
2) The LCM of 12 and 3 is 12
5
1
5
1× 4
5
4
1
– =
–
=
–
=
12
12
12
12
12
3
3× 4
4) 3 1 – 1 3
The LCM of 8 and 4 is 8
4
8
25
7
=
–
4
8
25
7×2
=
–
4× 2
8
25
14
11
3
=
–
=
=1
8
8
8
8
Exercise 2.6
Calculate without a calculator. Write your answers as mixed numbers where possible.
1)
2)
3)
4)
5)
6)
7)
8)
9)
10)
1
1
+ = ………………………………………………………………………………….
2
8
3
5
+ = ………………………………………………………………………………….
4
8
4
1
– = …………………………………………………………………………………..
5
3
5
2
3
+ + = ……………………………………………………………………………
12
3
8
5
2
1 – = ………………………………………………………………………………..
6
3
5
2
3 – 1 = ……………………………………………………………………………….
8
3
1
5
1
1 – + = ……………………………………………………………………………
3
6
6
1
3
2 – 1 = ……………………………………………………………………………….
2
4
7
5
2 – 1 = ……………………………………………………………………………….
8
6
7
2
4 – 2 = ………………………………………………………………………….........
9
3
26
NCV2 – Chapter2: Fractions and Decimals
§ 2.7 MULTIPLICATION OF FRACTIONS
EXAMPLE
Find the following products:
1)
2
4
×
5
9
2)
3
3
×
9 14
3
5
3) 2 × 1
SOLUTION
1) 2 × 4 = 2 × 4 = 8
5
9
5× 9
2
13
4) 3
3 1
×
2
7
2)
45
3
3
3× 3
1
×
=
=
9 14
9 × 14
4
3 is a factor of 3 and 9, so cancel
3)
3
2
13 15
2 ×1 =
×
5
13
5
13
13 ×15
cancel numerator and denominator by 13 and 5
=
5 × 13
=3
4)
3
3 1
24 1
24 × 1
× =
× =
2
2
7× 2
7
7
=
12
5
=1
7
7
2 is a factor of 24 and 2, so you can cancel by 2
The answer is given as a mixed number
Exercise 2.7
Find the following products. Whenever possible, write your answers as mixed numbers.
1)
2)
3)
4)
5)
6)
7)
8)
9)
10)
11)
12)
9
3
×
= ……………………………………………………………………………….
2
10
2
3
× = ………………………………………………………………………………...
7
5
5
9
×
= ………………………………………………………………………………
6
10
13
4
×
= ………………………………………………………………………………
16
5
2
3
× 3 = ………………………………………………………………………………
4
5
1
2
3 ×
= ……………………………………………………………………………….
4
9
1
2
2
× 12 = ……………………………………………………………………………
19
3
3
1
1 × 2 = ………………………………………………………………………………
2
5
3
5
8
×
×
= ………………………………………………………………………….
4
9
25
3
2
1
× 2 × 3 = ………………………………………………………………………..
4
4
3
1
2
3
4
×
×
×
= …………………………………………………………………….
2
4
3
5
8
2
2 × 1 = ……………………………………………………………………………..
9
13
27
NCV2 – Chapter2: Fractions and Decimals
§ 2.8 THE RECIPROCAL
RECIPROCAL OF A NUMBER
When we multiply any number by its reciprocal, we get 1 as an answer.
1
1
, because 2 ×
= 1.
2
2
1
is 6, because
× 6=1
6
3
2
3
is , because
×
=1
2
2
3
The reciprocal of 2 is
1
6
2
The reciprocal of
3
The reciprocal of
EXAMPLES
What is the reciprocal of
1) 3?
2)
3
?
4
SOLUTIONS
The reciprocal of 3 is
The reciprocal of
1
1
, since 3 × = 1
3
3
3
4
3
4
is , since
×
=1
4
3
4
3
1
5
3) –5
The reciprocal of –5 is − , since -5 × −
1
8
3
5) 2 ?
8
1
=1
5
1
1
is –8, since − × -8 = 1
8
8
3
8
3
19
19
8
The reciprocal of 2 is , since 2 =
and
×
=1
8
19
8
8
8
19
4) − ?
The reciprocal of −
Exercise 2.8
1)
The reciprocal of 6 is ………… since …………………………………………………
2)
The reciprocal of
2
is ………… since ………………………………………………...
3
3)
The reciprocal of
−4
is ………… since ………………………………………………..
3
4)
The reciprocal of –2 is ………… since …………………………………………………
5)
The reciprocal of
5
is ………… since ………………………………………………….
9
6)
The reciprocal of
−6
is ………… since …………………………………………………
7
7)
The reciprocal of 3
8)
The reciprocal of –9 is ………… since ………………………………………………….
9)
The reciprocal of 2
1
is ………… since …………………………………………………
2
3
is ………… since ………………………………………………..
4
28
NCV2 – Chapter2: Fractions and Decimals
§ 2.9 DIVISION OF FRACTIONS
1) If we are asked to calculate 3 ÷
1
, we are actually being asked "how many sixths are
6
there in 3 wholes".
1
6
1
6
1
6
1
6
1
6
1
6
1
6
1
6
There are 18 sixths in 3 wholes, so 3 ÷
2) 6 ÷ 2 = 3 but 6 ×
1
6
1
6
1
= 18
6
1
6
1
6
OR
1
6
3 ÷
1
6
1
6
1
6
1
6
1
6
1
3
3× 6
18
=
=
=
= 18
1
1
1
6
×6
6
6
1
1
= 3, so division by 2 is the same as multiplying by , its reciprocal.
2
2
As a shortcut, when you divide by a fraction, you multiply by its reciprocal.
EXAMPLES
Calculate
SOLUTIONS
1)
2
1
÷
3
3
2
1
2
3
2 3/
÷ = × = × =2
3
1
3
3
3/ 1
2)
−6
3
÷
14
7
−6
3
−6 14
−2 6/ 14
÷
=
×
=
×
= –4
14
7
7
3
7/
3/
9
9
1
1
÷ 18 =
×
=
10
10 18
20
1
1
7
7
7
2
2
2 ÷3 = ÷ =
×
=
3
2
2
3
3
7
3
9
÷ 18
10
1
1
4) 2 ÷ 3
3
2
3)
Exercise 2.9
Find the following quotients and simplify your answers:
1)
2)
3)
4)
5)
6)
7)
8)
9)
–
5
5
÷
= ……………………………………………………………………………
12
6
2
÷ 6 = ………………………………………………………………………………
3
1
45 ÷ 7 = ……………………………………………………………………………
2
3
1
4 ÷
= …………………………………………………………………………….
4
4
3
–3 ÷
= ……………………………………………………………………………..
4
5
1
2 ÷ 3 = ……………………………………………………………………………
2
8
4
1
4 ÷ 1 = ……………………………………………………………………………
5
5
4
1
3 ÷1
= …………………………………………………………………………..
14
7
2
5
1
÷
÷
= ……………………………………………………………………….
7
7
10
29
NCV2 – Chapter2: Fractions and Decimals
§ 2.10 CONVERTING FRACTIONS TO DECIMALS
To convert a fraction to a decimal, divide the denominator into the numerator.
Note:
We can write any number of
• 2 = 2,0 = 2,00 = 2,0000….
zeros after the decimal comma.
• 3,5 = 3,50 = 3,500 = 3,5000 = 3,50000….
EXAMPLES
Convert the following fractions to decimals
1)
SOLUTIONS
1
2
1
1, 0
=
= 0,5
2
2
3
7
7, 000
1 = =
= 1,75
4
4
4
19
= 0,019
1000
3
4
19
3)
1000
2) 1
Exercise 2.10
1) Convert these fractions to decimals:
a) 2
b)
3
7
= ………..
10
c)
e)
f)
5
= …………..
8
g)
1
= ………..
2
7
= ………...
80
7
= …………..
8
10
= ………….
8
d)
h)
2
= ……….
100
45
= ………...
150
Recurring decimal notation:
Some answers will be a recurring or repeating decimal.
In a recurring decimal, a digit or group of digits, which occur after the decimal comma,
are repeated. The numbers which recur are written under a line, or under and between
dots.
EXAMPLES
Write as a decimal:
1)
SOLUTIONS
•
1
= 0,333333.... = 0,3 or 1,3
3
1
3
2) 4
23
99
4
3) 9
152
999
9
4) 3
97
666
• •
23
= 4,23232323..... = 4, 23 or 4, 23
99
• •
152
= 9,152 152 152 ..... = 9,152
999
97
3
= 3,1 456 456 456…. = 3,1456 , since the 1 does not recur
666
2) Write as a recurring decimal:
a) 3,88888……… = …………………..
c) 5,238238238238 …….. = ……………
e) 1
2
= ………………………………..
3
b) 12,45454545……….. = ……………….
d) 3,417171717…… = …………………..
f)
30
5
= ……………………………………
6
NCV2 – Chapter2: Fractions and Decimals
§ 2.11 COMPARING AND CONVERTING
Comparing the size of decimal numbers
1) Compare the signs:
1,2 > – 6,7 since a positive number is always greater than a negative number.
– 1,2 > – 3,8 since on the number line, the greater negative number is further from 0.
2) Compare the whole numbers:
3,49 is greater than 2,9 since 3 > 2.
3) Compare the numbers in the first decimal place:
If the integer parts are the same, then we compare the numbers in the first decimal place.
e.g. 4,69 is greater than 4,298 since in the first decimal place, 6 > 2.
4) Compare the second decimal place:
If the numbers in first decimal place are the same, compare the numbers in the second
decimal place.
e.g. 2,369 9 is less than 2,391 97 since 6 < 9. We can continue in this way.
5) Compare recurring numbers:
•
1, 3 > 1,3 since 1,3333… is greater than 1,3
Exercise 2.11
1) Insert = or > or <:
a) 3,1659 …... 3,1700
b) 0,00879 …... 0,00891
c) 5,1903 …... 5,1903000
d) 1,001 …... 0,009
e) –3,6 …… 0,1
f) –2,78 …… –2,99
h) –0,019 …… 0
i) 11,009 …… 11,0019
•
g) 2,5 …… 2, 5
Converting from a decimal to a fraction
We write one zero in the denominator for each decimal place
EXAMPLES
Convert to a fraction and simplify: 1) 0,5
SOLUTION
5
10
1)
0,5 =
4)
3,79 = 3
79
379
=
100
100
2) 0,34
34
100
2)
0,34 =
5
0,005 =
3) 0,056
4) 3,79
3) 0,056 =
5
1 000
2) Convert to a fraction and simplify:
a) 3,65 = ………………..
b)
c) 7,5 = ………………...
d) 2,04 = ………………..
e) 0,045 = ………………..
31
5) 0,005
0,1 = ………………..
56
1 000
NCV2 – Chapter2: Fractions and Decimals
§ 2.12 MULTIPLYING DECIMALS
EXAMPLES
Calculate:
1) 1,2 × 0,3
SOLUTIONS
1,2 × 0,3 =
12
3
36
×
=
= 0,36
10
10
100
Shortcut:
Multiply the numbers ignoring the decimal comma. 3 × 12 = 36.
There is 1 decimal place in each number, so there will be 1 + 1 = 2 decimal
places in the answer.
2) 0,04 × 200
4
200
×
= 8,00
1
100
0,04 × 200 =
Shortcut:
4 × 200 = 800.
There are 2 decimal places in the answer
13
1
×
= 0,130
10
10
3) 1,30 × 0,1
1,30 × 0,1 =
4) 0,0042 × 1,2
Shortcut:
130 × 1 = 130. There are 2 + 1 = 3 decimal places in the answer.
0,0042 × 1,2 = 0,00504
Shortcut:
42 × 12 = 504.
There are 4 + 1 = 5 decimal places in the answer.
Add zeros to the left of 504 in order to get the correct number of decimal places
Exercise 2.12
Simplify without using a calculator:
1) 0,25 × 4 = …………………………………………………………………………………..
2) 0,004 × 1,6 = ……………………………………………………………………………….
3) 14,5 × 1,1 = ………………………………………………………………………………...
4) 1,25 × 0,4 = ………………………………………………………………………………...
5) 21,5 × 0,02 = ……………………………………………………………………………….
6) 1,004 × 10 = ………………………………………………………………………………..
7) 0,02 × 500 = ………………………………………………………………………………..
8) 1,300 × 0,40 = ……………………………………………………………………………...
9) 0,005 × 1,02 = ……………………………………………………………………………...
10) 1,000 × 0,09 = ……………………………………………………………………………...
11) 250 × 4,0 = …………………………………………………………………………………
12) 2,5 × 0,002 = ……………………………………………………………………………….
32
NCV2 – Chapter2: Fractions and Decimals
§ 2.13 DIVIDING DECIMALS
To divide decimals:
Count the number of decimal places in the denominator.
Make the denominator a whole number by multiplying both the numerator and the
denominator by the same multiple of 10 as the number of decimal places in the
denominator.
EXAMPLE
Calculate
1) 4,2 ÷ 0,02
SOLUTION
4,2 ÷ 0,02 =
4, 2
4, 2 ×100
420
=
=
= 210.
2
0, 02
0, 02 × 100
Note: There are two decimal places in the denominator, so multiply both
numerator and denominator by 100
2) 32 ÷ 0,005
32 ÷ 0,005 =
32
32 ×1 000
32 000
=
=
= 64 000
0, 005 ×1 000
0, 005
5
Note: There are 3 decimal places in the denominator, so multiply both
numerator and denominator by 1 000
Exercise 2.13
1) Calculate the following quotients without using a calculator:
a) 350 ÷ 100 = ………………………...
b) 37,4 ÷ 11 = …………………………
c) 12,65 ÷ 0,05 = ……………………...
d) 300,05 ÷ 0,005 = …………………...
e) 1,004 ÷ 0,02 = ……………………...
f)
g) 3,411 ÷ 0,9 = ……………………….
h) 45,550 ÷ 0,5 = ……………………...
i)
39,3 ÷ 0,3 = ………………………...
j)
284,6 ÷ 1,2 = ……………………….
1,44 ÷ 1,2 = ………………………...
2) Calculate the following products and quotients without using a calculator:
a) 600 ÷ 0,003 = …………………………………………………………………………
b) 0,002 × 60 = …………………………………………………………………………..
c) 120 × 0,4 = ……………………………………………………………………………
d) 36 ÷ 1,2 = ……………………………………………………………………………..
e) 0,2 × 0,3 × 200 = ……………………………………………………………………..
f) 4 500 ÷ 0,5 ÷ 1 000 = ………………………………………………………………...
g) 2,4 ÷ 0,02 × 5 = ………………………………………………………………………
h) 900 ÷ 0,03 × 100 = …………………………………………………………………...
i) 1,08 ÷ 9 ÷ 0,4 = ………………………………………………………………………
j) 12,1 ÷ 0,11 × 0,002 = …………………………………………………………………
33
NCV2 – Chapter2: Fractions and Decimals
§ 2.14 ROUNDING OFF TO THE NEAREST WHOLE NUMBER
AND TO 1 OR 2 DECIMAL PLACES
1. Rounding off to the nearest whole number:
Underline the number in the units position. If the number to the right is 5 or greater, round
up to the next whole number. If it is less than 5, round down to the given whole number.
EXAMPLE
Round the following numbers to the
nearest whole number
1) 4,276
2) 12,53
3) 0,199
SOLUTION
4,276 ≈ 4, since 2 < 5
12,53 ≈ 13, since 5 ≥ 5
0,199 ≈ 0, since 1 < 5
Exercise 2.14
1) Round off to the nearest whole number:
a)
3,29 ≈ ……….
b)
21,51 ≈ ……... c)
0,482 ≈ ……..
d)
0,59 ≈ ……..
e)
109,51 ≈ …….
f)
120,49 ≈ …… g)
2 000,82 ≈ …
h)
199,67 ≈ …..
2. Rounding off or correcting to 1 decimal place:
Underline the number in the first decimal place. If the number to the right is 5 or greater,
round up the number in the first decimal place. If it is less than 5, round down.
Note:
To round off or correct to any number of decimal places, look at the next decimal place to
see whether it is 5 or greater. If so, round up. If not, round down.
EXAMPLE
Round the following numbers
to 1 decimal place:
1) 3,551
SOLUTION
3,551 ≈ 3,6 since 5 ≥ 5
2) 11,949
11,949 ≈ 11,9 since 4 < 5
3) 3,991
3,991 ≈ 4,0 since 9 ≥ 5
2) Round off to 1 decimal place:
a) 23,451 = …….. b)
36,623 = ……..
c)
0,025 = ……..
d)
329,95 = ……..
e) 145,459 = …… f)
6,019 = ……..
g)
0,009 = ……..
h)
26,951 = ……..
c) 99,995 = …….
d)
0,0051 = ……..
3) Round off to 2 decimal places:
a) 3,5608 = …….. b)
12,915 = ……..
34
NCV2 – Chapter2: Fractions and Decimals
§ 2.15 ROUNDING OFF AND ESTIMATING ANSWERS
EXAMPLE
1) Use a calculator to calculate 6,179 × 180,5
correct to 2 decimal places.
2) Check your answer by rounding each
number to the nearest whole number.
3) Check your answer by rounding each
number to 1 significant figure
SOLUTION
6,179 × 180,5 = 1 115,3095… = 1 115,31
correct to 2 decimal places.
6,179 × 180,5 ≈ 6 × 181 = 1 086
correct to the nearest whole number.
6,179 × 180,5 = 6 × 200 = 1 200
Exercise 2.15
1) a) Calculate
1, 26 × 25,3
using a calculator correct to the nearest whole number.
0,995 6 × 10, 729
b) Estimate the answer by rounding each number to one significant figure.
c) Does your answer seem to be correct?
2) a) Use a calculator to calculate 207,8 × 15,995 correct to 1 decimal place.
b) Check this answer by rounding each number to one significant figure.
c) Estimate the answer by rounding each number to the nearest whole number
d) Do these estimates show that your answer is reasonably correct?
35
NCV2 – Chapter2: Fractions and Decimals
§ 2.16 FRACTIONS AND DECIMALS ON THE SCIENTIFIC
CALCULATOR
Fraction keys on the calculator enable you to:
Simplify fractions (i.e. write them in their simplest form)
Convert between mixed numbers, improper fractions and decimals
Do calculations involving fractions
The CASIO fx-82ES PLUS Scientific Calculator:
has fraction keys marked
and
.
is shown above the
key and is used by pressing shift and then
The arrow keys
move the cursor around the blocks on the display.
.
Note:
• First get the calculator in the correct mode by entering the following keys:
MODE: Select 1: COMP SETUP: Select 1: Mth IO
• Repeated entering of S ⇔ D will enable you to go back and forth between a
decimal and an improper fraction, i.e. you toggle from one to the other.
EXAMPLE
1) Use your calculator to:
a) Simplify
b) Write
SOLUTION
KEY SEQUENCE
18
27
18
18
as a decimal
27
2) Use your calculator to write 2
ANSWER
27 =
2
3
0,6666
S⇔ D
5
9
a) as an improper fraction
2
b) as a decimal
5
9 =
S⇔ D
23
9
2,5555..
3) Use your calculator to:
a) Simplify
36
15
36
36
as a mixed number
15
36
c) Write
as a decimal
15
b) Write
shift
15 =
S⇔ D
S⇔ D
12
5
2
2
5
2,4
Exercise 2.16
Use your calculator to:
24
…………………………….
18
24
as a decimal …
18
1)
Simplify
3)
Write
5)
Write 2,54 as a mixed number ……………………………………………………..
24
as a mixed number …………..
18
2) Write
4) Write 2,54 as an improper fraction …….
36
NCV2 – Chapter2: Fractions and Decimals
§ 2.17 ADDING AND SUBTRACTING FRACTIONS ON A
SCIENTIFIC CALCULATOR
CALCULATOR
EXAMPLES
1) Calculate
4
2
+ , and write the answer as a) an improper fraction, b) a decimal and
5
3
c) a proper fraction.
Key Sequence
4
5
+
2) Calculate 1
2
3 =
(a)
KS
(b)
KS
22
15
S⇔ D
1,466...
shift S ⇔ D
5
3) Calculate
7
+
3
7
15
Display
KS
Display
69
28
shift S ⇔ D
2 13
Display
KS
Display
36
5
shift S ⇔ D
4 =
28
3
as a proper fraction
2 1
−
3 4
Key sequence
3
1
5
3
+ . Write your answer as a proper fraction.
4
7
Key sequence
1
(c)
2
3
–
1
4
=
7
1
5
Exercise 2.17
12
as a decimal (correct to 1 decimal place) ................……….…
13
23
2) Use a calculator to write
as
14
1) Use a calculator to write
a) a mixed number ………………………………….
b) a decimal (correct to 1 decimal place) …………………………………………
3) Use a calculator to write each answer as
i) a decimal (correct to 2 decimal places) and ii) a mixed number.
a)
3
4
+ = ….……………………………………………………………………………
4
7
b) 7
11
7
– 5 = ……………………………………………………………………………
12
8
c) 1
6
5
+ 4 = …………………………………………………………………………….
7
8
d) 3
5
1
– 2 = …………………………………………………………………………….
4
8
37
NCV2 – Chapter2: Fractions and Decimals
§ 2.18 USING A SCIENTIFIC CALCULATOR TO CONVERT
FROM RECURRING DECIMALS
DECIMALS TO FRACTIONS
EXAMPLE
Use your calculator to:
•
5
1) Prove that 3,5 = 3
9
•
SOLUTION
KEY SEQUENCE
3,55555…… (until the display is full)
= shift S ⇔ D
•
ANSWER
5
9
3
145
999
2) Convert 1, 145 to a mixed number
1,145145145 ……
= shift
S⇔ D
1
3) Convert 2,3 45 to a mixed number
2,34545454545…
= shift
S⇔ D
2
19
55
Note:
• In all three examples it is necessary to fill the screen with the recurring decimals
• In example 3, the 3 in the first decimal place does not recur
Exercise 2.18
Use a calculator to write each recurring decimal as a fraction. Where possible, write as a
mixed number.
•
1) 0,3 = ………………………………………
2) 2,8 = ………………………………………
•
3) 3,5 = ……………………………………….
4) 12, 46 = …………………………………….
5) 1,444 ….. = ………………………………..
• •
6) 0, 35 = ……………………………………..
•
•
7) 5, 153 = …………………………………….
•
8) 1,2 4 = ……………………………………...
38
NCV2 – Chapter 3: Rational Numbers and Real Numbers
CHAPTER 3
Rational numbers and real
numbers
In this chapter you will:
• Find ratios of two or more quantities
• Find equivalent ratios
• Divide a quantity in a given ratio
• Calculate rates
• Work with ratios in direct proportion
• Work with ratios in indirect proportion
• Calculate percentages
• Multiply and divide exponents
• Work with zero exponents and negative exponents
• Simplify products and quotients raised to a power
• Raise an exponent to a power
• Work with number sequences
• Find the value of missing terms in a sequence
• Decide whether numbers are rational numbers or not
• Decide whether numbers are irrational numbers or not
• Simplify surds
• Decide whether numbers are real or not
• Use the real number diagram to answer questions
This chapter covers material from Topic 1: Numbers
SUBJECT OUTCOME 1.1:
Use Computational Tools and Strategies and Make Estimates and Approximations
Use a scientific calculator competently and efficiently
Learning Outcome 1:
Execute algorithms appropriately in calculations
Learning Outcome 2:
SUBJECT OUTCOME 1.2:
Demonstrate understanding of numbers and relationships among numbers and number systems and
represent numbers in different ways
Identify rational numbers and convert between terminating or recurring decimals like
Learning Outcome 1:
a
; a; b ∈ Z; b ≠ 0
b
Learning Outcome 2:
Learning Outcome 3:
Learning Outcome 4:
Know, understand and apply the laws of exponents
Convert surds into rational forms
Identify and work with arithmetic progressions, sequences and series
39
NCV2 – Chapter 3: Rational Numbers and Real Numbers
§ 3.1 RATIO
A ratio compares two or more quantities. In the nine identical rectangles below, three are
shaded and six are unshaded.
We say that:
• the ratio of the shaded to the unshaded rectangles is 3 to 6 which we write 3 : 6
• the ratio of the unshaded to the shaded rectangles is 6 to 3 which we write 6 : 3
• the ratio of the shaded rectangles to all the rectangles is 3 to 9 which we write 3 : 9
• the ratio of the unshaded rectangles to all the rectangles is 6 to 9 which we write 6 : 9
Note:
• The ratios remain the same no matter which three of the nine rectangles are shaded
• The order in which the ratio is written is important
• The ratio is written without units, since we are comparing rectangles with rectangles
• The ratio 3 : 6 can be written as the fraction
3
1
= =1:2
2
6
• 3 : 6 = 1 : 2 are equivalent ratios
EXAMPLE
Fruit juice is made by mixing 1 cup of
concentrate with 3 cups of water.
Find the ratio of the number of cups of:
1) concentrate to water
2) concentrate to mixture
3) mixture to water
4) water to mixture
5) water to concentrate
6) mixture to concentrate
SOLUTION
1:3
1:4
4:3
3:4
3:1
4:1
Note:
• the unit here is the cup
• A ratio is always written without units
Exercise 3.1
In a class of 30 learners, 12 are boys. Find the following ratios and write in simplest form:
1) number of boys : number of girls …………………………………………
2) number of girls : number in class ………………………………………….
3) number of boys : number in class ………………………………………….
4) number of girls : number of boys …………………………………………..
5) number in class : number of boys …………………………………………..
6) number in class : number of girls …………………………………………..
7) number of boys and girls : number in class …………………………………
40
NCV2 – Chapter 3: Rational Numbers and Real Numbers
§ 3.2 EQUIVALENT RATIOS
EXAMPLE
SOLUTION
1
1) A recipe uses 1 cups of
2
Flour : sugar : milk
flour, 1 cup of sugar and
3
4
of a cup of milk. Express
the ratio of flour to sugar to
milk in its simplest form.
2) Palesa receives R 90 pocket
money per month. The ratio
of the pocket money
received by Palesa and
Thabo is 3 : 4. How much
does Thabo receive per
month?
1
3
:1:
2
4
3
3
= :1:
The LCM of 2 and 4 = 4, so multiply by 4.
2
4
=1
=6:4:3
Note:
• The quantities to be compared have the same unit (in
this case cups)
• A ratio has no units
Let Thabo receive Rx.
P : T = 3 : 4 = 90 : x
3 90
=
4
x
3 × 4 x 90 × 4 x
=
(The LCM of 4 and x is 4x)
4
x
3x = 4 × 90 = 360
x = 360
3
= R 120
Thabo receives R 120.
Exercise 3.2
1) Mortar (for bricklaying) is made by mixing 3 wheelbarrows of cement with 2
wheelbarrows of sand and 1 wheelbarrow of water.
a) Find the ratio of cement : sand : water ………………………………
b) How much sand will I need to order to mix with 9 wheelbarrows of cement?
………………………………………………………………………………………….
c) How much cement and water will I need to order to mix with 8 wheelbarrows of sand?
………………………………………………………………………………………….
d) How much cement and sand will I need to order to mix with 6 wheelbarrows of water?
…………………………………………………………………………………………..
2) John, who is 16 years old and Thandi, who is 11 years old, receive pocket money in the
ratio of their ages. John is given R 80 per month. How much is Thandi given?
………………………………………………………………………………………………
………………………………………………………………………………………………
………………………………………………………………………………………………
………………………………………………………………………………………………
41
NCV2 – Chapter 3: Rational Numbers and Real Numbers
§ 3.3
DIVIDING A QUANTITY IN A GIVEN RATIO
To divide something in a given ratio, find
a) The total number of parts from the ratio
b) What one part is
EXAMPLE
Divide 66 sweets between Jim and Busi in the ratio of 6 : 5.
SOLUTION
Suppose that the sweets are divided into 6 + 5 = 11 equal shares.
6
× 66 = 36 sweets.
11
5
Busi gets
× 66 = 30 sweets.
11
Jim gets
Note:
• As a calculation check, 36 + 30 = 66 sweets
Exercise 3.3
1) Divide R650 between Susan and James in the ratio of 7 : 3.
2) The perimeter of a triangle is 612 cm. The sides of the triangle are in the ratio 3 : 4 : 5.
What is the length of the longest side?
3) A green paint is mixed from blue and yellow paint in the ration 3 : 5. How much of each
colour is needed to make 40 litres of this green paint?
42
NCV2 – Chapter 3: Rational Numbers and Real Numbers
§ 3.4 RATE
A rate links two quantities. It tells us how one quantity compares or changes with
another.
The unit of a rate is usually given as “somethings” per “something” e.g. kilometres per
hour, cents per litre, revolutions per minute.
“Per” means “for each” or “for every”. It is often shortened to p or /.
EXAMPLE
1) A box of 15
naartjies costs
R16,65. A bag of
8 naartjies of the
same size costs
R9,10. Which is
the best deal?
2) A motorist
travels 56 km in
29 minutes.
What is his
average speed in
km/hour?
SOLUTION
16,65
= R1,11
15
9,10
BAG: 8 naartjies cost R9,10 so 1 naartjie costs R
= R1,14
8
BOX: 15 naartjies cost R16,65 so 1 naartjie costs R
The bag is the best deal. The larger quantity is more expensive.
Note: In each case you calculated the price per 1 naartjie or the unit price,
which is a rate.
In 29 minutes, 56 km is travelled.
So in 1 minute, 56 ÷ 29 = 1,931….km are travelled (do not round off)
In 60 minutes 60 × 2,115….= 115,862… km are travelled.
The average speed ≈ 116 km per hour.
Note: Average speed =
distance
time
Exercise 3.4
1) A 500g tin of jam is marked R15,98. The 300g tin of the same jam costs R9,50. Which size
is the better buy?
2) A car travels at a speed of 120 km per hour. How long will a trip of 2 520 km take if he
travels at the same speed all the way?
3) My car has a petrol consumption of 8,5 litres per 100 km.
a) How many litres of petrol will I need to travel 1 km?
b) How many litres of petrol will I need to travel 420 km?
c) How far, to the nearest km, can I travel on 20 litres?
43
NCV2 – Chapter 3: Rational Numbers and Real Numbers
§ 3.5
DIRECT (or linear) PROPORTION
When two ratios are equal to each other, we say that they are in proportion.
When quantities are in direct proportion, one quantity increases in the same ratio as the
other quantity increases, or one quantity decreases in the same ratio as the other quantity
decreases.
EXAMPLE
1) The price of a sweet is
R3,45. What is the price of
two sweets?
SOLUTION
Two sweets will cost 2 × R3,45 = R6,90.
2) Andile can run 5 km in 20
minutes. How far can he
run at the same speed in 15
minutes?
In 20 minutes Andile runs 5 km
Note:
The number of sweets and the money both doubled – they increased in
the same ratio, so they are in direct (linear) proportion
5
km (less distance)
20
5
In 15 minutes (more time) he runs
× 15 = 3,75 km
20
In 1 minute (less time) he runs
….. (more km)
Note:
The distance (km) decreased in the same ratio as the time (minutes)
decreased, so the distance and the time are in direct (linear) proportion
Exercise 3.5
1) Ben’s computer can print 1 800 words in 5 minutes. Working at the same rate, how many
words can he print in
a) 1 minute?
…………………………………………………………………………..
…………………………………………………………………………..
b) 7 minutes?
…………………………………………………………………………..
………………………………………………………………………….
2) My diesel motorcar has a petrol consumption of 6,5 litres per 100 km.
a) How many litres of petrol will I need to travel 1 km? ……………………………….…
…………………………………………………………………………………………..
b) How many litres of petrol will I need to travel 350 km? ……………………………….
…………………………………………………………………………………………..
c) How far can I travel on 10 litres (correct to 1 decimal place)? …………………………
…………………………………………………………………………………………..
3) A factory produces 600 washers every 3 minutes. How many washers will it produce in 8
minutes? …………………………………………………………………………………..
………………………………………………………………………………………………
……………………………………………………………………………………………...
44
NCV2 – Chapter 3: Rational Numbers and Real Numbers
§ 3.6 INDIRECT (inverse) PROPORTION
When quantities are in indirect proportion, one quantity decreases in the same ratio as
the other quantity increases.
EXAMPLE
Betty has enough sandwiches
to give 5 children, 4
sandwiches each.
However, 10 children arrive.
How many sandwiches will
each child receive, if they are
shared equally?
SOLUTION
The total number of sandwiches available is 5 × 4 = 20.
The number of sandwiches per child will be 20 ÷ 10 = 2.
Note:
The number of children has doubled and the number of
sandwiches has halved, so the number of children and the
number of sandwiches are in indirect (inverse) proportion.
Exercise 3.6
1) 3 campers have enough food to last them for 15 days. 2 others join them. How many days
will they take to finish the food if all eat the same quantity?
2) Seven builders complete a building project in 4 weeks.
a) How long will 14 builders, working at the same rate, take to complete the project?
b) How long will 2 builders, working at the same rate, take to complete the project?
3) It is estimated that a group of 6 plasterers can plaster a house in 9 days. Only 4 arrive for
work. How long will it take these plasterers to finish the job?
45
NCV2 – Chapter 3: Rational Numbers and Real Numbers
§ 3.7 PERCENTAGES
“Percent” means “per hundred”, and is written %. We can convert a percentage to a
fraction or a percentage to a decimal number.
EXAMPLE
1) Convert 6% to a fraction
SOLUTION
6% means 6 equal parts per 100 equal parts,
so 6% =
6
3
=
.
100
50
2) Convert to a decimal number
a) 9%
9% means
b) 4,5%
4,5% =
9
100
or 9 ÷ 100 = 0,09.
4,5
45
=
= 0,045
100
1000
It is useful to know the following percentage to decimal conversions off by heart:
1% = 0,01
10% = 0,1
25% = 0,25
100% = 1
5% = 0,05
50% = 0,5
75% = 0,75
12,5% = 0,125
We can also convert any fraction to a percentage, even those whose denominators do
not divide exactly into 100.
EXAMPLE
SOLUTION
1
1) Express as a percentage
4
25
1
1
25
=
×
=
= 25%
4
4
25
100
2) Use a calculator to write the
fraction
15
as a percentage,
43
rounded to 1 decimal place.
This was easy since 100 is exactly divisible by 4
Key Sequence
Display
Answer
38.888…… 38,9 %
15
43 % =
or
15 ÷ 43 × 100 =
Exercise 3.7
1) Convert 24% to a fraction ………………………………………………………………….
2) Convert 46,5% to a fraction ………………………………………………………………..
3) Convert 45% to a decimal ………………………………………………………………….
4) Convert 32,9% to a decimal ………………………………………………………………..
5) Convert
3
to a percentage ………………………………………………………………..
25
6) Convert
34
to a percentage correct to 2 decimal places …………………………………..
77
7) Convert 2
5
to a percentage correct to 1 decimal place ……………………………………
9
46
NCV2 – Chapter 3: Rational Numbers and Real Numbers
§ 3.8 EXPONENTS
3
In 2 , 2 is the base and 3 is the exponent
24 = 2 × 2 × 2 × 2 . There are 4 factors. We have expanded 24.
If we contract 3 × 3 × 3 …. 7 factors, we have 37.
Multiplication:
2 3 × 2 4 = (2 × 2 × 2) × (2 × 2 × 2 × 2) = 2 7 . But 3 + 4 = 7. We can add the exponents rather
than expand, provided the bases are the same.
Note:
• 4 = 41
EXAMPLE
Simplify
1) 23 × 32 × 24 × 3
2) a3 × a4
SOLUTION
3) 2 b × 2 3 b × 2 4 × 2
2 b × 2 3 b × 2 4 × 2 = 2 b + 3b + 4 + 1 = 2 4 b + 5
4) x −4 × x5 × x 0 × x
x −4 × x 5 × x 0 × x = x − 4 + 5 + 0 + 1 = x 2
23 × 32 × 24 × 31 = 23+4 × 32+1 = 27 × 33
a 3 × a 4 = a 3+ 4 = a 7
Division:
You know that
25 2× 2× 2× 2× 2
=
= 2 2 . But 5 – 3 = 2. We can subtract the exponents
23
2× 2× 2
rather than expand, provided the bases are the same.
EXAMPLE
8
SOLUTION
EXAMPLE
8
SOLUTION
1)
5
56
5
= 58 – 6 = 52
6
5
2)
a
a2
a6
= a 6− 2 = a 4
2
a
3)
23 × 28
2 4× 2
23 × 28
= 2 3+8− 4 −1 = 26
2 4× 2
4)
23
25
23
1
1
= 5 −3 = 2
5
2
2
2
9x3 ÷ 3x2 × x
= 3x3 – 2 + 1 = 3x2
6)
d 3c 4 e
c d 2e2
d 3c 4 e d 3 − 2 c 4 − 1 d c 3
=
=
e
c d 2 e2
e 2−1
5) 9x3 ÷ 3x2 × x
6
Exercise 3.8
Simplify:
1) 52 × 54 ÷ 53 = ………………………………………………………………………………..
2) 2c2 × 4c4 ÷ 8c5 = …………………………………………………………………………..
3)
42 × 4 5
= ……………………………………………………………………………………
48
4) a 5 ×
5)
a2
÷ a = ………………………………………………………………………………..
a4
24 2
× × 3 = …………………………………………………………………………………
32 23
6) 32 × 34 ÷ 33 × 36 × 3 ÷ 37 = ……………………………………………………………….
47
NCV2 – Chapter 3: Rational Numbers and Real Numbers
§ 3.9 ZERO EXPONENTS AND NEGATIVE EXPONENTS
You know that
42
42
2−2
0
=
4
=
4
.
But
= 1 , since any non-zero number divided by itself is
42
42
1, so 4 0 = 1 .
Note:
• 0 0 is undefined. The base of a zero exponent may not be zero
• Any number other than zero raised to the power of zero is 1
• Zero raised to any other power is zero, e.g. 03 = 0
24
24
2× 2× 2× 2
1
1
1
=
.
But
= 2 4 − 7 = 2 − 3 . So 2 − 3 = 3
=
=
7
7
3
2
2× 2× 2× 2× 2× 2× 2 2× 2× 2 2
2
2
In this way we can convert negative exponents to positive exponents.
EXAMPLE
Write with positive
exponents:
1) 3 − 2
SOLUTION
3) p − a
5)
x2
x5
EXAMPLE
SOLUTION
3− 2 =
2) x − 3
x− 3 =
p−a
4)
( − 4) −2
( − 4) −2 =
6)
1
c −1
1
1
c
= 1 ÷ = 1× = c
−1
c
1
c
1
32
1
= a
p
x2
1
= x −3 = 3
5
x
x
1
x3
1
1
=
2
( − 4 ) 16
Exercise 3.9
Simplify, giving your answers with positive exponents:
3)
32
= …………………
36
6)
–30 = …………………
8)
( −3) − 2 = ………………. 9)
−( 3) − 2 = …………….
11)
1
= …………………..
3−2
12)
1
1
+ −1 = ………….
−1
x
y
( a + b ) −1 = ……………. 14)
23
= …………………..
23
15)
a2 b3 c
= ……………
a4 b2 c0
17) 5-4 × 52 × 5 = …………
18)
2−3 × 22
= …………….
2−4
1)
03 = …………………..
2)
4)
c3 × c
= …………….
c2 × c6
5)
x−4
7)
3
4
 
1
= …………………
−1
= ……………….
10) 45 a − a = ……………….
13)
0
( −7 ) = ………………..
16) 4– 3 × 4 3 = ……………
48
NCV2 – Chapter 3: Rational Numbers and Real Numbers
§ 3.10 PRUDUCTS AND QUOTIENTS RAISED TO A POWER
You know that ( 5 × 3) 2 = ( 5 × 3) × ( 5 × 3) = 5 2 × 3 2 .
In the same way, ( a × b ) 3 = ( a × b ) × ( a × b ) × ( a × b ) = a 3 × b 3
2
4
2 2 2 2
24
Also   =   ×   ×   ×   = 4
3 3 3 3 3 3
EXAMPLE
1) Write without brackets
2
a) ( 3.4 )
3
b)
(x × y)
c)
(x
2
y 3z
x
d)  
 y
)
2
y 3 z1
)
4
= x 2 × 4 y 3 × 4 z 1 × 4 = x 8 y 12 z 4
3
x
x3
  = 3
y
 y
2
2
 2x 
2 2 x 2 4x 2
=
= 2
 
2 2
9y
 3y  3 y
2) Write in brackets
a) a 2 × b 2
b) 3 4 × 2 4
c) (x + y) × (x + y)
d) a6 × b4 × c8
3
2 3
e)   ×  
3 4
3
2
f)
2
( 3.4 ) = 32.42
3
( x × y ) = x3 × y 3
(x
4
3
 2x 
e)  
 3y 
SOLUTION
 3a 2   a 3 

 × 
 2b   b 
a 2 × b 2 = (a b) 2
3 4 × 2 4 = (3 × 2 ) 4
(x + y) × (x + y) = (x + y)2
a6 × b4 × c8 = (a3 b2 c4)2
3
3
3
2 3
 2×3 
1
 3  ×  4  =  3× 4  =  2 
 


   
2
2
2
2
3
 3a 2   a 3   3a 2 a 3   3a 5 
×

 ×  =
 =

b   2b 2 
 2b   b   2b
2
Exercise 3.10
1) Remove the brackets and simplify where possible:
a) (4 + 5)2 = ………………………………. b)
c) (a3 b2 c)3 = ……………………………... d)
e) (2p4 × 6q2)2 = …………………………. f)
3
g)
 3y2 
 3  = ………………………………..
 4x 
i)
 2x2 
 3  = ………………………………..
 4x 
(2 × 3)2 = ……………………………..
(p × q)3 = ……………………………..
– (–4a3b)2 = ……………………………
( 5a )
2
h)
2
2
= ……………………………….
5a 2
3
j)
 10 z 4 
 2  = ………………………………
 5z 
2) Contract to 1 bracket and simplify where possible:
a)
23 × 53 = ………………………………
c)
2
3
 3  ×  2  = …………………………
 
 
4
4
49
b) 24 × 54 × 54 × 24 = …………………..
2
 72 
2
 
d)   ×  2  = …………………….
 7 
 25 
25
NCV2 – Chapter 3: Rational Numbers and Real Numbers
§ 3.11 POWER OF AN EXPONENT
4
From the exponent definition, ( 3 2 ) = (3 × 3) × (3 × 3) × (3 × 3) × (3 × 3) = 3 8
As a shortcut, we can multiply the exponents, keeping the existing base.
Note:
• 3 2 × 3 4 = 3 6 , the bases are both 3, so we add the exponents, but
(3 )
2
4
= 3 8 , the base of the 2 is 3 and the base of the 4 is 3 squared, so the bases are
different and we cannot add the exponents.
•
2
( x × y ) = x 2 × y 2 So (2 × 3)2 = 62 = 36 = 4 × 9 = 22 × 32
2
But ( x + y ) ≠ x 2 + y 2 So (2 + 3)2 = 52 = 25, but 22 + 32 = 4 + 9 = 13
We can factorise the base of an exponent and write is as the product of prime factors
For example: 63 = (2 × 3)3 = 23 × 33 Check: 63 = 216 and 23 × 33 = 8 × 27 = 216
EXAMPLE
SOLUTION
Remove the brackets and simplify:
1)
(2 )
2)
(x )
4
2
3
(2 )
4
3
(a )
b
2
3
2×3
= x6
x2 × x3 = x5
c
 24 
2 
3 
= 2 4 × 3 = 2 12
(x ) = x
3) x2 × x3
4)
3
(a )
b
c
= ab×c = abc
5
 2 4  2 4 × 5 2 20
 2  = 2 × 5 = 10
3
3  3
5
3
 3a 2 
 3
 5b 
5) 
6)
 3a 2 
 3
 5b 
7)
32 × 2
62
3
=
27a 6
33 a 2×3
=
125b9
53 b3×3
32 × 2
32 × 2
1
= 2 2 =
2
2
3 ×2
6
Exercise 3.11
Simplify the following. Where necessary write the base as the product of prime factors.
1)
4)
7)
10)
11)
12)
2
(3 x ) = ……………..
(8 ) = ……………….
5
0
5
 3a 2 


 2b 
(s
5)
(-43)2 = ………………
2
×t5
)
3
2)
= ………….
3c) (22 + 32)2 = ……………
(( 2 ) )
9)
 52 b 4 
 5 
 5b 
4
= ……………..
8)
3
(4x – x) = ……………
2 3
6)
2
= ………………
3
= ……………….
122
= ……………………………….……………………………………………………..
82
63
= …………………………….……………………………………………………..
27 × 42
182 × 2
= ………………………………………………………………………………….
42 × 33
50
NCV2 – Chapter 3: Rational Numbers and Real Numbers
§ 3.12 NUMBER SEQUENCES
Look at the number sequence: 1; 4; 7; 10; ….. ; ……. ; …
Each number MINUS the preceding number gives three: 4 – 1 = 3, 7 – 4 = 3, 10 – 7 = 3,
OR, each number PLUS 3 gives us the next number in the sequence.
Sequences like the one above, where adding the same quantity each time gives us the
terms of the sequence, are called arithmetic sequences.
EXAMPLE
SOLUTION
1) Is the following an arithmetic To find out whether this is an arithmetic sequence,
sequence: 13; 8; 3; –2; –7; … ? calculate Term 2 – Term 1 and then calculate
Term 3 – Term 2. If both answers are the same, the
sequence is an arithmetic sequence.
Term 2 – Term 1 = 8 – 13 = –5
Term 3 – Term 2 = 3 – 8 = –5
So this is an arithmetic sequence.
2) If so, write down the next three We subtract 5 in order to find the next three terms in
numbers in the sequence.
the sequence.
–7 – 5 = – 12; –12 – 5 = –17 and – 17 – 5 = –22
So the next three terms are –12; –17 and –22.
Exercise 3.12
Determine whether the following are arithmetic sequences. If they are, write down the next
three numbers in the sequence.
1) 2; 7; 12; ……………………………………………………………………………………
…………………………………………………………………………………………
2) 3; 6; 12; ..………………………………………………………………………………….
........................................................................................................................................
3) 0; -4; -8; ……………………………………………………………………………………
…………………………………………………………………………………………
4) 1,2; 2,3; 3,4; ………………………………………………………………………………
…………………………………………………………………………………………
5)
1 1
; ; 0; ……………………………………………………………………………………
2 4
…………………………………………………………………………………………
6) 1; 4; 9; 16; ………………………………………………………………………………….
…………………………………………………………………………………………
51
NCV2 – Chapter 3: Rational Numbers and Real Numbers
§ 3.13 TERMS OF A SEQUENCE
Each number in a sequence is called a term. T1 is the first term of the sequence, T2 is the
second term and so on.
In the sequence 1; 4; 7; 10; …….. T1 = 1, T2 = 4, T3 = 7 and T4 = 10.
The dots show that there are an infinite number of terms.
Sometimes a general term (Tx) of a sequence is given. You have to use it to find a specific
term or the number of a term.
EXAMPLE
1) Find the 1st, 2nd, 4th, 8th and 12th terms of a sequence with the general term Tx = 3x – 1
T1
Term
T x = 3x – 1
T2
T4
T8
T12
17
29
2) Which terms have the values 17 and 29?
3) Fill the missing values on the table
SOLUTION
T2 = 3(2) – 1
T4 = 3(4) – 1
T8 = 3(8) – 1
1) T1 = 3(1) – 1
=2
=5
= 11
= 23
Tx = 3 x – 1
Tx = 3 x – 1
2)
17 = 3x – 1
29 = 3x – 1
17 + 1 = 3x – 1 + 1
29 + 1 = 3x – 1 + 1
18 = 3x
30 = 3x
30 3 x
=
3
3
18 3x
=
3
3
10 = x
6=x
3)
T12 = 3(12) – 1
= 35
Term
T1
T2
T4
T6
T8
T10
T12
T x = 3x - 1
2
5
11
17
23
29
35
Exercise 3.13
Complete the following two tables:
1)
n
1
2
7
10
13
T n = 2n + 3
27
2)
x
T x = 3x – 4
1
4
5
5
10
20
52
NCV2 – Chapter 3: Rational Numbers and Real Numbers
§ 3.14 THE SET OF RATIONAL
RATIONAL NUMBERS
The letter that stands for the set of rational numbers is Q.
We define rational numbers in the following way:
All rational number can be written in the form
a
where a and b are integers and b
b
≠ 0.
Things to remember:
• Division of a number by zero is undefined in any number system
• Every rational number can be written as an integer divided by a non-zero integer.
• If you can't write the number in this form, it is not rational.
EXAMPLE
If possible, write each of the following numbers in the form
a
where a and b are integers and
b
b ≠ 0. Hence say whether the number is rational or not.
Write in the form
Number
1) 2
3
4
2) 1,5
3) –
12
5
4)
25
5)
3
−27
6) – 7
a
b
3
11
=
4
4
15 3
=
1,5 =
10 2
12
−12
12
–
=
or
5
5
−5
5
25 = 5 =
1
−3
3
3
−27 = –3 =
=
1
−1
−7
–7=
1
2
a, b ∈ Z, b ≠ 0
yes/no
Rational
yes/no
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Exercise 3.14
Use the table to help you decide whether the following numbers are rational numbers.
Number
Write number in the form
a
b
1) – 2,5
2)
22
7
3) – 49
4) 1
5) 0,05
6) 4⅔
7) 0
8) – 53
9)
8
53
Are a, b ∈ Z, b ≠ 0?
Rational?
NCV2 – Chapter 3: Rational Numbers and Real Numbers
§ 3.15 IRRATIONAL NUMBERS
Real numbers that are not rational are called irrational.
We use the rational number definition to decide whether numbers are rational or irrational.
EXAMPLE
Decide whether the following numbers are rational or irrational:
Number
Test for a rational number
1)
49
49 = 7 =
2)
7
7 =
3
3)
64
3
7
1
7
but
1
4
64 = 4 =
1
Rational / Irrational
Rational
7 is not an integer
Irrational
Rational
Other examples of irrational numbers are 3 , 11 , 3 24 etc. They are the square roots of
non-perfect squares or the cube roots of non-perfect cubes. We call these irrational
numbers are surds.
Find 5 on your calculator. 5 = 2,236 067 978 ……..
On your calculator, the digits after the decimal comma seem to come to an end.
This is because your screen can only hold a certain number of digits.
In actual fact, these digits never end and do not recur.
These surds are irrational numbers.
Note:
• π = 3,141 592 654 ……… is irrational and is a non-terminating, non-recurring decimal.
22
is an approximation for
7
•
π , and is rational. π = 3,142 857 142 … and is a recurring
decimal.
Exercise 3.15
Are the following numbers rational or irrational?
Number
Test for a rational number
•
1)
9,8
2)
5 2
3)
4
4)
7,148 148 148 ...
5)
1,54
6)
2,14
7)
π
8)
22
7
9)
4
16
54
Rational / irrational
NCV2 – Chapter 3: Rational Numbers and Real Numbers
§ 3.16
SURDS
Each irrational number lies between two rational numbers, and so can be shown
approximately on a number line.
You know that 16 < 18 < 25. By taking the square root of these numbers, 4 < 18 < 5.
Check this by finding 18 on your calculator. 18 = 4,242 640…….
EXAMPLE
1) Simplify
20
a)
b) ( 7 )3
c) 1,21
2) Simplify
a) 7 + 7
b) 3 2 – 2 2
c) 7 × 7
d) 2 × 3
e) –2 5 × 3 5
f)
6 × 12
g)
SOLUTION
20 =
5×4 = 2 5
3
( 7) = 7 × 7 × 7 =7 7
1,21 = 1,1× 1,1 = 1,1
7+
7 =2 7
3 2 –2 2 = 2
7 × 7 = 49 = 7
2 × 3 = 6
–2 5 × 3 5 = – 6 × 5 = – 30
6 × 12 = 6 × 6 × 2 = 6 2
8
8
32
32
=
2 2
4 2
=
1
2
Rationalisation of the denominator:
If the denominator of a faction is a surd, we can make the denominator a rational number.
Example:
4
=
2
=
4
2
×
2
2
Multiply numerator and denominator by the same surd
4 2
=2 2
2
Exercise 3.16
1) Simplify:
a) 8 × 8 = …………………………….
c) 4 3 –
3 = …………………………….
e) 4 3 × 3 3 = ………………………….
g)
18
6 2
= …………………………………..
b) 2 3 × 3 2 = ………………………
d) 3 5 + 4 5 = …………………………
f)
h)
6 ×
27 = ………………………….
625 ÷
3
125 = ……………………..
2) Rationalise the denominators:
a)
6
3
= ……………………………………
b)
14
7
= ………………………………….
3) Between which two integers does 7 lie? ………………………………………………..
4) Between which two integers does
3 lie? ………………………………………………..
55
NCV2 – Chapter 3: Rational Numbers and Real Numbers
§ 3.17 THE SET OF REAL NUMBERS
All the rational numbers together with all the irrational numbers form the set of Real
numbers, R. Here are some of them on a number line.
- 5
-3,5
-4
-3
-1
-2
3
4
1
2
•
- 0, 3
-1
0
11
3
2
1
2
3
4
Note:
•
−4 is not a real number, since 2 × 2 = 4 and –2 × –2 = 4. It is a complex number.
EXAMPLE
On the table, mark with an X the number system/s to which each number belongs:
SOLUTION
Natural
Whole
Rational
Irrational
Real
Number
Integer
Number
Number
Number
Number
Number
–56
1)
0
2)
13⅛
3)
5
4)
3
5)
27
6)
–
12
5
7)
π
8)
22
7
Exercise 3.17
On the table, mark with an X the number system/s to which each number belongs:
Natural
Whole
Rational Irrational
Number
Integer
Number
Number
Number
Number
•
1)
19, 2
2)
3)
4)
5)
π
–3
2
3
2
22
7
6) 1,148796…
7) 4 16
8) 15,78
9) 0
10) 144
56
Real
Number
NCV2 – Chapter 3: Rational Numbers and Real Numbers
§ 3.18 THE REAL NUMBER DIAGRAM
Here is a diagram to help you remember what you have learned about the real number
system.
Real
numbers
Rational
numbers
Integers
Whole
numbers
Irrational
numbers
Fractions
Negative
integers
0
Natural
numbers
Exercise 3.18
Use the number sets diagram to help you answer these questions:
Statement
1)
25 is rational.
2)
13 is rational.
3)
4)
Any recurring decimal number is rational.
3
−27 is irrational
56
99
5)
4,56 = 4
6)
0,6 = 0,6
7)
3
4 is rational because 4 is a perfect square
8)
All surds are rational
9)
3,11111.... = 3,1
•
10) 8,2 = 8
2
10
11) All rational numbers are real numbers
12) All real numbers are rational numbers
13) All whole numbers are rational numbers
14) All irrational numbers are real numbers.
15) All real numbers are whole numbers.
16) Zero is a real number
17) Some real numbers are irrational
57
True (T) / False (F)

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