Significant Figures - University of Michigan

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Determining
Significant
Figures
A4
Science Learning Center
University of Michigan,
Dearborn
Directions
This module consists of a written explanation and
examples problems. As you read through the
material, be sure to complete all example problems.
When you have completed reading the module and
you feel you fully understand the material, ask for a
posttest. If you pass the posttest, make sure your
name is recorded in the SLC database. If you do not
pass it, you may review the module and retake the
test as many times as needed.
Objectives
 Know what significant figures are.
 Know which numbers are significant.
 Be able to express the result of a measurement with the
correct number of significant figures.
 Distinguish between significant and non-significant zeros.
 Be aware that the number of figures in scientific notation
indicates precision.
 Be able to express the result of a calculation with the correct
number of significant figures.
What are Significant Figures?
Significant Figures and Measurements
A significant figure is one that has some significance but
does not necessarily denote a certainty.
Whenever you estimate any kind of measurement, for
example the length or weight of an object, there is always
a limit to the number of digits you can read.
The number of significant figures in a measurement is
the number of digits that are known with certainty plus
the last one that is not absolutely certain.
What are Significant Figures?
Significant Figures and Measurements
As a general rule you should attempt to read any scale to one
tenth of its smallest division by visual interpolation.
In the case below, you would read to + 0.01cm. This estimated
figure will always be your last significant figure with the implied
accuracy of + 1. Therefore, the measurement is written as
4.63 + 0.01cm.
Generally, read any scale to
1/10 of smallest division.
4.63 + 0.01 cm
What are Significant Figures?
Significant Figures and Measurements
 A length measurement of 5.63 cm contains three significant figures. The first two,
the 5 and 6, are certain. The last digit, the 3, is uncertain. The uncertainty in the
last significant figure is usually + 1. The result is 5.63 + 0.01cm.
 An Analytical Balance is precise to four decimal places with an uncertainty of +1 in
the last significant figure. Therefore, the measurement 13.7654g is written as
13.7654 + 0.0001 g and has six significant figures.
Precision of Two Instruments
Ruler
5.63 + 0.01 cm
3 significant figures
Analytical
Balance
13.7654 + 0.0001 g
6 significant figures
What are Significant Figures?
Significant Figures and Measurements
Here we see another kind of measurement, the reading of the position of a buret meniscus, (the curved
liquid surface in a buret). The liquid level is always read at the bottom of the meniscus for transparent
liquids.
The reading in this buret is 12.75. Four significant figures are implied. The last significant figure, 5, is
obtained by visual interpolation between the 0.1 milliliter divisions.
All observers should agree with the first three significant figures but not necessarily with the last figure
recorded here. Disagreements of +1 in the last digit are expected with visual interpolations. The
measurement is, therefore, written as 12.75 + 0.01ml and has four significant figures.
Measurement with a Buret
Read the bottom of meniscus at eye level
12.75 + 0.01 ml
4 significant figures
What are Significant Figures?
Significant Figures and Measurements
When reading a measurement from a meter, you should also
read to one digit past the smallest division on the meter.
On this meter, the reading should be 1.27. The 1 and the 2 are
certain. The 7 must be estimated visually. The measurement is
written as 1.27 + 0.01g.
1.00
2.00
0.00
grams
Reading: 1.27 + 0.01 g
The 7 is estimated and, therefore, uncertain.
Writing Significant Figures
During any calculation—addition, subtraction, multiplication, or division—
your answer could be expressed with too few or too many significant figures.
These numeric values may imply a precision that does not exist in the
experiment being evaluated.
If you round off incorrectly, your answer will have an incorrect number of
significant figures and will lose precision.
EXAMPLE
1.024 x 1.2 = 1.2288
Too many numerals
Too precise
1.024 x 1.2 = 1
Too few numerals
Not precise enough
We, therefore, have developed rules for determining the correct number of
significant figures in a number and apply these rules to calculations.
Writing Significant Figures
First we need to learn how to evaluate the number of significant figures a given number
contains. This is necessary for calculations such as addition, subtraction, multiplication and
division.
 Any written number that is not a zero is significant.
In this table the significant figures are underlined:
3 significant
figures
4 significant
figures
5 significant
figures
23.4
2.34
0.234
345.6
3.456
0.03456
678.90
Note: The zeros in 0.234 and 0.03456 are not significant, but the
zero in 678.90 is a significant figure. Zeros have special rules as
we will discuss in the next several slides.
Writing Significant Figures
Standard Exponential Notation and Zeros
 Zeros appearing between nonzero numbers are
significant.
Examples
40.7 L
87,009 km
3 significant figures
5 significant figures
Writing Significant Figures
Standard Exponential Notation and Zeros
 Zeros appearing in front of nonzero numbers
are not significant.
Examples
0.095987 m
0.0009 kg
5 significant figures
1 significant figure
Writing Significant Figures
Standard Exponential Notation and Zeros
 Zeros appearing at the end of a number and to
the right of a decimal point are significant.
Examples
850.00 g
9.500000000 mm
5 significant figures
10 significant figures
Writing Significant Figures
Standard Exponential Notation and Zeros
Without a decimal, large numbers containing zeros, such as 45,600 grams, pose a special problem.
As the number is written, we cannot tell whether the two zeros indicate the precision of the
measurement or whether the zeros merely locate the decimal point.
 If the zeros indicate precision, they are significant and the implied uncertainty is + 1. This
means that the measurement lies between 45,599 and 45,601.
 If, however, the zeros merely locate the decimal point and are not significant, the implied
uncertainty is + 100. Then we know the measurement lies between 45,500 and 45,700.
Thus a number written in this form is ambiguous.
Example
45,600 grams
Zeros Significant
45,600 grams + 1 gram
45,599
45,601
5 significant figures
Zeros Not Significant
45,600 grams + 100 grams
45,500
45,700
3 significant figures
**For the purposes of this module and its posttest, assume the zeros are not
significant for ambiguous numbers.**
Writing Significant Figures
Standard Exponential Notation and Zeros
Zeros appearing after a nonzero number which
are not followed by another significant figure or
a decimal point are not significant.
Examples
85,000
85,000,000
85,000,000.0
85,000,000.
2 significant
figures
2 significant
figures
9 significant
figures
8 significant
figures
Note: The decimal point after
the zeros indicates that all
the numbers are significant.
Writing Significant Figures
Summary: Rules for Zeros
 Zeros appearing between two nonzero numbers are significant.
Examples: The number 2.035 has 4 significant figures
The number 3,0007 has 5 significant figures.
 Zeros appearing in front of nonzero numbers are not significant.
Example: The number 0.0567 has only 3 significant figures.
 Zeros appearing at the end of a number and to the right of a decimal point indicate
precision and are significant.
Example: 4.700 has 4 significant figures.
 Zeros appearing after a nonzero number which are not followed by another significant
figure or a decimal point are not significant
Example: The number 1200 has 2 significant figures, the number 1200. has 4 significant figures, and the
number 1200.0 has 5 significant figures.
Writing Significant Figures
Standard Exponential Notation and Zeros
Ambiguity about the precision and number of significant figures for
a particular number may be avoided by expressing the number in
standard exponential notation or scientific notation.
 If the number 45,600.0 contains 5
significant figures, the number would be
expressed as 4.5600 x 104. This notation
implies 5 significant figures.
 If only 3 numbers are significant, the
number would be expressed as 4.56 x 104.
Normal notation
45,600
Scientific notation
4.5600 x 104
4,5600
4.56 x 104
Writing Significant Figures
Standard Exponential Notation and Zeros
Scientific notation is also useful for clearly expressing very large and
very small numbers with the correct precision and number of
significant figures.
 The number 0.001230 can be expressed
as 1.230 x 10-3 which contains 4 significant
figures.
 The number 900.00 can be expressed as
9.0000 x 102 which contains 5 significant
figures.
 Finally, the number 2000. has a decimal
on the end indicating precision and that
all of the zeros are significant. Thus, 2000.
can be expressed as 2.000 x 103 which
contains 4 significant figures.
Normal notation
Scientific notation
0.001230
1.230 x 10-3
900.00
9.0000 x 102
2000.
2.000 x 103
Practice Problems 1
How many significant figures are in each of the
following numbers?
a)
b)
c)
d)
e)
f)
g)
h)
1.234
1.2340
1.234 x 10-3
1.2340 x 10-3
1234
12340
0.012340
10234
Solutions to Practice Problems 1
a) 1.234: 4
b) 1.2340: 5
c) 1.234 x 10-3: 4
d) 1.2340 x 10-3: 5
e) 1234: 4
f) 12340: 4
g) 0.012340: 5
h) 10234: 5
Calculations
You have learned how to determine how many
significant figures are in a number. Now you will
learn how many significant figures should be
expressed in the result of a calculation.
Calculations
Adding and Subtracting
When adding or subtracting quantities, the rule
is to determine which number in the calculation
has the least number of digits to the right of
the decimal point. Your result will have that
same number of digits to the right of the
decimal point.
Calculations
Adding and Subtracting
Example
26.46
+ 4.123
30.583
Least number of digits to the
right of the decimal = 2
26.46
- 4.123
22.337
30.58
Round to 2 decimals
22.34
 For example, when adding 26.46 to 4.123, the calculated sum is 30.583.
The original number 26.46 has the least number of digits to the right of the
decimal point, two, so the calculated sum is rounded to 30.58.
 In subtracting 4.123 from 26.46, the calculated difference is 22.337.
The original number 26.46 has the least number of digits to the right of the
decimal point, two, so the calculated difference is rounded to 22.34.
Calculations
Adding and Subtracting
Example
2.634
+ 0.02
2.654
Least digits to the right = 2
2.634
- 0.02
2.614
2.65
Round to 2 decimals
2.61
Another example of addition is 2.634 plus 0.02. The calculated sum,
2.654, will be rounded off to 2.65.
**The number 0.02 has the least number of digits to the right of the
decimal point, two, regardless of the fact that only one of the digits is
significant.**
The result will have two digits to the right of the decimal point.
When subtracting 2.634 minus 0.02, the calculated difference, 2.614, is
rounded off to 2.61.
Practice Problems 2
Complete the following arithmetic operations and
express the answer with the correct number of
significant figures:
a)
b)
c)
d)
e)
f)
g)
h)
1.421+ 0.4372 =
0.0241 + 0.11 =
0.14 + 1.2243 =
760.0 + 0.011 =
1.0123 - 0.002 =
123.69 - 20.1 =
463.231 - 14.0 =
47.2 - 0.01 =
Solutions to Practice Problems 2
a) 1.421+ 0.4372 = 1.858
b) 0.0241 + 0.11 = 0.13
c) 0.14 + 1.2243 = 1.36
d) 760.0 + 0.011 = 760.0
e) 1.0123 - 0.002 = 1.010
f) 123.69 - 20.1 = 103.6
g) 463.231 - 14.0 = 449.2
h) 47.2 - 0.01 = 47.2
Calculations
Multiplying and Dividing
For the multiplication and division of numbers,
there is a different rule for determining the
number of significant figures. When multiplying
or dividing, determine which number entering
the calculation has the smallest total number
of significant figures regardless of the position
of the decimal point. Your calculated result will
have that same number of significant figures.
Calculations
Multiplying and Dividing
Example
Least number of significant figures
2.61 x 1.2 = 3.132
2.61 / 1.2 = 2.175
Round to: 3.1
Round to: 2.2
 In the example above, 2.61 multiplied by 1.2, the result is rounded
off to 3.1.
The number 1.2 has the least number of significant figures, two. So the
calculated answer will also have two significant figures.
 When 2.61 is divided by 1.2, the result is also rounded off to two
significant figures.
The calculated answer, 2.175, is rounded off to 2.2.
Practice Problems 3
Perform the indicated operations. Express your
answers with the correct number of significant
figures:
a)
b)
c)
d)
42.3 x 2.61 =
0.61 x 42.1 =
46.1 / 1.21 =
23.2 / 4.1 =
Solutions to Practice Problems 3
a) 42.3 x 2.61 = 110.
b) 0.61 x 42.1 = 26
c) 46.1 / 1.21 = 38.1
d) 23.2 / 4.1 = 5.7
Review of Rules for Calculations
Addition/ Determine which number in the calculations
subtraction has the least number of digits to the right of
the decimal point. Your result will also have
the same number of digits to the right of the
decimal point.
234.7
+ 1.623
236.323
Result: 236.3
Multiplicati Determine which number in the calculations
on /
has the least total number of significant
Division
figures (regardless of the decimal point’s
position). Your result will also have that same
number of significant figures.
44.2
x 2.662
117.6604
Result: 117
Additional Principles
As you begin to apply the principles of significant figures to actual problems or laboratory
experiments, three additional principles should be presented.
Principle 1:
If you are using exact constants, they do not affect the number of significant figures in your
answer.
For example, you might need to calculate how many feet equal 26.1 yards. The conversion factor
you would need to use, 3 ft/yard, is an exact constant and does not affect the number of
significant figures in your answer.
26.1 yards multiplied by 3 feet per yard equals 78.3 feet which has 3 significant figures.
Example
EXACT CONSTANT: 3 ft = 1 yd
26.1 yd
3 significant
figures
X
3 ft / 1 yd
EXACT
=
78.3 ft
3 significant
figures
Additional Principles
Principle 2:
If you are using constants which are not exact (such as pi = 3.14 or 3.142 or 3.14159) select the
constant that has at least one or more significant figures than the smallest number of significant
figures in your original data. This way the number of significant figures in the constant will not
affect the number of significant figures in your answer.
For example, if you multiply 4.136 ft., which has four significant figures, times pi, you should use
3.1416 which has 5 significant figures for pi and your answer will have 4 significant figures.
EXAMPLE
4.316 ft. x π = 4.316 ft. x 3.1415 = 13.56
4 significant
figures
5 significant
figures for π
answer in 4
significant
figures
Additional Principles
Principle 3:
When you are doing several calculations, carry out all of the calculations to at least one more
significant figure than you need. When you get the final result, then round off.
For example, you would like to know how many meters per second equals 55 miles per hour.
The conversion factors you would use are: 1 mile = 1.61 x 103 meters and 1 hour = 3600 seconds
Your answer should have two significant figures.
Your result would be 88.55 divided by 3600 which equals 24.59 m/sec. This rounds off to 25 m/sec.
By carrying this calculation out to at least one extra significant figure, we were able to round off and
give the correct answer of 25 m/sec rather than 24 m/sec.
EXAMPLE
How many meters per second is 55 miles per hour?
1 mile = 1.61 x 103 m (not exact constant; 3 sig. figs.)
1 hour = 3600 seconds (exact constant; 4 sig. figs.)
55 miles / hour =
55 miles / 1 hr. x 1.61 x 103 m / 1 mile
x
1 hour / 3600 sec
= 24.597 m / 1 sec = 25 m/s
Take the Posttest!
You are now finished with this module. If you
haven't already done the practice problems, we
recommend you try them.
When you're done, obtain a posttest from the
Science Learning Center personnel and
complete it.
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