Chapter 1
Chemical Foundations
1.5
Significant Figures and
Calculations
Copyright © 2005 by Pearson Education, Inc.
Publishing as Benjamin Cummings
1
Significant Figures in
Measured Numbers
Significant figures
• obtained from a measurement include all
of the known digits plus the estimated
digit.
• reported in a measurement depend on the
measuring tool.
2
Significant Figures
3
Counting Significant Figures
All non-zero numbers in a measured number are
significant.
Measurement
38.15 cm
5.6 ft
65.6 lb
122.55 m
Number of
Significant Figures
4
2
3
5
4
Sandwiched Zeros
Sandwiched zeros
• occur between nonzero numbers.
• are significant.
Measurement
50.8 mm
2001 min
0.0702 lb
0.40505 m
Number of
Significant Figures
3
4
3
5
5
Trailing Zeros
Trailing zeros
• follow non-zero numbers in numbers without
decimal points.
• are usually place holders.
• are not significant.
Number of
Measurement
Significant Figures
25 000 cm
2
200 kg
1
48 600 mL
3
25 005 000 g
5
6
Leading Zeros
Leading zeros
• precede non-zero digits in a decimal number.
• are not significant.
Measurement
0.008 mm
0.0156 oz
0.0042 lb
0.000262 mL
Number of
Significant Figures
1
3
2
3
7
Learning Check
State the number of significant figures in each of
the following measurements.
A.
B.
C.
D.
0.030 m
4.050 L
0.0008 g
2.80 m
8
Solution
State the number of significant figures in each of
the following measurements.
A.
B.
C.
D.
0.030 m
4.050 L
0.0008 g
2.80 m
2
4
1
3
9
Significant Figures in
Scientific Notation
In scientific notation all digits including zeros in the
coefficient are significant.
Measurement
8 x 104 m
8.0 x 104 m
8.00 x 104 m
Number of
Significant Figures
1
2
3
10
Learning Check
A. Which answer(s) contain 3 significant figures?
1) 0.4760
2) 0.00476
3) 4.76 x 103
B. All the zeros are significant in
1) 0.00307
2) 25.300
3) 2.050 x 103
C. The number of significant figures in 5.80 x 102 is
1) one
3) two
3) three
11
Solution
A. Which answer(s) contain 3 significant figures?
2) 0.00476
3) 4.76 x 103
B. All the zeros are significant in
2) 25.300
3) 2.050 x 103
C. The number of significant figures in 5.80 x 102 is
3) three
12
Learning Check
In which set(s) do both numbers contain the
same number of significant figures?
1) 22.0 and 22.00
2) 400.0 and 40
3) 0.000015 and 150 000
13
Solution
In which set(s) do both numbers contain the same
number of significant figures?
3) 0.000015 and 150 000
Both numbers contain two (2) significant figures.
14
Examples of Exact Numbers
An exact number is obtained
• when objects are counted.
Counted objects
2 soccer balls
4 pizzas
• from numbers in a defined relationship.
Defined relationships
1 foot = 12 inches
1 meter = 100 cm
15
Exact Numbers
16
Learning Check
A. Exact numbers are obtained by
1. using a measuring tool.
2. counting.
3. definition.
B. Measured numbers are obtained by
1. using a measuring tool.
2. counting.
3. definition.
17
Solution
A. Exact numbers are obtained by
2. counting.
3. definition.
B. Measured numbers are obtained by
1. using a measuring tool.
18
Learning Check
Classify each of the following as (1) exact or (2) measured
numbers.
A.__Gold melts at 1064°C.
B.__1 yard = 3 feet
C.__The diameter of a red blood cell is 6 x 10-4 cm.
D.__There are 6 hats on the shelf.
E.__A can of soda contains 355 mL of soda.
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Solution
Classify each of the following as (1) exact or (2) measured
numbers.
A. 2
A measuring tool is required.
B. 1
This is a defined relationship.
C. 2
A measuring tool is used to determine
length.
D. 1
The number of hats is obtained by counting.
E. 2
The volume of soda is measured.
20
Rounding Off Calculated Answers
In calculations,
•
answers must have the same
number of significant figures
as the measured numbers.
•
often, a calculator answer
must be rounded off.
•
rounding rules are used to
obtain the correct number of
significant figures.
Copyright © 2005 by Pearson Education, Inc.
Publishing as Benjamin Cummings
21
Rounding Off Calculated
Answers
When the first digit dropped is 4 or less,
• the retained numbers remain the same.
45.832 rounded to 3 significant figures
drops the digits 32 = 45.8
When the first digit dropped is 5 or greater,
• the last retained digit is increased by 1.
2.4884 rounded to 2 significant figures
drops the digits 884 = 2.5 (increase by 0.1)
22
Adding Significant Zeros
•
Sometimes a calculated answer requires more
significant digits. Then, one or more zeros are
added.
Calculated
Answer
4
1.5
0.2
12
Zeros Added to
Give 3 Significant Figures
4.00
1.50
0.200
12.0
23
Learning Check
Adjust the following calculated answers to give
answers with three significant figures.
A. 824.75 cm
B. 0.112486 g
C. 8.2 L
24
Solution
Adjust the following calculated answers to give answers
with three significant figures
A. 825 cm
First digit dropped is greater than 5.
B. 0.112g
First digit dropped is 4.
C. 8.20 L
Significant zero is added.
25
Calculations with Measured
Numbers
In calculations with
measured numbers,
significant figures or
decimal places are
counted to determine
the number of figures in
the final answer.
Copyright © 2005 by Pearson Education, Inc.
Publishing as Benjamin Cummings
26
Multiplication and Division
When multiplying or dividing use
•
the same number of significant figures as the
measurement with the fewest significant figures.
•
rounding rules to obtain the correct number of
significant figures.
Example:
110.5
4 SF
x
0.048 = 5.304
2 SF
=
calculator
5.3 (rounded)
2 SF
27
Learning Check
Give an answer for the following with the correct
number of significant figures.
A. 2.19 x 4.2
1) 9
=
2) 9.2
3) 9.198
B. 4.311 ÷ 0.07 =
1) 61.59
2) 62
3) 60
C. 2.54 x 0.0028 =
0.0105 x 0.060
1) 11.3
2) 11
3) 0.041
28
Solution
A. 2.19 x 4.2
B. 4.311 ÷ 0.07
C. 2.54 x 0.0028
0.0105 x 0.060
= 2) 9.2
= 3) 60
= 2) 11
On a calculator, enter each number followed by
the operation key.
2.54 x 0.0028 ÷ 0.0105 ÷ 0.060 = 11.28888889
= 11 (rounded)
29
Addition and Subtraction
When adding or subtracting use
•
the same number of decimal places as the
measurement with the fewest decimal places.
•
rounding rules to adjust the number of digits in the
answer.
25.2
+ 1.34
26.54
26.5
one decimal place
two decimal places
calculated answer
answer with one decimal place
30
Learning Check
For each calculation, round the answer to give the
correct number of significant figures.
A. 235.05 + 19.6 + 2 =
1) 257
2) 256.7
B.
58.925 - 18.2 =
1) 40.725 2) 40.73
3) 256.65
3) 40.7
31
Solution
A. 235.05
+19.6
+ 2
256.65 rounds to 257
B.
58.925
-18.2
40.725 round to 40.7
Answer (1)
Answer (3)
32