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1
1.2Measured Numbers and
Significant Figures
Chapter 1 Measurements
Copyright © 2005 by Pearson Education, Inc.Publishing as Benjamin Cummings
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Measured Numbers
A measuring tool
• is used to determine a quantity such as height or the mass of an object.
• provides numbers for a measurement called measured numbers.
Copyright © 2005 by Pearson Education, Inc.Publishing as Benjamin Cummings
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. l2. . . . l . . . . l3 . . . . l . . . . l4. . cm
• The markings on the meter stick at the end of the orange line are read as
the first digit 2
plus the second digit 2.7 • The last digit is obtained by estimating. • The end of the line might be estimated between 2.7–
2.8 as half-way (0.5) or a little more (0.6), which gives a reported length of 2.75 cm or 2.76 cm.
Reading a Meter Stick
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Known & Estimated Digits
In the length reported as 2.76 cm,
• The digits 2 and 7 are certain (known).• The final digit 6 was estimated (uncertain).• All three digits (2.76) are significant including the
estimated digit.
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. l8. . . . l . . . . l9. . . . l . . . . l10. . cm
What is the length of the orange line?
1) 9.0 cm
2) 9.03 cm
3) 9.04 cm
Learning Check
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. l8. . . . l . . . . l9. . . . l . . . . l10. . cm
The length of the orange line could be reported as
2) 9.03 cm
or 3) 9.04 cm
The estimated digit may be slightly different. Both readings are acceptable.
Solution
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. l3. . . . l . . . . l4. . . . l . . . . l5. . cm
• For this measurement, the first and second known digits are 4.5.
• Because the line ends on a mark, the estimated digit in the hundredths place is 0.
• This measurement is reported as 4.50 cm.
Zero as a Measured Number
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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.
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Significant Figures
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All non-zero numbers in a measured number are significant.
Number of Measurement Significant Figures
38.15 cm 4
5.6 ft 2
65.6 lb 3
122.55 m 5
Counting Significant Figures
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Sandwiched zeros• occur between nonzero numbers.• are significant.
Number of Measurement Significant Figures50.8 mm 32001 min 40.0702 lb 30.40505 m 5
Sandwiched Zeros
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Trailing zeros• follow non-zero numbers in numbers without
decimal points.• are usually place holders. • are not significant.
Number of Measurement Significant Figures25 000 cm 2
200 kg 1 48 600 mL 3
25 005 000 g 5
Trailing Zeros
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Leading zeros • precede non-zero digits in a decimal number. • are not significant.
Number of Measurement Significant Figures0.008 mm 10.0156 oz 30.0042 lb 20.000262 mL 3
Leading Zeros
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State the number of significant figures in each of the following measurements.
A. 0.030 m
B. 4.050 L
C. 0.0008 g
D. 2.80 m
Learning Check
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State the number of significant figures in each of the following measurements.
A. 0.030 m 2
B. 4.050 L 4
C. 0.0008 g 1
D. 2.80 m 3
Solution
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Significant Figures in Scientific Notation
In scientific notation all digits including zeros in the coefficient are significant.
Number of Measurement Significant Figures
8 x 104 m 1
8.0 x 104 m 2
8.00 x 104 m 3
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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
Learning Check
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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
Solution
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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
Learning Check
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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.
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Examples of Exact Numbers
An exact number is obtained
• when objects are counted.Counted objects
2 soccer balls4 pizzas
• from numbers in a defined relationship. Defined relationships
1 foot = 12 inches1 meter = 100 cm
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Exact Numbers
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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.
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Solution
A. Exact numbers are obtained by
2. counting.
3. definition.
B. Measured numbers are obtained by
1. using a measuring tool.
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Learning Check
Classify each of the following as (1) exact or (2) measurednumbers.
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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Classify each of the following as (1) exact or (2) measurednumbers.
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.
Solution