Transcript 1_3 Measured Numbers and Significant Figures

Chapter 1
Measurements
1.3
Measured Numbers and
Significant Figures
Copyright © 2009 by Pearson Education, Inc.
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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 © 2009 by Pearson Education, Inc.
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Reading a Meterstick
. l2. . . . l . . . . l3 . . . . l . . . . l4. .
cm
• The markings on the meterstick 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 may 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.
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Known & Estimated Digits
If the length is reported as 2.76 cm,
• the digits 2 and 7 are certain (known).
• the final digit, 6, is estimated (uncertain).
• all three digits (2, 7, and 6) are significant, including
the estimated digit.
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Learning Check
. l8. . . . l . . . . l9. . . . l . . . . l10. . cm
What is the length of the orange line?
1) 9.0 cm
2) 9.04 cm
3) 9.05 cm
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Solution
. l8. . . . l . . . . l9. . . . l . . . . l10. . cm
The length of the orange line could be reported as
2) 9.04 cm
or
3) 9.05 cm
The estimated digit may be slightly different. Both
readings are acceptable.
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Zero as a Measured Number
. l3. . . . l . . . . l4. . . . l . . . . l5. . cm
• For this measurement, the first and second known
digits are 4 and 5.
• When a measurement ends on a mark, the estimated
digit in the hundredths place is 0.
• This measurement is reported as 4.50 cm.
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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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Counting Significant Figures
All nonzero 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
10
Sandwiched Zeros
Sandwiched Zeros
• occur between nonzero numbers.
• are significant.
Measurement
50.8 mm
2001 min
0.0702 lb
0.405 05 m
Number of
Significant Figures
3
4
3
5
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Trailing Zeros
Trailing Zeros
• follow nonzero numbers in numbers without
decimal points.
• are usually placeholders.
• are not significant.
Number of
Measurement
Significant Figures
25 000 cm
2
200 kg
1
48 600 mL
3
25 005 000 g
5
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Leading Zeros
Leading Zeros
• precede nonzero digits in a decimal number.
• are not significant.
Measurement
0.008 mm
0.0156 oz
0.0042 lb
0.000 262 mL
Number of
Significant Figures
1
3
2
3
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Learning Check
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
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Solution
State the number of significant figures in each of
the following measurements.
A. 0.030 m
B. 4.050 L
2
4
C. 0.0008 g
D. 2.80 m
1
3
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Significant Figures in
Scientific Notation
In scientific notation all digits in the coefficient
including zeros are significant.
Measurement
8 x 104 m
8.0 x 104 m
8.00 x 104 m
Number of
Significant Figures
1
2
3
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Study Tip: Significant Figures
The significant figures in a measured number are
• all the nonzero numbers.
12.56 m
4 significant figures
• zeros between nonzero numbers.
4.05 g
3 significant figures
• zeros that follow nonzero numbers in a decimal
number.
25.800 L
5 significant figures
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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 (1).
2) two (2).
3) three (3).
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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 (3).
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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) 0.000 015 and 150 000
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Solution
In which set(s) do both numbers contain the same
number of significant figures?
3) 0.000 015 and 150 000
Both numbers contain 2 significant figures.
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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
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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) 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.
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