# 1_3 Measured Numbers and Significant Figures

```Chapter 1
Measurements
1.3
Measured Numbers and
Significant Figures
1
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.
2
. l2. . . . l . . . . l3 . . . . l . . . . l4. .
cm
• The markings on the meterstick at the end of the
•
•
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.
3
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.
4
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
5
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
6
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.
7
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.
8
Significant Figures
9
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
11
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
12
• 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
13
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
14
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
15
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
16
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
17
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).
18
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).
19
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
20
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.
21
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
22
Exact Numbers
23
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.
24
Solution
A. Exact numbers are obtained by
2. counting.
3. definition.
B. Measured numbers are obtained by
1. using a measuring tool.
25
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.
26
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.
27
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