The document focuses on measured numbers and significant figures, explaining how to determine and report measurements accurately using proper estimation. It discusses the importance of significant figures, including rules for non-zero digits, leading and trailing zeros, and scientific notation. Additionally, it differentiates between exact and measured numbers through examples and classification exercises.

1. 1
1.2
Measured Numbers and
Significant Figures
Chapter 1 Measurements
Copyright © 2005 by Pearson Education, Inc.
Publishing as Benjamin Cummings

2. 2
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

3. 3
. 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

4. 4
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.

5. 5
. 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

6. 6
. 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

7. 7
. 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

8. 8
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.

9. 9
Significant Figures

10. 10
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

11. 11
Sandwiched zeros
• occur between nonzero numbers.
• are significant.
Number of
Measurement Significant Figures
50.8 mm 3
2001 min 4
0.0702 lb 3
0.40505 m 5
Sandwiched Zeros

12. 12
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
Trailing Zeros

13. 13
Leading zeros
• precede non-zero digits in a decimal number.
• are not significant.
Number of
Measurement Significant Figures
0.008 mm 1
0.0156 oz 3
0.0042 lb 2
0.000262 mL 3
Leading Zeros

14. 14
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

15. 15
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

16. 16
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

17. 17
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

18. 18
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

19. 19
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

20. 20
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.

21. 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. 22
Exact Numbers

23. 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. 24
Solution
A. Exact numbers are obtained by
2. counting.
3. definition.
B. Measured numbers are obtained by
1. using a measuring tool.

25. 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. 26
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.
Solution