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1.3Measured Numbers and

Significant Figures

Chapter 1 Measurements

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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. 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.

Reading a Meterstick

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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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. 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

Learning Check

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. 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.

Solution

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. 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.

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 nonzero 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 30.405 05 m 5

Sandwiched Zeros

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Trailing Zeros• follow nonzero numbers in numbers without

decimal points.• are usually placeholders. • 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 nonzero digits in a decimal number. • are not significant.

Number of Measurement Significant Figures0.008 mm 10.0156 oz 30.0042 lb 20.000 262 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 in the coefficient

including zeros 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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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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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).

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 (3).

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) 0.000 015 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) 0.000 015 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 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