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1.8Significant Figures

Chapter 1 Matter, Measurements, &

Calculations

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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Rounding Off Calculated Answers

In calculations,

• answers must have the same number of significant figures as the measured numbers.

• often, a calculator answer must be rounded off.

• rounding rules are used to obtain the correct number of significant figures.

Copyright © 2005 by Pearson Education, Inc.Publishing as Benjamin Cummings

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Rounding Off Calculated Answers

When the first digit dropped is 4 or less, • the retained numbers remain the same.

45.832 rounded to 3 significant figures

drops the digits 32 = 45.8

When the first digit dropped is 5 or greater, • the last retained digit is increased by 1.

2.4884 rounded to 2 significant figures

drops the digits 884 = 2.5 (increase by 0.1)

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Adding Significant Zeros

• Sometimes a calculated answer requires more significant digits. Then, one or more zeros are added.

Calculated Zeros Added to Answer Give 3 Significant Figures

4 4.001.5 1.500.2 0.200

12 12.0

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Learning Check

Adjust the following calculated answers to giveanswers with three significant figures.

A. 824.75 cm

B. 0.112486 g

C. 8.2 L

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Solution

Adjust the following calculated answers to give answers

with three significant figures

A. 825 cm First digit dropped is greater than 5.

B. 0.112g First digit dropped is 4.

C. 8.20 L Significant zero is added.

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Calculations with Measured Numbers

In calculations with measured numbers, significant figures ordecimal places arecounted to determinethe number of figures inthe final answer.

Copyright © 2005 by Pearson Education, Inc.Publishing as Benjamin Cummings

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When multiplying or dividing use

• the same number of significant figures as the measurement with the fewest significant figures.

• rounding rules to obtain the correct number of significant figures.

Example:

110.5 x 0.048 = 5.304 = 5.3 (rounded)

4 SF 2 SF calculator 2 SF

Multiplication and Division

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Give an answer for the following with the correct number of significant figures.

A. 2.19 x 4.2 = 1) 9 2) 9.2 3) 9.198

B. 4.311 ÷ 0.07 = 1) 61.59 2) 62 3) 60

C. 2.54 x 0.0028 = 0.0105 x 0.060

1) 11.3 2) 11 3) 0.041

Learning Check

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A. 2.19 x 4.2 = 2) 9.2

B. 4.311 ÷ 0.07 = 3) 60

C. 2.54 x 0.0028 = 2) 11 0.0105 x 0.060

On a calculator, enter each number followed by the operation key.

2.54 x 0.0028 0.0105 0.060 = 11.28888889= 11 (rounded)

Solution

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When adding or subtracting use

• the same number of decimal places as the measurement with the fewest decimal places.

• rounding rules to adjust the number of digits in the answer.

25.2 one decimal place

+ 1.34 two decimal places

26.54 calculated answer

26.5 answer with one decimal place

Addition and Subtraction

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For each calculation, round the answer to give thecorrect number of significant figures.

A. 235.05 + 19.6 + 2 = 1) 257 2) 256.7 3) 256.65

B. 58.925 - 18.2 =1) 40.725 2) 40.73 3) 40.7

Learning Check

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A. 235.05 +19.6 + 2 256.65 rounds to 257 Answer (1)

B. 58.925 -18.2

40.725 round to 40.7 Answer (3)

Solution