Chapter 1 Measurements 1 2 Measured Numbers and

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 2

Reading a Meter Stick. l 2. . . . l 3. . . . l 4. . 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. 3

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. 4

Learning Check. l 8. . . . l 9. . . . l 10. . cm What is the length of the orange line? 1) 9. 0 cm 2) 9. 03 cm 3) 9. 04 cm 5

Solution. l 8. . . . l 9. . . . l 10. . 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. 6

Zero as a Measured Number. l 3. . . . l 4. . . . l 5. . 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. 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 non-zero 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. 40505 m Number of Significant Figures 3 4 3 5 11

Trailing Zeros 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 m. L 3 25 000 g 5 12

Leading Zeros Leading zeros • precede non-zero digits in a decimal number. • are not significant. Measurement 0. 008 mm 0. 0156 oz 0. 0042 lb 0. 000262 m. L Number of Significant Figures 1 3 2 3 13

Learning Check State the number of significant figures in each of the following measurements. A. B. C. D. 0. 030 m 4. 050 L 0. 0008 g 2. 80 m 14

Solution State the number of significant figures in each of the following measurements. A. B. C. D. 0. 030 m 4. 050 L 0. 0008 g 2. 80 m 2 4 1 3 15

Significant Figures in Scientific Notation In scientific notation all digits including zeros in the coefficient are significant. Measurement 8 x 104 m 8. 00 x 104 m Number of Significant Figures 1 2 3 16

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 3) two 3) three 17

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 18

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. 000015 and 150 000 19

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. 20

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 21

Exact Numbers 22

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. 23

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

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 m. L of soda. 25

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. 26