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Chapter 1 Measurements 1.3 Measured Numbers and Significant Figures Copyright © 2009 by Pearson Education, Inc. 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. Copyright © 2009 by Pearson Education, Inc. 2 Reading a Meterstick . 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. 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 readings are acceptable. 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 Leading Zeros Leading Zeros • 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