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Chapter 1 Matter, Measurements, & Calculations 1.8 Significant Figures Copyright © 2005 by Pearson Education, Inc. Publishing as Benjamin Cummings 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 © 2005 by Pearson Education, Inc. Publishing as Benjamin Cummings 2 Reading a Meter Stick . 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. 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 . 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 5 Solution . 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. 6 Zero as a Measured Number . 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. 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 mL 3 25 005 000 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 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 including zeros in the coefficient 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 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 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 21 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) 22 Adding Significant Zeros • Sometimes a calculated answer requires more significant digits. Then, one or more zeros are added. Calculated Answer 4 1.5 0.2 12 Zeros Added to Give 3 Significant Figures 4.00 1.50 0.200 12.0 23 Learning Check Adjust the following calculated answers to give answers with three significant figures. A. 824.75 cm B. 0.112486 g C. 8.2 L 24 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. 25 Calculations with Measured Numbers In calculations with measured numbers, significant figures or decimal places are counted to determine the number of figures in the final answer. Copyright © 2005 by Pearson Education, Inc. Publishing as Benjamin Cummings 26 Multiplication and Division 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 4 SF x 0.048 = 5.304 2 SF = calculator 5.3 (rounded) 2 SF 27 Learning Check 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 28 Solution A. 2.19 x 4.2 B. 4.311 ÷ 0.07 C. 2.54 x 0.0028 0.0105 x 0.060 = 2) 9.2 = 3) 60 = 2) 11 On a calculator, enter each number followed by the operation key. 2.54 x 0.0028 0.0105 0.060 = 11.28888889 = 11 (rounded) 29 Addition and Subtraction 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 + 1.34 26.54 26.5 one decimal place two decimal places calculated answer answer with one decimal place 30 Learning Check For each calculation, round the answer to give the correct number of significant figures. A. 235.05 + 19.6 + 2 = 1) 257 2) 256.7 B. 58.925 - 18.2 = 1) 40.725 2) 40.73 3) 256.65 3) 40.7 31 Solution A. 235.05 +19.6 + 2 256.65 rounds to 257 B. 58.925 -18.2 40.725 round to 40.7 Answer (1) Answer (3) 32