1. 1 1.8 Significant Figures Chapter 1 Matter, Measurements, & Calculations Copyright © 2005 by Pearson Education, Inc. Publishing as Benjamin Cummings

2. 2 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

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

4. 4 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.

5. 5. l 8.... l.... l 9.... l.... l 10.. cm What is the length of the orange line? 1) 9.0 cm 2) 9.03 cm 3) 9.04 cm Learning Check

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

7. 7. l 3.... l.... l 4.... l.... 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. Zero as a Measured Number

8. 8 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.

9. 9 Significant Figures

10. 10 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

11. 11 Sandwiched zeros occur between nonzero numbers. are significant. Number of Measurement Significant Figures 50.8 mm 3 2001 min 4 0.0702 lb 3 0.40505 m 5 Sandwiched Zeros

12. 12 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 Trailing Zeros

13. 13 Leading zeros precede non-zero digits in a decimal number. are not significant. Number of Measurement Significant Figures 0.008 mm 1 0.0156 oz 3 0.0042 lb 2 0.000262 mL 3 Leading Zeros

14. 14 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

15. 15 State the number of significant figures in each of the following measurements. A. 0.030 m2 B. 4.050 L4 C. 0.0008 g1 D. 2.80 m3 Solution

16. 16 Significant Figures in Scientific Notation In scientific notation all digits including zeros in the coefficient are significant. Number of Measurement Significant Figures 8 x 10 4 m 1 8.0 x 10 4 m 2 8.00 x 10 4 m 3

17. 17 A. Which answer(s) contain 3 significant figures? 1) 0.4760 2) 0.00476 3) 4.76 x 10 3 B. All the zeros are significant in 1) 0.00307 2) 25.300 3) 2.050 x 10 3 C. The number of significant figures in 5.80 x 10 2 is 1) one3) two3) three Learning Check

18. 18 A. Which answer(s) contain 3 significant figures? 2) 0.00476 3) 4.76 x 10 3 B. All the zeros are significant in 2) 25.300 3) 2.050 x 10 3 C. The number of significant figures in 5.80 x 10 2 is 3) three Solution

19. 19 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

20. 20 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.

21. 21 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

22. 22 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)

23. 23 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 44.00 1.51.50 0.20.200 12 12.0

24. 24 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

25. 25 Solution Adjust the following calculated answers to give answers with three significant figures A. 825 cm First digit dropped is greater than 5. B. 0.112gFirst digit dropped is 4. C. 8.20 LSignificant zero is added.

26. 26 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

27. 27 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

28. 28 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) 11.32) 11 3) 0.041 Learning Check

29. 29 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

30. 30 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 26.54calculated answer 26.5 answer with one decimal place Addition and Subtraction

31. 31 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 3) 256.65 B. 58.925 - 18.2= 1) 40.725 2) 40.73 3) 40.7 Learning Check

32. 32 A. 235.05 +19.6 + 2 256.65 rounds to 257 Answer (1) B. 58.925 -18.2 40.725 round to 40.7Answer (3) Solution