Chapter 1 Measurements

1. Chapter 1Measurements Measured Numbers and Significant Figures

2. Measured Numbers When you use a measuring tool is used to determine a quantity such as your height or weight, the numbers you obtain are called measured numbers.

3. Reading a Meterstick . l2. . . . I . . . . I3 . . . .I . . . . I4. . cm First digit (known) = 2 2.?? cm Second digit (known) = 0.7 2.7?cm Third digit (estimated)between 0.05- 0.07 Length reported = 2.75 cm or 2.76 cm or 2.77 cm

4. Known + Estimated Digits • Known digits 2 and 7 are 100% certain • The third digit 6 is estimated (uncertain) • In the reported length, all three digits (2.76 cm) are significant including the estimated one

5. Learning Check . l8. . . . I . . . . I9. . . .I . . . . I10. . cm What is the length of the line? 1) 9.2 cm 2) 9.22 cm 3) 9.23 cm How your answer compare with your neighbor’s answer? Why or why not?

6. Solution . l8. . . . I . . . . I9. . . .I . . . . I10. . cm Estimate to the hundredth's place (0.01 cm) 2) 9.22 cm 3) 9.23 cm The estimated digit may be slightly different. Both readings are acceptable.

7. Learning Check l5. . . . I . . . . I6. . . .I . . . . I7. . cm What is the length of the line? 1) 6.0 cm 2) 6.06 cm 3) 6.60 cm

8. Solution l5. . . . I . . . . I6. . . .I . . . . I7. . cm Estimate to the hundredth's place (0.01 cm). The reading is between 6.0 and 6.1. You may have estimated 6.07 cm as well. 2) 6.06 cm

9. Zero as a Measured Number . l3. . . . I . . . . I4 . . . . I . . . . I5. . cm What is the length of the line? First digit 4.?? cm Second digit 4.5? cm Last (estimated) digit is 4.50 cm (not to the left or right of .5)

10. Exact Numbers • Obtained when you count objects 2 soccer balls 1 watch 4 pizzas • Obtained from a defined relationship 1 foot = 12 inches 1 meters = 100 cm • Not obtained with measuring tools

11. Learning Check A. Exact numbers are obtained by 1. measuring 2. counting 3. definition B. Measured numbers are obtained by 1. measuring 2. counting 3. definition

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

13. Learning Check Classify each of the following as an exact (1) or a measured (2) number. A.___Gold melts at 1064°C B.___1 yard = 3 feet C.___A red blood cell with diameter 6 x 10-4 cm D.___There were 6 hats on the shelf E.___A can of soda contains 355 mL of soda

14. Solution Classify each of the following as an exact (1) or a measured(2) number. Give reason. A. 2 Requires a thermometer(measuring tool) B. 1 From a definition in U.S. system C. 2 Need measuring tool to determine D. 1 Counted the hats E. 2 Measured

15. Significant Figures in Measurement • The numbers reported in a measurement are limited by the measuring tool • Significant figures in a measurement include the known digits plus one estimated digit

16. Counting Significant Figures Number of Significant Figures 38.15 cm 4 5.6 ft 2 65.6 lb ___ 122.55 m ___ Complete: All non-zero digits in a measured number are (significant or not significant).

17. Leading Zeros Number of Significant Figures 0.008 mm 1 0.0156 oz 3 0.0042 lb ____ 0.000262 mL ____ Complete: Leading zeros in decimal numbers are (significant or not significant).

18. Sandwiched Zeros Number of Significant Figures 50.8 mm 3 2001 min 4 0.702 lb ____ 0.00405 m ____ Complete: Zeros between nonzero numbers are (significant or not significant).

19. Trailing Zeros Number of Significant Figures25,000 in. 2 200 yr 1 48,600 gal 3 25,005,000 g ____ Complete: Trailing zeros in numbers without decimals are (significant or not significant) if they are serving as place holders.

20. Learning Check A. Which answers contain 3 significant figures? 1) 0.4760 2) 0.00476 3) 4760 B. All the zeros are significant in 1) 0.00307 2) 25.300 3) 2.050 x 103 C. 534,675 rounded to 3 significant figures is 1) 535 2) 535,000 3) 5.35 x 105

21. Solution A. Which answers contain 3 significant figures? 2) 0.00476 3) 4760 B. All the zeros are significant in 2) 25.300 3) 2.050 x 103 C. 534,675 rounded to 3 significant figures is 2) 535,000 3) 5.35 x 105

22. 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

23. Solution In which set(s) do both numbers contain the same number of significant figures? 3) 0.000015 and 150,000

24. Learning Check SF3 State the number of significant figures in each of the following: A. 0.030 m 1 2 3 B. 4.050 L 2 3 4 C. 0.0008 g 1 2 4 D. 3.00 m 1 2 3 E. 2,080,000 bees 3 5 7

25. Solution A. 0.030 m 2 B. 4.050 L 4 C. 0.00008 g 1 D. 3.00 m 3 E. 2,080,000 bees 3

26. Significant Numbers in Calculations • A calculated answer cannot be more precise than the measuring tool. • A calculated answer must match the least precise measurement. • Significant figures are needed for final answers from 1) adding or subtracting 2) multiplying or dividing

27. Adding and Subtracting The answer has the same number of decimal places as the measurement with the fewest decimal places. 25.2one decimal place + 1.34two decimal places 26.54 answer 26.5 one decimal place

28. Learning Check In each calculation, round the answer to the correct number of significant figures. A. 235.05 + 19.6 + 2.1 = 1) 256.75 2) 256.8 3) 257 B. 58.925 - 18.2 = 1) 40.725 2) 40.73 3) 40.7

29. Solution A. 235.05 + 19.6 + 2.1 = 2) 256.8 B. 58.925 - 18.2 = 3) 40.7

30. Multiplying and Dividing Round (or add zeros) to the calculated answer until you have the same number of significant figures as the measurement with the fewest significant figures.

31. Learning Check A. 2.19 X 4.2 = 1) 9 2) 9.2 3) 9.198 B. 4.311 ÷ 0.07 = 1)61.582) 62 3) 60 C. 2.54 X 0.0028 = 0.0105 X 0.060 1) 11.3 2) 11 3) 0.041

32. Solution 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 Continuous calculator operation = 2.54 x 0.0028  0.0105  0.060