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DIFFERENTIATE: ACCURACY AND PRECISION Can you hit the bull's-eye? Three targets with three arrows each to shoot. How do they compare? Both accurate and precise Precise but not accurate Neither accurate nor precise Can you define accuracy and precision? What would a bullseye with accuracy and no precision look like? Significant Figures What is a significant figure? • There are 2 kinds of numbers: APPROXIMATE (OR MEASURED): weight, height, mass, distance— anything MEASURED needs significance attached. No measurement is perfect. EXACT (counted items and conversion factors have unlimited significant digits): the amount of money in your account, jelly beans in a jar, etc. And #m in a km or in in a ft Reading a Meterstick . l2. . . . I . . . . I3 . . . .I . . . . I4. . First digit (known) =2 cm 2.?? cm Second digit (known) = 0.7 2.7? cm Third digit (estimated) between 0.05- 0.07 Length reported = 2.75 cm or 2.74 cm or 2.76 cm Known + Estimated Digits In 2.76 cm… • 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 When to use Significant figures To a mathematician 21.70, or 21.700 is the same. But, to a scientist 21.7cm and 21.70cm is NOT the same When to use Significant figures •When a measurement is recorded only those digits that are dependable are written down. Always estimate ONE place past the smallest mark! Zero as a Measured Number . l 3. . . . I . . . . I 4 . . . . I . . . . I 5. . What is the length of the line? First digit Second digit Last (estimated) digit is cm 5.?? cm 5.0? cm 5.00 cm Enter question text... . l8. . . . I . . . . I9. . . .I . . . . I10. . cm What is the length of the line? 1) 9.6 cm 2) 9.62 cm 3) 9.63 cm 0% 3 1 How does your answer compare with your neighbor’s answer? Why or why not? 0% 2 0% Significant Figures 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 Counting Significant Figures RULE 1. All non-zero digits in a measured number are significant. Only a zero could indicate that rounding occurred. Number of Significant Figures 38.15 cm 5.6 m 65.6 kg 122.55 m 4 2 ___ ___ How many sig figs in 65.8 kg 0% 0% 5 0% 3 0% 2 1 0% 4 1. 1 2. 2 3. 3 4. 4 5. 5 How many sig figs in 122.55 m 0% 0% 5 0% 3 0% 2 1 0% 4 1. 1 2. 2 3. 3 4. 4 5. 5 Leading Zeros RULE 2. Leading zeros in decimal numbers are NOT significant. Number of Significant Figures 0.008 mm 1 0.0156 mg 3 0.0042 kg ____ 0.000262 mL ____ How many sig figs in 0.0042 kg 0% 0% 5 0% 3 0% 2 1 0% 4 1. 1 2. 2 3. 3 4. 4 5. 5 How many sig figs in 0.000262 mL 0% 0% 5 0% 3 0% 2 1 0% 4 1. 1 2. 2 3. 3 4. 4 5. 5 Sandwiched Zeros RULE 3. Zeros between nonzero numbers are significant. (They can not be rounded unless they are on an end of a number.) Number of Significant Figures 50.8 mm 3 2001 min 4 0.702 kg ____ 0.00405 m ____ How many sig figs in 0.702 kg 0% 0% 5 0% 3 0% 2 1 0% 4 1. 1 2. 2 3. 3 4. 4 5. 5 How many sig figs in 0.00405 m 0% 0% 5 0% 3 0% 2 1 0% 4 1. 1 2. 2 3. 3 4. 4 5. 5 Trailing Zeros RULE 4. Trailing zeros in numbers without decimals are NOT significant. They are only serving as place holders. Number of Significant Figures 25,000 cm 2 200. yr 3 48,600 L ____ 25,005,000 g ____ How many sig figs in 48,600 L 0% 0% 5 0% 3 0% 2 1 0% 4 1. 1 2. 2 3. 3 4. 4 5. 5 How many sig figs in 48,600. L 0% 0% 5 0% 3 0% 2 1 0% 4 1. 1 2. 2 3. 3 4. 4 5. 5 How many sig figs in 25,005,000 g 0% 0% 5 0% 3 0% 2 1 0% 4 1. 1 2. 2 3. 3 4. 4 5. 5 Which answers contain 3 significant figures? 1. ) 0.4760 2. ) 0.004706 3. ) 4760 47 60 0% ) 0. 00 47 06 0% ) ) 0. 47 60 0% All the zeros are significant in .00370 25,300 2.050 x 103 0% 2. 05 0 x1 0. .. 0% 25 30 0 0% .0 03 7 1. 2. 3. 534,675 rounded to 3 significant figures is 1. 535 2. 535,000 3. 5.35 x 105 0% 5. 35 x1 05 0% 53 50 00 53 5 0% 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 0% nd 15 a 0. 00 00 an d 40 1. .. 0% 40 0. 0 22 .0 an d 22 .0 ... 0% 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 Adding and Subtracting The answer has the same number of decimal places as the measurement with the fewest decimal places. 25.2 one decimal place + 1.34 two decimal places 26.54 answer 26.5 one decimal place 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 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. Learning Check A. 2.19 X 4.2 = 1) 9 B. C. 2) 9.2 3) 9.198 4.311 ÷ 0.07 = 1) 61.58 2) 62 3) 60 2.54 X 0.0028 = 0.0105 X 0.060 1) 11.3 2) 11 3) 0.041 Learning Check 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 How many sig figs? •7 • 40 • 0.5 • 0.00003 5 • 7 x 10 •1 •1 •1 •1 •1 How many sig figs here? • 1.2 • 2100 • 56.76 • 4.00 • 0.0792 • 7,083,000,00 0 •2 •2 •4 •3 •3 •4 How many sig figs here? • 3401 • 2100 • 2100.0 • 5.00 • 0.00412 • 8,000,050,000 •4 •2 •5 •3 •3 •6 Sig Figs?? 640 0% 0% 6 0% 4 0% 3 0% 2 0% 5 1 2 3 4 5 6 1 a) b) c) d) e) f) Sig Figs?? 200.0 0% 0% 6 0% 4 0% 3 0% 2 0% 5 1 2 3 4 5 6 1 a) b) c) d) e) f) Sig Figs?? 0.5200 0% 0% 6 0% 4 0% 3 0% 2 0% 5 1 2 3 4 5 6 1 a) b) c) d) e) f) Sig Figs?? 1.005 0% 0% 6 0% 4 0% 3 0% 2 0% 5 1 2 3 4 5 6 1 a) b) c) d) e) f) Sig Figs?? 10,000 0% 0% 6 0% 4 0% 3 0% 2 0% 5 1 2 3 4 5 6 1 a) b) c) d) e) f) How many sig figs? 700 700. 700.00 .007 .00700 •1 •3 •5 •1 •3 How many sig figs here? • 1.02 x 2.3 • 210 x 200 • 210. x 210 • .070 x .910 • 0.0791 x 33.1 • 2.3x105 x 200 •2 •1 •2 •2 •3 •1 How many sig figs here? • 23 x 2 x 231 • 455 x 21 x 25.2 • 2100.0 x .0005 • 5.00 x 311.22 • 0.00412 x 9.1 •1 •2 •1 •3 •2 Metric Prefixes