Chapter 3: Measurement: Sigfigs

1. Chapter 3: Measurement: Sigfigs

2. Objectives • Determine the number of significant figures in a measurement and in a calculated answer

3. Olympic 100 Meter Results ‘48-60 • 1948 Time • GOLD Harrison Dillard, USA 10.3 • SILVER Barney Ewell, USA 10.4 • BRONZE Lloyd LaBeach, PAN 10.4 • 1952 • GOLD Lindy Remigino, USA 10.79 • SILVER Herbert McKenley, JAM 10.80 • BRONZE Emmanuel Bailey, GBR 10.83 • 1956 • GOLD Bobby Morrow, USA 10.62 • SILVER Thane Baker, USA 10.77 • BRONZE Hector Hogan, AUS 10.77 • 1960 • GOLD Armin Hary, GER 10.32 • SILVER David Sime, USA 10.35 • BRONZE Peter Radford, GBR 10.42

4. Uncertainty in measurements • All measurements are limited by the equipment and/or people performing the measurement • Data must express the uncertainty caused by how it was obtained/measured • First, we must know how to properly measure using the tools given to us. • Second, we must understand significant figures in order to properly express the uncertainty to others

5. Proper measurement • Always measure one more digit than your tool allows • The last digit in a measurement is always uncertain

6. Sigfig rules: counting • Nonzero integers always count as a sigfig • 47.9 has 3 sigfigs • Leading zeros do not count as sigfigs • 0.076 has 2 sigfigs • Captive zeros count as sigfigs • 1703 has 4 sigfigs • 1.0008 has 5 sigfigs • Trailing zeros only count if they come after a decimal • 9300 has 2 sigfigs • 74.00 has 4 sigfigs • 2.6180 has 5 sigfigs

7. Sigfig rules: exact numbers • Sometimes, there are numbers which are considered exact numbers • Example: 2.54 cm = 1 in, exactly • Exact numbers have an infinite number of sigfigs • These include • counts (counting 28 people in the room for example) • exactly defined ratios such as the example above or something like 60 min = 1 hr, 100 cm = 1 m, 5280 ft = 1 mi

8. Counting practice How many significant figures in each of the following? 1.0070 m  5 sig figs 17.10 kg  4 sig figs 5 sig figs 100,890 L  3.29 x 103 s  3 sig figs 0.0054 cm  2 sig figs 3,200,000  2 sig figs

9. Sigfig rules: multiplication/division • The number of sigfigs allowed in the result is the same as the least precise measurement (the one with the least sigfigs) in the calculation. • Example: 2.0 x 6.38 = 12.76 (4 sigfigs) • 2.0 has 2 sigfigs • 6.38 has 3 sigfigs • The least precise measurement has 2 sigfigs so the answer must be rounded to 2 sigfigs • The correct answer is 13 (2 sigfigs)

10. Multiplication/division practice Calculation Calculator says: Answer 22.68 m2 3.24 m x 7.0 m 23 m2 100.0 g ÷ 23.7 cm3 4.22 g/cm3 4.219409283 g/cm3 0.02 cm x 2.371 cm 0.05 cm2 0.04742 cm2 710 m ÷ 3.0 s 236.6666667 m/s 240 m/s 5870 lb·ft 1818.2 lb x 3.23 ft 5872.786 lb·ft 0.358885017 g/mL 0.359 g/mL 1.030 g ÷ 2.87 mL

11. Sigfig rules: addition/subtraction • The number of decimal places in the result equals the same number as the measurement with the fewest decimal places. • If there are no decimal places, then go with the measurement with the fewest place values extending right. • Example: 6.8 + 11.934 = 18.734 • 6.8 has 1 decimal place • 11.934 has 3 decimal places • The measurement with the fewest decimal places has 1, so the answer must be rounded to 1 decimal • The correct answer is 18.7 (1 decimal place)

12. Addition/subtraction practice Calculation Calculator says: Answer 10.24 m 3.24 m + 7.0 m 10.2 m 100.0 g - 23.73 g 76.3 g 76.27 g 0.02 cm + 2.371 cm 2.39 cm 2.391 cm 710 L - 3.872 L 706.128 L 710 L 1821 lb 1818 lb + 3.37 lb 1821.37 lb 0.160 mL 0.16 mL 2.030 mL - 1.870 mL