Significant Digits (or Significant Figures)

1. Significant Digits (or Significant Figures) Jim Pulickeel SPH 3U1 September 2009

2. What’s the point of Significant Digits? • The radius of a circle is 6.2 cm. What is the Area? • But pi is.... • 3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067982148086513282306647093844609550582231725359408128481117450284102701938521105559644622948954930381964428810975665933446128475648233786783165271201909145648566923460348610454326648213393607260249141273724587006606315588174881520920962829254091715364367892590360011330530548820466521384146951941511609 • Which answer is more accurate?

3. The original measurement of the radius was far too crude to produce a result of that precision. The real area of the circle could easily be as high as 128, or as low as 113 • Sig Figs are a way of describing the precision of a measurement. 6.2 cm has two digits of precision - two significant digits. Anything beyond that is in the range of round off errors - further digits are artefacts of the calculation, which shouldn't be treated as meaningful.

4. What time is it? • Someone might say “1:30” or “1:28” or “1:27:55” • Each is appropriate for a different situation • In science we describe a value as having a certain number of “significant digits” • The # of significant digits in a value includes all digits that are certain and one that is uncertain • “1:30” likely has 2, 1:28 has 3, 1:27:55 has 5

5. Significant Errors with Significant Digits • The Hubble Space Telescope has a lens, one of the most precisely figured mirror ever made, but it was too flat at the edges by about 2.2 microns (10-6m) because of rounding and conversion errors during the grinding process. • During the Iraq War I (1991), 24% of American friendly fire deaths were attributed to rounding errors in missile launching software.

6. What are Significant Digits? • Significant Digits indicate the precision of a measurement • When recording Sig. Figs. we include the known (certain) digits plus a final estimated digit • What can we say about the following paper clip?

7. What is the length of this thing?

11. What About When Numbers are Given to You? RULES FOR SIGNIFICANT DIGITS

13. Rounding...

15. Significant Digits • It is better to represent 100 as 1.00 x 102 • Alternatively you can underline the position of the last significant digit. E.g. 100. • This is especially useful when doing a long calculation or for recording experimental results • Don’t round your answer until the last step in a calculation.

16. Adding with Significant Digits • Howfaris it fromToronto to room 229? To 225? • Adding a value that is much smaller than the last sig. digit of another value is irrelevant • When adding or subtracting, the # of sig. digits is determined by the sig. digit furthest to the left when #s are aligned according to their decimal.

17. Adding with Significant Digits • Howfaris it fromToronto to room 229? To 225? • Adding a value that is much smaller than the last sig. digit of another value is irrelevant • When adding or subtracting, the # of sig. digits is determined by the sig. digit furthest to the left when #s are aligned according to their decimal. • E.g. a) 13.64 + 0.075 + 67 b) 267.8 – 9.36 13.64 267.8 + 0.075 – 9.36 + 67. 80.715 81 258.44

18. i) ii) iii) 83.25 4.02 0.2983 – 0.1075 + 0.001 + 1.52 Sample Questions 83.14 4.02 1.82

19. Multiplication and Division • Determining sig. digits for questions involving multiplication and division is slightly different • For these problems, your answer will have the same number of significant digits as the value with the fewest number of significant digits. • E.g. a) 608.3 x 3.45 b) 4.8  392 a) 3.45 has 3 sig. digits, so the answer will as well 608.3 x 3.45 = 2098.635 = 2.10 x 103 b) 4.8 has 2 sig. digits, so the answer will as well 4.8  392 = 0.012245 = 0.012 or 1.2 x 10–2

20. Sample Problems

21. i) 7.255  81.334 = 0.08920 ii) 1.142 x 0.002 = 0.002 iii) 31.22 x 9.8 = 3.1 x 102 iv) 6.12 x 3.734 + 16.1  2.3 22.85208 + 7.0 = 29.9 vii) 1700 v) 0.0030 + 0.02 = 0.02 + 134000 vi) 33.4 135700 =1.36 x105 + 112.7 + 0.032 146.132  6.487 = 22.5268 = 22.53 Note: 146.1  6.487 = 22.522 = 22.52

22. Unit conversions • Sometimes it is more convenient to express a value in different units. • When units change, basically the number of significant digits does not. E.g. 1.23 m = 123 cm = 1230 mm = 0.00123 km • Notice that these all have 3 significant digits • This should make sense mathematically since you are multiplying or dividing by a term that has an infinite number of significant digits E.g. 123 cm x 10 mm / cm = 1230 mm

23. i) 1.0 cm = 0.010 m ii) 0.0390 kg = 39.0 g iii) 1.7 m = 1700 mm or 1.7 x 103 mm