Significant Figures

1. Significant Figures The Rules of the Game

2. Over hundreds of years ago, physicists and other scientists developed a traditional way of expressing their observations. • International System of Units (SI) – metric system.

3. Precision • The amount of information that the measuring instrument can provide. • The precision of a measuring tool depends on how readable the tool is. • Ruler example • 14.5 m is all you can read off of a cm ruler • To say that the line was 14.5678 would give the impression that a very precise tool had been used not a ruler.

4. A few notes • When measuring with a mechanical instrument (ruler, triple beam balance etc), record all the digits that are marked on the instrument’s scale and estimate (and only one) more digit. • When measuring with an electronic instrument, record all the digits on the readout. Consider the last digit to be approximate. • Round calculated answers only once, at the end of the calculation, so that the number of significant digits reflects the precision of the original measurements.

5. Measurement and Significant Figures • Every experimental measurement has a degree of uncertainty. • The volume, V, at right is certain in the 10’s place, 10mL<V<20mL • The 1’s digit is also certain, 17mL<V<18mL • A best guess is needed for the tenths place.

6. What is the Length? • We can see the markings between 1.6-1.7cm • We can’t see the markings between the .6-.7 • We must guess between .6 & .7 • We record 1.67 cm as our measurement • The last digit an 7 was our guess...stop there

7. Learning Check What is the length of the wooden stick? 1) 4.5 cm 2) 4.54 cm 3) 4.547 cm

8. ? 8.00 cm

9. Measured Numbers • Do you see why Measured Numbers have error…you have to make that Guess! • All but one of the significant figures are known with certainty. The last significant figure is only the best possible estimate. • To indicate the precision of a measurement, the value recorded should use all the digits known with certainty.

10. Below are two measurements of the mass of the same object. The same quantity is being described at two different levels of precision or certainty.

11. Significant Figures: RULES 1) All non- zero digits are significant Ex. (34 cm --- 2 sig fig) 2) Zeros in zero sandwiches are significant Ex. (2009 --- 4 sig fig) 3) In order for zeros at the end of a measurement to be significant, there must be a decimal point. Ex. (23.000 -- 5 sig fig but 2300 ---2 sig fig) 4) Zeros to the left of the first non-zero are NOT significant. They are just place holders. Ex. (0.00124 ---3 sig fig)

13. Sig. Figs. • When we are using measurements in different calculations in Chemistry and Physics and even Biology, we need to account for the level of precision. To do so, scientists use significant figures. • The last sig fig in a measurement is always the doubtful digit. But that is not always clear. For example • 12340cm or 12340.cm or 12340.0cm

14. QUESTIONS • If Jenn measures a line to be 12.0 cm, what number is she doubtful about and how many sig. figs. are there? • If Darren measures a mass to be 1300 g, what number is he doubtful about and how many sig. figs. are there?

15. Scientific Notation • Scientists use scientific notation as a method to show the proper amount of sig. figs. • 300 km can be written as 3 x 102 km. (102 is 100 – so 3 x 100 =300) • 300 km (with 2 sig. figs) can be written as 3.0 x 102 km. Notice we clearly have 2 sig. figs.

16. Scientific Notation • When multiplying by 10+x we can move the decimal x times to make the number bigger. Ex. 4 x 105 N = 400000 N Ex. 3.00 x 104 kg = 30000 kg (3 sig. figs – shown in sci. notation)

17. Scientific Notation • When multiplying by 10-x, we can move the decimal x times to make the number smaller. • 6.01 x 10-2 m = 0.0601 m (3 sig. figs) • 3.10 x 10-5 N = 0.0000310 N (3 sig. figs.)

18. Rounding Numbers • Often when doing arithmetic on a pocket calculator, the answer is displayed with more significant figures than are really justified. • How do you decide how many digits to keep? • Simple rules exist to tell you how.

19. Once you decide how many digits to retain, the rules for rounding off numbers are straightforward: • RULE 1. If the first digit you remove is 4 or less, drop it and all following digits. 2.4271 becomes 2.4 when rounded off to two significant figures because the first dropped digit (a 2) is 4 or less.

20. RULE 2. If the first digit removed is 5 or greater, round up by adding 1 to the last digit kept. 4.5832 is 4.6 when rounded off to 2 significant figures since the first dropped digit (an 8) is 5 or greater. • If a calculation has several steps, it is best to round off at the end.

21. RULE 1. In carrying out a multiplication or division, the answer cannot have more significant figures than either of the original numbers.

22. RULE 2. In carrying out an addition or subtraction, the answer cannot have more digits after the decimal point than either of the original numbers.

23. Practice Rule #2 Rounding Make the following into a 3 Sig Fig number Your Final number must be of the same value as the number you started with, 129,000 and not 129 1.5587 .0037421 1367 128,522 1.6683 106 1.56 .00374 1370 129,000 1.67 106

24. Examples of Rounding For example you want a 4 Sig Fig number 0 is dropped, it is <5 8 is dropped, it is >5; Note you must include the 0’s 5 is dropped it is = 5; note you need a 4 Sig Fig 4965.03 780,582 1999.5 4965 780,600 2000.

25. Addition/Subtraction 25.5 32.72 320 +34.270‑ 0.0049+ 12.5 59.770 32.7151 332.5 59.8 32.72 330

26. Addition and Subtraction Look for the last important digit .56 + .153 = .713 82000 + 5.32 = 82005.32 10.0 - 9.8742 = .12580 10 – 9.8742 = .12580 .71 82000 .1 0

27. Multiplication and division 49.7 46.4 .05985 1.586 107 1.000 32.27  1.54 = 49.6958 3.68  .07925 = 46.4353312 1.750  .0342000 = 0.05985 3.2650106 4.858 = 1.586137  107 6.0221023 1.66110-24= 1.000000