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# Significant Figures

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### Significant Figures

1. 1. Significant Figures► When using our calculators we must determine the correct answer; our calculators are mindless drones and don’t know the correct answer.► There are 2 different types of numbers – Exact – Measured► Exact numbers are infinitely important► Measured number = they are measured with a measuring device (name all 4) so these numbers have ERROR.► When you use your calculator your answer can only be as accurate as your worst measurement…Doohoo  Chapter Two 1
2. 2. Exact NumbersAn exact number is obtained when you count objects or use a defined relationship. Counting objects are always exact 2 soccer balls 4 pizzas Exact relationships, predefined values, not measured 1 foot = 12 inches 1 meter = 100 cm For instance is 1 foot = 12.000000000001 inches? No 1 ft is EXACTLY 12 inches. 2
3. 3. Learning CheckA. Exact numbers are obtained by 1. using a measuring tool 2. counting 3. definitionB. Measured numbers are obtained by 1. using a measuring tool 2. counting 3. definition 3
4. 4. SolutionA. Exact numbers are obtained by 2. counting 3. definitionB. Measured numbers are obtained by 1. using a measuring tool 4
5. 5. Learning CheckClassify each of the following as an exact or ameasured number. 1 yard = 3 feet The diameter of a red blood cell is 6 x 10-4 cm. There are 6 hats on the shelf. Gold melts at 1064°C. 5
6. 6. SolutionClassify each of the following as an exact (1) or ameasured(2) number.This is a defined relationship.A measuring tool is used to determine length.The number of hats is obtained by counting.A measuring tool is required. 6
7. 7. 2.4 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. Chapter Two 7
8. 8. What is the Length? 1 2 3 4 cm►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 8
9. 9. Learning CheckWhat is the length of the wooden stick? 1) 4.5 cm 2) 4.54 cm 3) 4.547 cm
10. 10. ? 8.00 cm or 3 (2.2/8) 10
11. 11. 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. 11
12. 12. Below are two measurements of the mass of thesame object. The same quantity is being describedat two different levels of precision or certainty. Chapter Two 12
13. 13. Note the 4 rules When reading a measured value, all nonzero digits should be counted as significant. There is a set of rules for determining if a zero in a measurement is significant or not. ► RULE 1. Zeros in the middle of a number are like any other digit; they are always significant. Thus, 94.072 g has five significant figures. ► RULE 2. Zeros at the beginning of a number are not significant; they act only to locate the decimal point. Thus, 0.0834 cm has three significant figures, and 0.029 07 mL has four.Chapter Two 13
14. 14. ► RULE 3. Zeros at the end of a number and after the decimal point are significant. It is assumed that these zeros would not be shown unless they were significant. 138.200 m has six significant figures. If the value were known to only four significant figures, we would write 138.2 m.► RULE 4. Zeros at the end of a number and before an implied decimal point may or may not be significant. We cannot tell whether they are part of the measurement or whether they act only to locate the unwritten but implied decimal point. Chapter Two 14
15. 15. Practice Rule #1 Zeros45.8736 6 •All digits count.000239 3 •Leading 0’s don’t.00023900 5 •Trailing 0’s do 5 •0’s count in decimal form48000. 2 •0’s don’t count w/o decimal48000 4 •All digits count3.982×106 6 •0’s between digits count as well1.00040 as trailing in decimal form
16. 16. 2.5 Scientific Notation► Scientific notation is a convenient way to write a very small or a very large number.► Numbers are written as a product of a number between 1 and 10, times the number 10 raised to power.► 215 is written in scientific notation as:215 = 2.15 x 100 = 2.15 x (10 x 10) = 2.15 x 102 Chapter Two 16
17. 17. Two examples of converting standard notation toscientific notation are shown below. Chapter Two 17
18. 18. Two examples of converting scientific notation back tostandard notation are shown below. Chapter Two 18
19. 19. ► Scientific notation is helpful for indicating how many significant figures are present in a number that has zeros at the end but to the left of a decimal point.► The distance from the Earth to the Sun is 150,000,000 km. Written in standard notation this number could have anywhere from 2 to 9 significant figures.► Scientific notation can indicate how many digits are significant. Writing 150,000,000 as 1.5 x 108 indicates 2 and writing it as 1.500 x 108 indicates 4.► Scientific notation can make doing arithmetic easier. Rules for doing arithmetic with numbers written in scientific notation are reviewed in Appendix A. Chapter Two 19
20. 20. 2.6 Rounding Off 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. Chapter Two 20
21. 21. ► 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.► 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. Chapter Two 21
22. 22. Practice Rule #2 Rounding Make the following into a 3 Sig Fig number Your Final number1.5587 1.56 must be of the same.0037421 .00374 value as the number you started with,1367 1370 129,000 and not 129128,522 129,0001.6683 ×106 1.67 ×106
23. 23. Examples of RoundingFor example you want a 4 Sig Fig number4965.03 4965 0 is dropped, it is <5780,582 780,600 8 is dropped, it is >5; Note you must include the 0’s1999.5 2000. 5 is dropped it is = 5; note you need a 4 Sig Fig
24. 24. RULE 1. In carrying out a multiplication or division,the answer cannot have more significant figures thaneither of the original numbers. Chapter Two 24
25. 25. ►RULE 2. In carrying out an addition orsubtraction, the answer cannot have more digitsafter the decimal point than either of theoriginal numbers. Chapter Two 25
26. 26. Multiplication and division32.27 × 1.54 = 49.6958 49.73.68 ÷ .07925 = 46.4353312 46.41.750 × .0342000 = 0.05985 .059853.2650×106 × 4.858 = 1.586137 × 107 1.586 ×1076.022×1023 × 1.661×10-24 = 1.000000 1.000
27. 27. Addition/Subtraction 25.5 32.72 320+34.270 ‑ 0.0049 + 12.5 59.770 32.7151 332.5 59.8 32.72 330
28. 28. Addition and Subtraction.56 + .153 = .713 __ ___ __ .71 Look for the last important82000 + 5.32 = 82005.32 82000 digit10.0 - 9.8742 = .12580 .110 – 9.8742 = .12580 0
29. 29. Mixed Order of Operation8.52 + 4.1586 × 18.73 + 153.2 == 8.52 + 77.89 + 153.2 = 239.61 = 239.6(8.52 + 4.1586) × = 2179.692 = = = 12.68 × 171.9 (18.73 + 153.2) 2180.
30. 30. TryFind the standard deviation for the followingnumbers: 7.691 g, 7.23 g, 7.892 g 1  2  2  ∑ x − x    xi  i i  x=∑ s=   i n  n −1     
31. 31. Try xix=∑ 7.691 g, 7.23 g, 7.892 g i n 7.691 g 22.81 g = 7.603 g 7.23 g 3 7.892 g22.813 g
32. 32. Try 1   2  2  ∑ x − x    = 7.603 g  i i  s=   n −1  7.691 g – 7.603 g = .088 g     7.23 g – 7.603 g = -.37 g 7.892 g – 7.603 g = .289 g .01 g
33. 33. Try .25 1   2  2  ∑ x − x   .01 = .01 g  i i  s=   n −1       ( .01)  2 1/ 2 s=  = .007  2 