1. 1 INTRODUCTION IV. Significant Figures

2. A. Purpose of Sig Figs Units of Measurement: Measurements indicate the magnitude of something Must include: –A number –A unit

3. A. Purpose of Significant Figures There are 2 different types of numbers in our world –Exact numbers (counting numbers) –Measured numbers Measured numbers are measured with a tools; these numbers have ERRORS and are not exact. Errors (called the degree of uncertainty) are a result of: –Human error reading the instrument (not intentional error) –Limitation in precision of instruments 3

4. 4 1. Exact Numbers An 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.

5. 5 Learning Check A. Exact numbers are obtained by 1. using a measuring tool 2. counting 3. definition B. Measured numbers are obtained by 1. using a measuring tool 2. counting 3. definition

6. 6 Solution A. Exact numbers are obtained by 2. counting 3. definition B. Measured numbers are obtained by 1. using a measuring tool

7. 7 Learning Check Classify each of the following as an exact or a measured number. 1. 1 yard = 3 feet 2. The diameter of a red blood cell is 7.8 um. 3. There are 6 hats on the shelf. 3. Gold melts at 1064°C.

8. 8 Classify each of the following as an exact (1) or a measured(2) number. 1. This is a defined relationship. 2. A measuring tool is used to determine length. 3. The number of hats is obtained by counting. 4. A measuring tool is required. Solution

9. 2. Measurement and Significant Figures Every measurement has a degree of uncertainty. The volume, V, at right is certain in the 1’s place: 17mL<V<18mL A best guess is needed for the tenths place. Two people may estimate to the tenth differently 9

10. 10 We can see the measurement is between 1 and 2 cm We can’t see the markings between We must guess between 1 and 2 I estimate the measurement to be 1.6 cm The last digit in a measurement is an estimate You may estimate differently

11. 11 What is the Length? We can see the measurement is between 1.6- 1.7cm We can’t see the markings between 1.6-1.7 We must guess between.6 &.7 I estimate the measurement to be 1.67 cm The last digit in a measurement is an estimate You cannot make a more precise estimate than to the hundreth using this ruler

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

13. 13 ?

14. Measured Numbers Measurements have a degree of uncertainty that results from –_________________ Significant figures are the numbers represented in a measurement. “Sig Figs” include: –All of the numbers recorded are known with certainty –One estimated digit. To indicate the precision of a measurement, the value recorded should use all the digits known with certainty. 14

15. 15 The mass of an object is measured using two different scales. The same quantity is being described at two different levels of precision or certainty.

16. B. Using Significant Figures Timberlake lecture plus16

17. 17 1. Counting Significant Figures Non Zero Numbers Number of Significant Figures 38.15 cm4 5.6 ft2 65.6 lb___ 122.55 m___ All non-zero digits in a measured number are significant

18. 18 Leading Zeros Number of Significant Figures 0.008 mm1 0.0156 oz3 0.0042 lb____ 0.000262 mL ____ Leading zeros in decimal numbers (less than 1) are not significant.

19. 19 Sandwiched Zeros Number of Significant Figures 50.8 mm3 2001 min4 0.702 lb____ 0.00405 m ____ Zeros between nonzero numbers are significant Zeros between a non zero and followed by a decimal are significant

20. 20 Trailing Zeros Number of Significant Figures 25,000 in. 2 200 yr1 48,600 gal3 25,005,000 g ____ Trailing zeros in numbers without decimals are not significant They are significant if followed by or preceded by a decimal

21. 21 Learning Check A. Which answers contain 3 significant figures? 1) 0.4760 2) 0.00476 3) 4760 B. All the zeros are significant in 1) 0.00307 2) 25.300 3) 2.050 x 10 3 C. 534,675 rounded to 3 significant figures is 1) 535 2) 535,000 3) 5.35 x 10 5

22. 22 Solution A. Which answers contain 3 significant figures? 2) 0.00476 3) 4760 B. All the zeros are significant in 2) 25.300 3) 2.050 x 10 3 C. 534,675 rounded to 3 significant figures is 2) 535,000 3) 5.35 x 10 5

23. 23 Learning Check 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

24. 24 Solution In which set(s) do both numbers contain the same number of significant figures? 3) 0.000015 and 150,000

25. 25 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 Learning Check

26. 26 A. 0.030 m2 B. 4.050 L4 C. 0.00008 g1 D. 3.00 m 3 E. 2,080,000 bees3 Solution

27. 27 2. 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

28. 28 a. Adding and Subtracting The answer has the same number of decimal places as the measurement with the fewest decimal places. The uncertain digit must reflect precision of instruments. 25.2 one decimal place + 1.34 two decimal places 26.54 answer 26.5 one decimal place

29. 29 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) 256.83) 257 B. 58.925 - 18.2= 1) 40.725 2) 40.733) 40.7

30. 30 Solution A. 235.05 + 19.6 + 2.1 = 2) 256.8 B. 58.925 - 18.2= 3) 40.7

31. 31 b. Multiplying and Dividing In carrying out a multiplication or division, the answer cannot have more significant figures than either of the original numbers.

32. 32 Learning Check A. 2.19 X 4.2 = 1) 9 2) 9.2 3) 9.198 B. 4.311 ÷ 0.07 = 1) 61.58 2) 62 3) 60 C. 2.54 X 0.0028 = 0.0105 X 0.060 1) 11.32) 11 3) 0.041

33. 33 Solution A. 2.19 X 4.2 = 2) 9.2 B. 4.311 ÷ 0.07 = 3) 60 C.2.54 X 0.0028 = 2) 11 0.0105 X 0.060 Continuous calculator operation = 2.54 x 0.0028  0.0105  0.060

34. Review 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. 34

35. 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. 35

36. Practice Rule #1 Zeros 45.8736 0.000239 0.00023900 48000. 48000 3.982  10 6 1.00040 6 3 5 5 2 4 6 All digits count Leading 0’s don’t Trailing 0’s do 0’s count in decimal form 0’s don’t count w/o decimal All digits count 0’s between digits count as well as trailing in decimal form

37. Practice Rounding Make the following into a 3 Sig Fig number 1.5587 0.0037421 1367 128,522 1.6683  10 6 Your Final number must be of the same value as the number you started with, 129,000 and not 129

38. Practice Rounding Make the following into a 3 Sig Fig number 1.5587 0.0037421 1367 128,522 1.6683  10 6 1.56 0.00374 1370 129,000 1.67  10 6 Your Final number must be of the same value as the number you started with, 129,000 and not 129

39. Examples of Rounding For example you want a 4 Sig Fig number 4965.03 780,582 1999.5

40. Examples of Rounding For example you want a 4 Sig Fig number 4965.03 780,582 1999.5 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 780,600 2000.

41. Multiplication and division 32.27  1.54 = 3.68 .07925 = 1.750 .0342000 = 3.2650  10 6  4.858 = 6.022  10 23  1.661  10 -24 =

42. Multiplication and division 32.27  1.54 = 49.6958 3.68 .07925 = 46.4353312 1.750 .0342000 = 0.05985 3.2650  10 6  4.858 = 1.586137  10 7 6.022  10 23  1.661  10 -24 = 1.000000 49.7 46.4.05985 1.586  10 7 1.000

43. __ ___ __ Addition and Subtraction.56 +.153 = 82000 + 5.32 = 10.0 - 9.8742 = 10 – 9.8742 = Look for the last important digit

44. __ ___ __ Addition and Subtraction.56 +.153 =.713 82000 + 5.32 = 82005.32 10.0 - 9.8742 =.12580 10 – 9.8742 =.12580.71 82000.1 0 Look for the last important digit