1. The Return of GUSS Featuring Significant Digits

2. A Justification for “Sig Digs” Measurements are not perfect.

3. A Justification for “Sig Digs” Measurements are not perfect. They always include some degree of uncertainty because no measuring device is perfect. Each is limited in its precision.

4. A Justification for “Sig Digs” Measurements are not perfect. They always include some degree of uncertainty because no measuring device is perfect. Each is limited in its precision. Note that we are not talking about human errors here.

5. Precision We indicate the precision to which we measured our quantity in how we write our measurement.

6. Precision For example, which measurement is more precise? 15 cm 15 cm 15.0 cm 15.0 cm

7. Precision We indicate the precision to which we measured our quantity in how we write our measurement. For example, which measurement is more precise? 15 cm 15 cm 15.0 cm  This one, obviously. 15.0 cm  This one, obviously. Scientists wouldn’t bother to write the.0 if they didn’t mean it.

8. What we mean When we write 15 cm, we mean that we’ve measured the quantity to be closer to 15 cm than to 14 cm or 16 cm BUT When we write 15.0 cm, we mean that we’ve measured the quantity to be closer to 15 cm than to 14.9 cm or 15.1 cm.

9. Significant Digits In any measurement the significant digits are the digits that we’ve measured: the digits we know for certain plus the single last digit that is estimated or uncertain.

10. For Example The measurement 21.6 cm has three sig digs

11. For Example The measurement 21.6 cm has three sig digs, and the “6” is estimated or uncertain, by which we mean that the measurement is closer to 21.6 cm than to 21.5 or 21.7 cm, but may actually be 21.58 cm or 21.62 cm if measured more precisely.

12. The following rules are used to determine if a digit is significant: All non-zero digits are significant All non-zero digits are significant e.g. 42.5 N has three significant digits

13. The following rules are used to determine if a digit is significant: All non-zero digits are significant All non-zero digits are significant Any zeroes placed after other digits and behind a decimal are significant Any zeroes placed after other digits and behind a decimal are significant e.g. 0.50 kg has two significant digits

14. The following rules are used to determine if a digit is significant: All non-zero digits are significant All non-zero digits are significant Any zeroes placed after other digits and behind a decimal are significant Any zeroes placed after other digits and behind a decimal are significant Any zeroes placed between significant digits are significant Any zeroes placed between significant digits are significant e.g. 30.07 m has four significant digits

15. The following rules are used to determine if a digit is significant: All non-zero digits are significant All non-zero digits are significant Any zeroes placed after other digits and behind a decimal are significant Any zeroes placed after other digits and behind a decimal are significant Any zeroes placed between significant digits are significant Any zeroes placed between significant digits are significant All other zeroes are not significant All other zeroes are not significant e.g. both 100 cm and 0.004 kg each have only one significant digit

16. How can you say those digits are not significant? Both 100 cm and 0.004 kg each have only one sig dig? The zeros here are placeholders – they’re just there to show in which place the non-zeros belong. If the measurements are rewritten 1 m and 4 g, it becomes apparent that there’s only one sig dig.

17. How can you say those digits are not significant? Both 100 cm and 0.004 kg each have only one sig dig? The zeros here are placeholders – they’re just there to show in which place the non-zeros belong. If the measurements are rewritten 1 m and 4 g, it becomes apparent that there’s only one sig dig. But what if you measured 100 cm exactly?

18. Making Zeros Significant But what if you measured 100 cm exactly? You can show that a zero is significant by either: underscoring or overscoring the zero: 100 cm underscoring or overscoring the zero: 100 cm (if the measurement is in a table) rewriting the measurement in scientific notation: rewriting the measurement in scientific notation: 1.00 x 10 2 cm

19. Making Zeros Significant And yes, if you measure a zero, you must write it. Your lab tables should not look like this: Time (s) Distance (cm) 0.15.4 0.28.7 0.310 0.413.1

20. Making Zeros Significant They should look like this: Time (s) Distance (cm) 0.15.4 0.28.7 0.310.0 0.413.1

21. No ½ measurements Your tables also should not look like this: Time (s) Distance (cm) 0.112 0.214 0.316.5 0.418

22. No ½ measurements If you can clearly measure.5 in one case, surely you could measure to the tenths place in the other cases too? We don’t use.5 to substitute for “about ½”:.5 means closer to.5 than to.4 or.6. Be exact. Time (s) Distance (cm) 0.112 0.214 0.316.5 0.418

23. And now for some practice....

24. How many significant digits are there in each of the following? 12 m/s 12 m/s 60 W 60 W 305 K 305 K 9.5 kg 9.5 kg 2.0 T 2.0 T 0.8 N 0.8 N 20450 cal 20450 cal

25. How many significant digits are there in each of the following? 12 m/s2 s.d. 12 m/s2 s.d. 60 W 60 W 305 K 305 K 9.5 kg 9.5 kg 2.0 T 2.0 T 0.8 N 0.8 N 20450 cal 20450 cal

26. How many significant digits are there in each of the following? 12 m/s2 s.d. 12 m/s2 s.d. 60 W1 s.d. 60 W1 s.d. 305 K 305 K 9.5 kg 9.5 kg 2.0 T 2.0 T 0.8 N 0.8 N 20450 cal 20450 cal

27. How many significant digits are there in each of the following? 12 m/s2 s.d. 12 m/s2 s.d. 60 W1 s.d. 60 W1 s.d. 305 K3 s.d. 305 K3 s.d. 9.5 kg 9.5 kg 2.0 T 2.0 T 0.8 N 0.8 N 20450 cal 20450 cal

28. How many significant digits are there in each of the following? 12 m/s2 s.d. 12 m/s2 s.d. 60 W1 s.d. 60 W1 s.d. 305 K3 s.d. 305 K3 s.d. 9.5 kg2 s.d. 9.5 kg2 s.d. 2.0 T 2.0 T 0.8 N 0.8 N 20450 cal 20450 cal

29. How many significant digits are there in each of the following? 12 m/s2 s.d. 12 m/s2 s.d. 60 W1 s.d. 60 W1 s.d. 305 K3 s.d. 305 K3 s.d. 9.5 kg2 s.d. 9.5 kg2 s.d. 2.0 T2 s.d. 2.0 T2 s.d. 0.8 N 0.8 N 20450 cal 20450 cal

30. How many significant digits are there in each of the following? 12 m/s2 s.d. 12 m/s2 s.d. 60 W1 s.d. 60 W1 s.d. 305 K3 s.d. 305 K3 s.d. 9.5 kg2 s.d. 9.5 kg2 s.d. 2.0 T2 s.d. 2.0 T2 s.d. 0.8 N1 s.d. 0.8 N1 s.d. 20450 cal 20450 cal

31. How many significant digits are there in each of the following? 12 m/s2 s.d. 12 m/s2 s.d. 60 W1 s.d. 60 W1 s.d. 305 K3 s.d. 305 K3 s.d. 9.5 kg2 s.d. 9.5 kg2 s.d. 2.0 T2 s.d. 2.0 T2 s.d. 0.8 N1 s.d. 0.8 N1 s.d. 20450 cal4 s.d. 20450 cal4 s.d.

32. How many significant digits are there in each of the following? 1.40  1.40  0.075 h 0.075 h 102.5 MHz 102.5 MHz 2500 J 2500 J 100.0 V 100.0 V 40.20 A 40.20 A 0.09030 km 0.09030 km

33. How many significant digits are there in each of the following? 1.40  s.d. 1.40  s.d. 0.075 h 0.075 h 102.5 MHz 102.5 MHz 2500 J 2500 J 100.0 V 100.0 V 40.20 A 40.20 A 0.09030 km 0.09030 km

34. How many significant digits are there in each of the following? 1.40  s.d. 1.40  s.d. 0.075 h2 s.d. 0.075 h2 s.d. 102.5 MHz 102.5 MHz 2500 J 2500 J 100.0 V 100.0 V 40.20 A 40.20 A 0.09030 km 0.09030 km

35. How many significant digits are there in each of the following? 1.40  s.d. 1.40  s.d. 0.075 h2 s.d. 0.075 h2 s.d. 102.5 MHz4 s.d. 102.5 MHz4 s.d. 2500 J 2500 J 100.0 V 100.0 V 40.20 A 40.20 A 0.09030 km 0.09030 km

36. How many significant digits are there in each of the following? 1.40  s.d. 1.40  s.d. 0.075 h2 s.d. 0.075 h2 s.d. 102.5 MHz4 s.d. 102.5 MHz4 s.d. 2500 J2 s.d. 2500 J2 s.d. 100.0 V 100.0 V 40.20 A 40.20 A 0.09030 km 0.09030 km

37. How many significant digits are there in each of the following? 1.40  s.d. 1.40  s.d. 0.075 h2 s.d. 0.075 h2 s.d. 102.5 MHz4 s.d. 102.5 MHz4 s.d. 2500 J2 s.d. 2500 J2 s.d. 100.0 V4 s.d. 100.0 V4 s.d. 40.20 A 40.20 A 0.09030 km 0.09030 km

38. How many significant digits are there in each of the following? 1.40  s.d. 1.40  s.d. 0.075 h2 s.d. 0.075 h2 s.d. 102.5 MHz4 s.d. 102.5 MHz4 s.d. 2500 J2 s.d. 2500 J2 s.d. 100.0 V4 s.d. 100.0 V4 s.d. 40.20 A4 s.d. 40.20 A4 s.d. 0.09030 km 0.09030 km

39. How many significant digits are there in each of the following? 1.40  s.d. 1.40  s.d. 0.075 h2 s.d. 0.075 h2 s.d. 102.5 MHz4 s.d. 102.5 MHz4 s.d. 2500 J2 s.d. 2500 J2 s.d. 100.0 V4 s.d. 100.0 V4 s.d. 40.20 A4 s.d. 40.20 A4 s.d. 0.09030 km4 s.d. 0.09030 km4 s.d.

40. Round each measurement to the required significant digits: 4080 J to 1 s.d. 4080 J to 1 s.d. 4080 J to 2 s.d. 4080 J to 2 s.d. 2.715 kg to 1 s.d. 2.715 kg to 1 s.d. 2.715 kg to 2 s.d. 2.715 kg to 2 s.d. 0.987 V to 1 s.d. 0.987 V to 1 s.d. 0.987 V to 2 s.d. 0.987 V to 2 s.d.

41. Round each measurement to the required significant digits: 4080 J to 1 s.d.4000 J 4080 J to 1 s.d.4000 J 4080 J to 2 s.d. 4080 J to 2 s.d. 2.715 kg to 1 s.d. 2.715 kg to 1 s.d. 2.715 kg to 2 s.d. 2.715 kg to 2 s.d. 0.987 V to 1 s.d. 0.987 V to 1 s.d. 0.987 V to 2 s.d. 0.987 V to 2 s.d.

42. Round each measurement to the required significant digits: 4080 J to 1 s.d.4000 J 4080 J to 1 s.d.4000 J 4080 J to 2 s.d.4100 J 4080 J to 2 s.d.4100 J 2.715 kg to 1 s.d. 2.715 kg to 1 s.d. 2.715 kg to 2 s.d. 2.715 kg to 2 s.d. 0.987 V to 1 s.d. 0.987 V to 1 s.d. 0.987 V to 2 s.d. 0.987 V to 2 s.d.

43. Round each measurement to the required significant digits: 4080 J to 1 s.d.4000 J 4080 J to 1 s.d.4000 J 4080 J to 2 s.d.4100 J 4080 J to 2 s.d.4100 J 2.715 kg to 1 s.d.3 kg 2.715 kg to 1 s.d.3 kg 2.715 kg to 2 s.d. 2.715 kg to 2 s.d. 0.987 V to 1 s.d. 0.987 V to 1 s.d. 0.987 V to 2 s.d. 0.987 V to 2 s.d.

44. Round each measurement to the required significant digits: 4080 J to 1 s.d.4000 J 4080 J to 1 s.d.4000 J 4080 J to 2 s.d.4100 J 4080 J to 2 s.d.4100 J 2.715 kg to 1 s.d.3 kg 2.715 kg to 1 s.d.3 kg 2.715 kg to 2 s.d.2.7 kg 2.715 kg to 2 s.d.2.7 kg 0.987 V to 1 s.d. 0.987 V to 1 s.d. 0.987 V to 2 s.d. 0.987 V to 2 s.d.

45. Round each measurement to the required significant digits: 4080 J to 1 s.d.4000 J 4080 J to 1 s.d.4000 J 4080 J to 2 s.d.4100 J 4080 J to 2 s.d.4100 J 2.715 kg to 1 s.d.3 kg 2.715 kg to 1 s.d.3 kg 2.715 kg to 2 s.d.2.7 kg 2.715 kg to 2 s.d.2.7 kg 0.987 V to 1 s.d.1 V 0.987 V to 1 s.d.1 V 0.987 V to 2 s.d. 0.987 V to 2 s.d.

46. Round each measurement to the required significant digits: 4080 J to 1 s.d.4000 J 4080 J to 1 s.d.4000 J 4080 J to 2 s.d.4100 J 4080 J to 2 s.d.4100 J 2.715 kg to 1 s.d.3 kg 2.715 kg to 1 s.d.3 kg 2.715 kg to 2 s.d.2.7 kg 2.715 kg to 2 s.d.2.7 kg 0.987 V to 1 s.d.1 V 0.987 V to 1 s.d.1 V 0.987 V to 2 s.d.0.99 V 0.987 V to 2 s.d.0.99 V

47. Round each measurement to the required significant digits: 13.5 N to 2 s.d. 13.5 N to 2 s.d. 12.5 N to 2 s.d. 12.5 N to 2 s.d. 12.51 N to 2 s.d. 12.51 N to 2 s.d. 100.5 km to 3 s.d. 100.5 km to 3 s.d.

48. Round each measurement to the required significant digits: 13.5 N to 2 s.d.14 N 13.5 N to 2 s.d.14 N 12.5 N to 2 s.d. 12.5 N to 2 s.d. 12.51 N to 2 s.d. 12.51 N to 2 s.d. 100.5 km to 3 s.d. 100.5 km to 3 s.d.

49. Round each measurement to the required significant digits: 13.5 N to 2 s.d.14 N 13.5 N to 2 s.d.14 N 12.5 N to 2 s.d.12 N 12.5 N to 2 s.d.12 N 12.51 N to 2 s.d. 12.51 N to 2 s.d. 100.5 km to 3 s.d. 100.5 km to 3 s.d.

50. Round each measurement to the required significant digits: 13.5 N to 2 s.d.14 N 13.5 N to 2 s.d.14 N 12.5 N to 2 s.d.12 N 12.5 N to 2 s.d.12 N The “Rule of 5”: If the first digit to be dropped is a lone 5 (or a 5 followed by zeroes), round down if the preceding digit is even and up if the preceding digit is odd.

51. Round each measurement to the required significant digits: 13.5 N to 2 s.d.14 N 13.5 N to 2 s.d.14 N 12.5 N to 2 s.d.12 N 12.5 N to 2 s.d.12 N 12.51 N to 2 s.d. 12.51 N to 2 s.d. 100.5 km to 3 s.d. 100.5 km to 3 s.d.

52. Round each measurement to the required significant digits: 13.5 N to 2 s.d.14 N 13.5 N to 2 s.d.14 N 12.5 N to 2 s.d.12 N 12.5 N to 2 s.d.12 N 12.51 N to 2 s.d.13 N 12.51 N to 2 s.d.13 N 100.5 km to 3 s.d. 100.5 km to 3 s.d.

53. Round each measurement to the required significant digits: 13.5 N to 2 s.d.14 N 13.5 N to 2 s.d.14 N 12.5 N to 2 s.d.12 N 12.5 N to 2 s.d.12 N 12.51 N to 2 s.d.13 N 12.51 N to 2 s.d.13 N 100.5 km to 3 s.d. 1.00 × 10 2 km 100.5 km to 3 s.d. 1.00 × 10 2 km

54. Write each of the following in scientific notation 20450 cal 20450 cal 1200 N 1200 N 0.235 J 0.235 J 0.09030 km 0.09030 km 0.0000007 s 0.0000007 s 20.5 kHz 20.5 kHz

55. Write each of the following in scientific notation 20450 cal 2.045 × 10 4 cal 20450 cal 2.045 × 10 4 cal 1200 N 1200 N 0.235 J 0.235 J 0.09030 km 0.09030 km 0.0000007 s 0.0000007 s 20.5 kHz 20.5 kHz

56. Write each of the following in scientific notation 20450 cal 2.045 × 10 4 cal 20450 cal 2.045 × 10 4 cal 1200 N 1.2 × 10 3 N 1200 N 1.2 × 10 3 N 0.235 J 0.235 J 0.09030 km 0.09030 km 0.0000007 s 0.0000007 s 20.5 kHz 20.5 kHz

57. Write each of the following in scientific notation 20450 cal 2.045 × 10 4 cal 20450 cal 2.045 × 10 4 cal 1200 N 1.2 × 10 3 N 1200 N 1.2 × 10 3 N 0.235 J 2.35 × 10 -1 J 0.235 J 2.35 × 10 -1 J 0.09030 km 0.09030 km 0.0000007 s 0.0000007 s 20.5 kHz 20.5 kHz

58. Write each of the following in scientific notation 20450 cal 2.045 × 10 4 cal 20450 cal 2.045 × 10 4 cal 1200 N 1.2 × 10 3 N 1200 N 1.2 × 10 3 N 0.235 J 2.35 × 10 -1 J 0.235 J 2.35 × 10 -1 J 0.09030 km 9.030 × 10 -2 km 0.09030 km 9.030 × 10 -2 km 0.0000007 s 0.0000007 s 20.5 kHz 20.5 kHz

59. Write each of the following in scientific notation 20450 cal 2.045 × 10 4 cal 20450 cal 2.045 × 10 4 cal 1200 N 1.2 × 10 3 N 1200 N 1.2 × 10 3 N 0.235 J 2.35 × 10 -1 J 0.235 J 2.35 × 10 -1 J 0.09030 km 9.030 × 10 -2 km 0.09030 km 9.030 × 10 -2 km 0.0000007 s 7 × 10 -7 s 0.0000007 s 7 × 10 -7 s 20.5 kHz 20.5 kHz

60. Write each of the following in scientific notation 20450 cal 2.045 × 10 4 cal 20450 cal 2.045 × 10 4 cal 1200 N 1.2 × 10 3 N 1200 N 1.2 × 10 3 N 0.235 J 2.35 × 10 -1 J 0.235 J 2.35 × 10 -1 J 0.09030 km 9.030 × 10 -2 km 0.09030 km 9.030 × 10 -2 km 0.0000007 s 7 × 10 -7 s 0.0000007 s 7 × 10 -7 s 20.5 kHz 2.05 × 10 1 kHz 20.5 kHz 2.05 × 10 1 kHz

61. The Final Answer When a measurement is used in a calculation, the final answer must take into consideration the uncertainty in the original measurements.

62. The Final Answer When a measurement is used in a calculation, the final answer must take into consideration the uncertainty in the original measurements. Why? You don’t want to suggest that you know something to a greater precision than you actually measured. For example, 4 m/3 s = 1.3333333333333333333333 m/s???

63. The Final Answer When a measurement is used in a calculation, the final answer must take into consideration the uncertainty in the original measurements. Why? You don’t want to suggest that you know something to a greater precision than you actually measured. For example, 4 m/3 s = 1 m/s.

64. The Final Answer When a measurement is used in a calculation, the final answer must take into consideration the uncertainty in the original measurements. Note: Exact numbers used in calculations (e.g. a factor such as ½ in the equation K=½mv 2 ) are not measurements and do not have any uncertainty.

65. Addition and Subtraction When adding or subtracting measurements, the final answer should be rounded off to the least number of decimals in the original measurements. e.g.5.124 cm (3 decimal places) +0.01 cm (2 decimal places) 5.13 cm (2 decimal places)

66. Practice 7 m + 7 m = 7 m + 7 m = 7 m + 7.0 m = 7 m + 7.0 m = 7.0 m + 7.0 m = 7.0 m + 7.0 m =

67. Practice 7 m + 7 m = 14 m 7 m + 7 m = 14 m 7 m + 7.0 m = 7 m + 7.0 m = 7.0 m + 7.0 m = 7.0 m + 7.0 m =

68. Practice 7 m + 7 m = 14 m 7 m + 7 m = 14 m 7 m + 7.0 m = 14 m 7 m + 7.0 m = 14 m 7.0 m + 7.0 m = 7.0 m + 7.0 m =

69. Practice 7 m + 7 m = 14 m 7 m + 7 m = 14 m 7 m + 7.0 m = 14 m 7 m + 7.0 m = 14 m 7.0 m + 7.0 m = 14.0 m 7.0 m + 7.0 m = 14.0 m

70. Multiplication and Division When multiplying or dividing measurements, the final answer should be rounded off to the same number of sig digs as are in the measurement with the least number of sig digs. e.g.5.124 cm (4 sig digs) x0.01 cm (1 sig dig) 0.05 cm 2 (1 sig dig)

71. Practice 7 m x 7 m = 7 m x 7 m = 7 m x 7.0 m = 7 m x 7.0 m = 7.0 m x 7.0 m = 7.0 m x 7.0 m =

72. Practice 7 m x 7 m = 50 m 2 7 m x 7 m = 50 m 2 (Yes, your math teacher would not approve.) 7 m x 7.0 m = 7 m x 7.0 m = 7.0 m x 7.0 m = 7.0 m x 7.0 m =

73. Practice 7 m x 7 m = 50 m 2 7 m x 7 m = 50 m 2 7 m x 7.0 m = 50 m 2 7 m x 7.0 m = 50 m 2 7.0 m x 7.0 m = 7.0 m x 7.0 m =

74. Practice 7 m x 7 m = 50 m 2 7 m x 7 m = 50 m 2 7 m x 7.0 m = 50 m 2 7 m x 7.0 m = 50 m 2 7.0 m x 7.0 m = 49 m 2 7.0 m x 7.0 m = 49 m 2

75. A Warning When solving multi-step problems, round-off error can occur if a rounded-off answer from one step is used in a later step. Record or store in your calculator all calculated digits for use in later steps.

76. A Warning When solving multi-step problems, round-off error can occur if a rounded-off answer from one step is used in a later step. Record or store in your calculator all calculated digits for use in later steps. SERIOUSLY!!!! DON’T ROUND TOO SOON

77. This is GUSS

78. GUSS has a procedure for solving problems. First, he identifies his Givens. Then he identifies his Unknown. Next, he Selects an equation that relates his Givens and his Unknown, rearranging it for the Unknown if necessary. Finally, he substitutes his Givens into the equation and Solves for his Unknown.

79. An Example How long does it take an object travelling at a speed of 3.0 m/s to travel a distance of 1.5 km?

80. An Example How long does it take an object travelling at a speed of 3.0 m/s to travel a distance of 1.5 km? Givens: v = 3.0 m/s d = 1.5 km

81. An Example How long does it take an object travelling at a speed of 3.0 m/s to travel a distance of 1.5 km? Givens: v = 3.0 m/s d = 1.5 km = 1500 m

82. An Example How long does it take an object travelling at a speed of 3.0 m/s to travel a distance of 1.5 km? Givens: v = 3.0 m/s d = 1.5 km = 1500 m Unknown:t = ?

83. An Example Select an Equation: v = d/t

84. An Example Select an Equation: v = d/t t = d/v

85. An Example Select an Equation: v = d/t t = d/v Substitute and Solve: t = (1500 m/3.0 m/s) =

86. An Example Select an Equation: v = d/t t = d/v Substitute and Solve: t = (1500 m/3.0 m/s) = 500 s

87. An Example Select an Equation: v = d/t t = d/v Substitute and Solve: t = (1500 m/3.0 m/s) = 500 s i.e. 5.0 x 10 2 s i.e. 5.0 x 10 2 s

88. An Example Select an Equation: v =  d/  t  t =  d/v Substitute and Solve:  t = (1500 m/3.0 m/s) = 500 s i.e. 5.0 x 10 2 s i.e. 5.0 x 10 2 s or 8.3 min or 8.3 min

89. An Example Select an Equation: v =  d/  t  t =  d/v Substitute and Solve:  t = (1500 m/3.0 m/s) = 500 s i.e. 5.0 x 10 2 s i.e. 5.0 x 10 2 s or 8.3 min or 8.3 min PLEASE NOTE THAT ALL OUR MEASUREMENTS ALWAYS HAVE UNITS!

90. More Practice Homework: Significant Digits and GUSS Handout