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The Return of GUSS

The Return of GUSS. Featuring Significant Digits. A Justification for “Sig Digs”. Measurements are not perfect. A Justification for “Sig Digs”. Measurements are not perfect. They always include some degree of uncertainty because no measuring device is perfect.

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The Return of GUSS

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  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 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 • 15.0 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 • 15.0 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 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 • 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 • Any zeroes placed after other digits and behind a decimal 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 • Any zeroes placed after other digits and behind a decimal are significant • Any zeroes placed between significant digits are 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 (if the measurement is in a table) • rewriting the measurement in scientific notation: 1.00 x 102 cm

  19. Making Zeros Significant And yes, if you measure a zero, you must write it. Your lab tables should not look like this:

  20. Making Zeros Significant They should look like this:

  21. No ½ measurements Your tables also should not look like this:

  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.

  23. And now for some practice. . . .

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

  25. How many significant digits are there in each of the following? • 12 m/s 2 s.d. • 60 W • 305 K • 9.5 kg • 2.0 T • 0.8 N • 20450 cal

  26. How many significant digits are there in each of the following? • 12 m/s 2 s.d. • 60 W 1 s.d. • 305 K • 9.5 kg • 2.0 T • 0.8 N • 20450 cal

  27. How many significant digits are there in each of the following? • 12 m/s 2 s.d. • 60 W 1 s.d. • 305 K 3 s.d. • 9.5 kg • 2.0 T • 0.8 N • 20450 cal

  28. How many significant digits are there in each of the following? • 12 m/s 2 s.d. • 60 W 1 s.d. • 305 K 3 s.d. • 9.5 kg 2 s.d. • 2.0 T • 0.8 N • 20450 cal

  29. How many significant digits are there in each of the following? • 12 m/s 2 s.d. • 60 W 1 s.d. • 305 K 3 s.d. • 9.5 kg 2 s.d. • 2.0 T 2 s.d. • 0.8 N • 20450 cal

  30. How many significant digits are there in each of the following? • 12 m/s 2 s.d. • 60 W 1 s.d. • 305 K 3 s.d. • 9.5 kg 2 s.d. • 2.0 T 2 s.d. • 0.8 N 1 s.d. • 20450 cal

  31. How many significant digits are there in each of the following? • 12 m/s 2 s.d. • 60 W 1 s.d. • 305 K 3 s.d. • 9.5 kg 2 s.d. • 2.0 T 2 s.d. • 0.8 N 1 s.d. • 20450 cal 4 s.d.

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

  33. How many significant digits are there in each of the following? • 1.40 W 3 s.d. • 0.075 h • 102.5 MHz • 2500 J • 100.0 V • 40.20 A • 0.09030 km

  34. How many significant digits are there in each of the following? • 1.40 W 3 s.d. • 0.075 h 2 s.d. • 102.5 MHz • 2500 J • 100.0 V • 40.20 A • 0.09030 km

  35. How many significant digits are there in each of the following? • 1.40 W 3 s.d. • 0.075 h 2 s.d. • 102.5 MHz 4 s.d. • 2500 J • 100.0 V • 40.20 A • 0.09030 km

  36. How many significant digits are there in each of the following? • 1.40 W 3 s.d. • 0.075 h 2 s.d. • 102.5 MHz 4 s.d. • 2500 J 2 s.d. • 100.0 V • 40.20 A • 0.09030 km

  37. How many significant digits are there in each of the following? • 1.40 W 3 s.d. • 0.075 h 2 s.d. • 102.5 MHz 4 s.d. • 2500 J 2 s.d. • 100.0 V 4 s.d. • 40.20 A • 0.09030 km

  38. How many significant digits are there in each of the following? • 1.40 W 3 s.d. • 0.075 h 2 s.d. • 102.5 MHz 4 s.d. • 2500 J 2 s.d. • 100.0 V 4 s.d. • 40.20 A 4 s.d. • 0.09030 km

  39. How many significant digits are there in each of the following? • 1.40 W 3 s.d. • 0.075 h 2 s.d. • 102.5 MHz 4 s.d. • 2500 J 2 s.d. • 100.0 V 4 s.d. • 40.20 A 4 s.d. • 0.09030 km 4 s.d.

  40. Round each measurement to the required significant digits: • 4080 J to 1 s.d. • 4080 J to 2 s.d. • 2.715 kg to 1 s.d. • 2.715 kg to 2 s.d. • 0.987 V to 1 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 2 s.d. • 2.715 kg to 1 s.d. • 2.715 kg to 2 s.d. • 0.987 V to 1 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 2 s.d. 4100 J • 2.715 kg to 1 s.d. • 2.715 kg to 2 s.d. • 0.987 V to 1 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 2 s.d. 4100 J • 2.715 kg to 1 s.d. 3 kg • 2.715 kg to 2 s.d. • 0.987 V to 1 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 2 s.d. 4100 J • 2.715 kg to 1 s.d. 3 kg • 2.715 kg to 2 s.d. 2.7 kg • 0.987 V to 1 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 2 s.d. 4100 J • 2.715 kg to 1 s.d. 3 kg • 2.715 kg to 2 s.d. 2.7 kg • 0.987 V to 1 s.d. 1 V • 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 2 s.d. 4100 J • 2.715 kg to 1 s.d. 3 kg • 2.715 kg to 2 s.d. 2.7 kg • 0.987 V to 1 s.d. 1 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. • 12.5 N to 2 s.d. • 12.51 N to 2 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 • 12.5 N to 2 s.d. • 12.51 N to 2 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 • 12.5 N to 2 s.d. 12 N • 12.51 N to 2 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 • 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.

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