2. How big? Measurements and Significant Digits How small? How accurate?

3. Agenda MORE PRACTICE ON SIGNIFICANT DIGITS HW: complete scientific notation, rounding, sig. digits worksheets

4. Using Scientific Measurements Precision and Accuracy 1. Precision – the closeness of a set of measurements of the same quantities made in the same way (how well repeated measurements of a value agree with one another). 2. Accuracy – is determined by the agreement between the measured quantity and the correct value. Ex: Throwing Darts ACCURATE = CORRECT PRECISE = CONSISTENT

5. Accuracy vs. Precision Random errors: reduce precision Good accuracy Good precision Poor accuracy Good precision Poor accuracy Poor precision Systematic errors: reduce accuracy (person)(instrument)

6. Precision Accuracy  reproducibility  check by repeating measurements  poor precision results from poor technique  correctness  check by using a different method  poor accuracy results from procedural or equipment flaws.

7. Percent Error is calculated by subtracting the experimental value from the accepted value, then dividing the difference by the accepted value. Multiply this number by 100. Accuracy can be compared quantitatively with the accepted value using percent error.

8. Measurement Exact number - results from counting items that cannot be subdivided - has an infinite number of significant digits. Approximate number -results from measuring -does not express absolute accuracy -has a defined number of significant digits that depends on the accuracy of the measuring device

9. What time is it? Someone might say “1:30” or “1:28” or “1:27:55” Each is appropriate for a different situation In science we describe a value as having a certain number of “significant digits” The # of significant digits in a value includes all digits that are certain and one that is uncertain “1:30” likely has 2, 1:28 has 3, 1:27:55 has 5

10. Reporting Measurements Using significant figures Report what is known with certainty Add ONE digit of uncertainty (estimation)

11. Counting Significant Figures When you report a measured value it is assumed that all the numbers are certain except for the last one, where there is an uncertainty of ±1. Example of nail: the nail is 6.36cm long. The 6.3 are certain values and the final 6 is uncertain! There are 3 significant figures in the value 6.36cm (2 certain and 1 uncertain). All measured values will have one (and one only) uncertain number (the last one) and all others will be certain. The reader can see that the 6.3 are certain values because they appear on the ruler, but the reader has to estimate the final 6.

12. Significant Figures Indicate precision of a measurement. Recording Significant Figures (SF) – Sig figs in a measurement include the known digits plus a final estimated digit 2.35 cm

13. Practice Measuring 4.5 cm 4.54 cm 3.0 cm cm 0 12345 0 12345 0 12345

14. 20 10 15 mL ? 15.0 mL? 1.50 x 10 1 mL

15. There are rules that dictate the number of significant digits in a value. 1.Read the handout up to A. 2.Try A 3.Bored? There are more: a. 38.4703 mL b. 0.00052 g c. 0.05700 s d. 500 g a. 6 b. 2 c. 4 d. 1

16. The rules for counting the number of significant figures in a value are: 1.All numbers other then zero will always be counted as significant figures. 2.Captive zeros always count. All zeros between two non- zero numbers are significant. 3.Leading zeros do not count. Zeros before a non-zero number after a decimal point are not significant. 4.Trailing zeros count only if there is a decimal. -All zeros after a non-zero number, after a decimal point are significant. -Zeros after a non zero number with no decimal point are not significant.

17. Answers to question A 1. 2.83 2. 36.77 3. 14.0 4. 0.0033 5. 0.02 6. 0.2410 7. 2.350 x 10 – 2 8. 1.00009 9. 3 10. 0.0056040 3 4 3 2 1 4 4 6 infinite 5

18. Rounding Rounding using the statistical approach: When a number ends in 5 and only 5 when you need to round: If the preceding number is even –leave it, don’t round up Ex. The number 21.45 rounded off to 3 significant figures becomes If the preceding number is odd – round up Ex. The number 21.350 rounded off to 3 significant figures becomes BUT If any nonzero digits follow the 5, raise the preceding digit by 1. Ex. The number 21.4501 rounded off to 3 significant figures becomes 21.4 21.5

19. Scientific notation All significant digits must be maintained Only one number is written before the decimal point and express the decimal points as a power of ten. 9.07 x 10 – 2 m0.0907m 5.06 x 10 – 4 cg0.000506cg 2.3 x 10 12 m2 300 000 000 000m 1.27 x 10 2 g127g Scientific notationDecimal notation

20. Scientific notation If your value is expressed in proper scientific notation, all of the figures in the pre-exponential value are significant, with the last digit being the least significant figure. “ 7.143 x 10 -3 grams ” contains 4 significant figures If that value is expressed as 0.007143, it still has 4 significant figures. Zeros, in this case, are placeholders. If you are ever in doubt about the number of significant figures in a value, write it in scientific notation.

21. Give the number of significant figures in the following values: a. 6.19 x 10 1 yearsb. 7 400 000 years c. 3.80 x 10 -19 J Helpful Hint :Convert to scientific notation if you are not certain as to the proper number of significant figures. When solving multiple step problems DO NOT ROUND OFF THE ANSWER UNTIL THE VERY END OF THE PROBLEM. ANS: a. 3 b. 3 c. 3

22. Significant Digits It is better to represent 100 as 1.00 x 10 2 Alternatively you can underline the position of the last significant digit. E.g. 100. This is especially useful when doing a long calculation or for recording experimental results Don’t round your answer until the last step in a calculation. Note that a line overtop of a number indicates that it repeats indefinitely. E.g. 9.6 = 9.6666… Similarly, 6.54 = 6.545454…

23. Fill in the table Ordinary Notation ( g)Scientific Notation (g)# of Significant Figures 0.0012 0.00102 0.00120 1.200 12.00 1200 1.2 x 10 -3 2 1.02 x 10 -3 1.20 x 10 -3 1.200 x 10 0 1.200 x 10 1 1.2 x 10 3 3 3 4 4 2 31.20 x 10 3

24. Fill in the table considering the number of significant figures. Previous Number (mL)Number ( mL)Following Number (mL) 120 120.0 120 110 119.9 11 9 130 120.1 121

25. Significant Figures in Calculations 1. In addition and subtraction, your answer should have the same number of decimal places as the measurement with the least number of decimal places. Example: 12.734mL - 3.0mL = __________ Solution: 12.734mL has 3 figures past the decimal point. 3.0mL has only 1 figure past the decimal point. Therefore your final answer should be rounded off to one figure past the decimal point. 12.734mL - 3.0mL 9.734 --------  9.7mL

26. Adding with Significant Digits How far is it from Warsaw to room C40? To B12? Adding a value that is much smaller than the last sig. digit of another value is irrelevant E.g. a) 13.64 + 0.075 + 67 b) 267.8 – 9.36 13.64 0.075 67. 80.71581 267.8 9.36 258.44 Try question B on the handout – + +

27. B) Answers 83.14 i) 83.25 0.1075 – 4.02 0.001+ ii) 1.82 0.2983 1.52+ iii)

28. Multiplying with Significant Digits 2. In multiplication and division, your answer should have the same number of significant figures as the least precise measurement (or the measurement with the fewest number of SF). Examples: a. 61cm x 0.00745cm = 0.45445 = = 2SF 3SF 2SF b. 608.3m x 3.45m = 2098.635 = 4SF 3SF 3SF c.4.8 g  392g = 0.012245 = 2SF 3SF 2SF Try question C and D on the handout (recall: for long questions, don’t round until the end) 0.45cm 2 2.10 x 10 3 m 2 0.012 or 1.2 x 10 – 2 4.5 x 10 -1 cm 2

29. C), D) Answers i) 7.255  81.334 = 0.08920 ii) 1.142 x 0.002 = 0.002 iii) 31.22 x 9.8 = 3.1 x 10 2 (or 310 or 305.956) i) 6.12 x 3.734 + 16.1  2.3 22.85208 + 7.0 = 29.9 ii) 0.0030 + 0.02 = 0.02 135700 =1.36 x10 5 1700 134000+ iii) iv) 33.4 112.7+ 0.032+ 146.132  6.487 = 22.5268 = 22.53 Note: 146.1  6.487 = 22.522 = 22.52

30. Calculations & Significant Digits In multiple step problems if addition or subtraction AND multiplication or division is used the rules for rounding are based off of multiplication and division (it “ trumps ” the addition and subtraction rules). There is no uncertainty in a conversion factor; therefore they do not affect the degree of certainty of your answer. The answer should have the same number of SF as the initial value. a. Convert 25 meters to millimeters. b. Convert 0.12L to mL. ?mm → 25 m X 1000 mm = 25 000 mm 1 1 m 2SF ?mL → 0.12L X 1000 m = 120 mL 1 1 L 2SF

31. Unit conversions & Significant Digits Sometimes it is more convenient to express a value in different units. When units change, basically the number of significant digits does not. E.g. 1.23 m = 123 cm = 1230 mm = 0.00123 km Notice that these all have 3 significant digits This should make sense mathematically since you are multiplying or dividing by a term that has an infinite number of significant digits. conversion factors= infinite # of sig. digits E.g. 123 cm x 10 mm / cm = 1230 mm Try question E on the handout

32. E) Answers A shocking number of patients die every year in United States hospitals as the result of medication errors, and many more are harmed. One widely cited estimate (Institute of Medicine, 2000) places the toll at 44,000 to 98,000 deaths, making death by medication "misadventure" greater than all highway accidents, breast cancer, or AIDS. If this estimate is in the ballpark, then nurses (and patients) beware: Medication errors are the forth to sixth leading cause of death in America. i) 1.0 cm = 0.010 m ii) 0.0390 kg = 39.0 g iii) 1.7 m = 1700 mm or 1.7 x 10 3 mm

33. Real World Connections : Information from the website “ Medication Math for the Nursing Student ” at http://www.alysion.org/dimensional/analysis.htm# problems http://www.alysion.org/dimensional/analysis.htm# problems