The result should be given to as many significant

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Unformatted text preview: gure 1. If we have a scale with ten etchings to every millimeter, we could use a microscope to measure the spacing to the nearest tenth of a millimeter 239 APPENDIX: SIGNIFICANT FIGURES The uncertain figures in each number are shown in bold-faced type. and guess at the one hundredth millimeter. Our measurement could be 5.814 cm with the uncertainty in the last figure, four significant figures instead of three. This is because our improved scale allowed our estimate to be more precise. This added precision is shown by more significant figures. The more significant figures a number has, the more precise it is. Multiplication and division When multiplying or dividing numbers, the number of significant figures must be taken into account. The result should be given to as many significant figures as the term in the product that is given to the smallest number of significant figures. How do I use significant figures in calculations? When using significant figures in calculations, you need to keep track of how the uncertainty propagates. There are mathematical procedures for doing this estimate in the most precise manner. This type of estimate depends on knowing the statistical distribution of your measurements. With a lot less effort, you can do a cruder estimate of the uncertainties in a calculated result. This crude method gives an overestimate of the uncertainty but it is a good place to start. For this course this simplified uncertainty estimate (described in Appendix C and below) will be good enough. The basis behind this rule is that the least accurately known term in the product will dominate the accuracy of the answer. As shown in the examples, this does not always work, though it is the quickest and best rule to use. When in doubt, you can keep track of the significant figures in the calculation as is done in the examples. Examples: Multiplication 15.84 17.27 x 2.5 x 4.0 7920 69.080 3168 39.600 40 69 Addition and subtraction When adding or subtracting numbers, the number of decimal places must be taken into account. The result should be given to as many decimal places as the term in the sum that is given to the smallest number of decimal places. Examples: Addition 6.242 +4.23 +0.013 10.485 10.49 Division 117 23)2691 23 39 23 161 161 1.2 x 102 Subtraction 5.875 -3.34 2.535 2.54 240 25 75)1875 150 375 375 2.5 x 101 APPENDIX: SIGNIFICANT FIGURES PRACTICE EXERCISES 1. Determine the number of significant figures of the quantities in the following table: Length (centimeters) Number of Significant Figures 17.87 0.4730 17.9 0.473 18 0.47 1.34 x 102 2.567x 105 2.0 x 1010 1.001 1.000 1 1000 1001 2. Add: 121.3 to 6.7 x 102: [Answer: 121.3 + 6.7 x 102 = 7.9 x 102] 3. Multiply: 34.2 and 1.5 x 104 [Answer: 34.2 x 1.5 x 104 = 5.1 x 105] 241 APPENDIX: SIGNIFICANT FIGURES 242 Appendix: Accuracy, Precision and Uncertainty - ERROR ANALYSIS Figure 1 How tall are you? How old are you? When you answered these everyday questions, you probably did it in round numbers such as "five foot, six inches" or "nineteen years, three months." But how true are these answers? Are you exactly 5' 6" tall? Probably not. You estimated your height at 5’ 6" and just reported two significant figures. Typically, you round your height to the nearest inch, so that your actual height falls somewhere between 5' 5½" and 5' 6½" tall, or 5' 6" ± ½". This ± ½" is the uncertainty, and it informs the reader of the precision of the value 5' 6". What is uncertainty? Whenever you measure something, there is always some uncertainty. There are two categories of uncertainty: systematic and random. (1) Systematic uncertainties are those that consistently cause the value to be too large or too small. Systematic uncertainties include such things as reaction time, inaccurate meter sticks, optical parallax and miscalibrated balances. In principle, systematic uncertainties can be eliminated if you know they exist. (2) Random uncertainties are variations in the measurements that occur without a predictable pattern. If you make precise measurements, these uncertainties arise from the estimated part of the measurement. Random uncertainty can be reduced, but never eliminated. We need a technique to report the contribution of this uncertainty to the measured value. Uncertainties cause every measurement you make to be distributed. For example, the key in Figure 2 is approximately 5.37cm long. For the sake of argument, pretend that it is exactly 5.37cm long. If you measure its length many times, you expect that most of the measurements will be close to, but not exactly, 5.37cm, and that there will be a few measurements much more than or much less than 5.37cm. This effect is due to random uncertainty. You can never know how accurate any single measurement is, but you expect that many measurements will cluster around the real length, so you can take the average as the “real” length, and more measurements will give you a better answer; see Figure 1. You must be...
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