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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 boldfaced 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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 Spring '14

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