Download T2 Uncertainties and significant figures

Uncertainties and significant figures
How big is the beetle?
Measure between the head
and the tail!
Between 1.5 and 1.6 in
Measured length: 1.54 in
The 1 and 5 are known
with certainty
The last digit (4) is
estimated between the
two nearest fine division
marks.
Copyright © 1997-2005 by Fred Senese
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Uncertainties and significant figures
Uncertainty in measurements
• A digit that must be estimated in a
measurement is called uncertain.
• A measurement always has some degree of
uncertainty. It is dependent on the precision of
the measuring device.
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Uncertainties and significant figures
Significant Figures
• In science, measured values are reported in terms
of significant figures.
• Significant figures in a measurement consist of all
the digits known with certainty plus one final
digit, which is uncertain or is estimated.
• The term significant does not mean certain.
• Insignificant digits are never reported.
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Uncertainties and significant figures
What value should be
recorded for the
length of this nail?
6.35
Uncertain or estimated number
but significant.
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Uncertainties and significant figures
Rules for Counting Significant Figures
1. Nonzero integers always count as significant
figures.
– 3456 has 4 sig figs (significant figures).
2. There are three classes of zeros.
a. Leading zeros are zeros that precede all the
nonzero digits. These do not count as significant
figures.
 0.048 has 2 sig figs.
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Uncertainties and significant figures
Rules for Counting Significant Figures
Classes of zeros.
b. Captive zeros are zeros between nonzero digits.
These always count as significant figures.
– 16.07 has 4 sig figs.
c. Trailing zeros are zeros at the right end of the
number. They are significant only if the number
contains a decimal point.
– 9.300 has 4 sig figs.
– 150 has 2 sig figs.
– 120. has 3 sig figs.
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Uncertainties and significant figures
Rules for Counting Significant Figures
3. Exact numbers have an infinite number of
significant figures.
– 1 inch = 2.54 cm, exactly.
– 9 pencils (obtained by counting).
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Uncertainties and significant figures
Sample Problem
How many significant figures are in each of the following
measurements?
a. 28.6 g
3 sig figs,
no zeros, all digits are significant
b. 3440. cm
4 sig figs,
zero significant, followed by a decimal point
c. 910 m
2 sig figs,
the zero is not significant, no decimal point
d. 0.046 04 L 4 sig figs, the first two zeros are not significant; the
third zero is significant.
e. 0.0067000 kg 5 sig figs, the first three zeros are not significant;
the last three zeros are significant.
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Uncertainties and significant figures
Significant Figures in Mathematical Operations
1. For multiplication or division, the number of
significant figures in the result is the same as the
number in the least precise measurement used in
the calculation.
1.342 × 5.5 = 7.381  7.4
2 sig figures
2 sig figures
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Uncertainties and significant figures
Rounding
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Uncertainties and significant figures
Significant Figures in Mathematical Operations
2. For addition or subtraction, the result has the same
number of decimal places as the least precise
measurement used in the calculation.
23.445

7.83
31.275
Corrected
31.28

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Uncertainties and significant figures
CONCEPT CHECK!
You have water in each graduated
cylinder shown. You then add both
samples to a beaker (assume that all
of the liquid is transferred).
How would you write the number
describing the total volume?
2.8 + 0.28 = 3.1 mL
What limits the precision of the total
volume?
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Uncertainties and significant figures
Exponential Notation
• Example
– 300. written as 3.00 × 102
– Contains three significant figures.
• Two Advantages
– Number of significant figures can be easily
indicated.
– Fewer zeros are needed to write a very large or
very small number.
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