Chapter 2 Significant Figures and Meaningful Data
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Chapter 2
(S 20)
SIGNIFICANT FIGURES AND MEANINGFUL DATA
Significant figures is the standard method used to round off an answer.
Rounding off should
NEVER interfere with solving a problem, and should only be considered after solving the problem.
The
rounded number indicates how important a number is, how much trust we can have in the number.
The more significant figures or digits, the more trust we have in the number.
A number with only a
few significant figures is not very trustworthy.
SIGNIFICANT FIGURES AND UNCERTAINTY
Significant figures are all measured digits in a reported number.
This includes all reported
integers. For example, 34.56 mm contains four significant figures.
Confusion may be encountered
when the integer is 0, which occurs one out of ten times.
For example, 34.560 mm contains five
significant figures.
The last zero was reported because it was measured and is thus significant.
In
contrast, 0.03456 mm contains four significant figures since the first two zeros are used only to
show where the decimal place is.
Putting such a number into scientific notation clears up any
confusion, 3.456 x 10
-2
mm clearly contains four significant figures.
The last significant figure in a
number contains uncertainty but is still significant.
Uncertainty is the +/- in a reported number.
Uncertainty is found in the last significant figure and
is understood to exist even if no one mentions it.
For example, the uncertainty in 34.56 mm occurs
in the hundreds place and is designated as
±
0.01 mm, or in scientific notation as 1 x 10
-2
mm.
How you write the number conveys information about how much trust you can put into the
number.
More significant figures and smaller uncertainty correlate with more trust.
When you
examine the following examples you will learn that 200.00 mm is a more trustworthy number than
200 mm:
number
uncertainty
meaning
# sig figs
200.00 mm
±
0.01 mm
199.99 mm, 200.00 mm, or 200.01 mm
5
200.0 mm
±
0.1 mm
199.9 mm, 200.0 mm, or 200.1 mm
4
200. mm
±
1 mm
199 mm, 200. mm, or 201 mm
3
200 mm
±
100 mm
100 mm, 200 mm, or 300 mm
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Notice the difference in meaning between 200. mm and 200 mm.
Without the terminal decimal
point, 200 mm is considered an almost useless number.
If a person doesn’t know about the meaning
of the decimal point and left it out, his number will be interpreted in the most conservative manner
and would be taken as insignificant or unworthy.
How would you show 200 m with 2 significant figures ?
Here are four methods:
_
i.
200 m, with the line over non-significant placeholders
ii.
20(0) m, with the parenthesis indicating non-significant integers
iii. 2.0 x 10
2
m, scientific notation leaves no doubt about the number of significant figures
iv. 200 m
±
10 m, the uncertainty indicates which are significant
Here are some examples using small numbers:
number
uncertainty
meaning
# sig figs
0.2 m
±
0.1 m
0.1 m, 0.2 m, or 0.3 m
1
0.002 m
±
0.001 m
0.001 m, 0.002 m, or 0.003 m
1
0.0020 m
±
0.0001 m
0.0019 m, 0.0020 m, or 0.0021 m
2
0.0002020 m
±
0.0000001 m 0.0002019 m, 0.0002020 m, or 0.0002021 m
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