Transcript Significant Figures

Significant Figures
Addition and Subtraction while minding the “sig figs.”
A quick review
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Recall that all measurements have inherent uncertainty
due to
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a) limits in the accuracy of the measuring device and
b) variations in the estimations made by those doing the
measuring.
Also recall how to count significant figures, based on
whether or not a decimal point is present.
Significant figures apply to MEASURED VALUES!
A problem…
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Suppose that you are told to find the
perimeter of a rectangular room.
Further suppose that the dimensions of
the room are measured by two
different methods (for some reason),
which are reported as 5.1 meters and
8.27 meters, respectively.
What are the estimated values in each
of these numbers? Which number
expresses more confidence?
The ‘1’ and the ‘7’ are estimated values
and the 8.27 measurement expresses
more confidence than the 5.1
measurement.
5.1 m
8.27 m
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Perimeter?
Problem Continued…
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So what’s the perimeter of the room?
If you punch into your calculator
8.27+8.27+5.1+5.1, your calculator will
report a perimeter of 26.74 meters.
But you are much smarter than your
calculator, which doesn’t understand a
thing about significant figures.
You know that the ‘7’ in 8.27 m is
estimated, which is to say that the
length could be as low as 8.26 or as
high as 8.28.
Likewise, “5.1 meters” represents a
range from 5.0 meters to 5.2 meters.
5.1 m
8.27 m
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Perimeter?
(sum of the
sides)
Why not to trust your calculator
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If you report your perimeter as 26.74 meters, then you
are expressing more confidence in you sum of numbers
than you are in either of your starting numbers (5.1 and
8.27).
You would be stating that the perimeter of the room is
certainly between 26.73 meters and 26.75 meters.
This is wrong, though. It could actually be as low as
26.52 meters or as high as 26.96 meters. How did I get
those numbers?
I simply added the low ends and high ends of the ranges
together, respectively.
So what to report?
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When adding and subtracting measured values, we need
to be mindful of the least certain starting number re:
decimal position.
In other words, we need to round our final answer to the
same number of decimal places as there are in the
measurement with the smallest number of decimal
places.
Doing so allows us to ‘hedge our bets’ about the
accuracy of the number we report. It casts a bigger net
in an attempt to capture the true/actual answer.
In the case of the perimeter of the room, we would
report a value of 26.7 meters because it is expressed to
the tenths place, as was 5.1 meters.
Some examples…
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Give the final answer of the following
addition/subtraction problems, assuming the values
represent measurements.
5.001 + 2.1 + 8.09 = ?
The least certain position in our numbers is the tenths
position in 2.1. Therefore, our answer needs to be
expressed to the tenths place.
15.2 (not 15.191, as your calculator reports)
What about 1,200 + 5.1 + 75?
1300 (because 1280.1 rounded to the same level as the
least certain starting number, is 1300).
The same rule applies to subtraction!
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Try these… answers on next slide
Q1: 23.5 – 18 – 1.46 =
Q2: 72.1 + 100.33 - 0.003=
Q3: 0.00058 + 0.00009 + 0.0002 =
Q4: 2,100 – 189 – 12 =
Answers to the previous slide…
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A1: 4 (because of the 18)
A2: 172.4 (because of the 72.1)
A3: 0.0009 (because of the 0.0002)
A4: 1,900
Some trickier ones… again, answers are on
next slide.
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Q5: 4010. + 25.33 – 8.2 =
Q6: 800. + 500 – 3.1 x 102 =
Q7: 12.2 – 13.50 – 0.0007 =
More answers…
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A5: 4027 (note that there is a decimal point present in
4010., which means that all four figures are significant.)
A6: 1000
A7: -1.3
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How fun was that?!?!?
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