A brief tutorial on significant figures
Suppose you want to measure the hypotenuse of a right triangle, and further suppose that you
cannot measure it directly.
Perhaps it is a few cm longer than your ruler, or it is too delicate
to put a ruler next to or some other contrived reason why you cannot measure it directly.
You
can calculate the length of that hypotenuse if you measure the two other legs of the triangle,
and we will suppose that these can be measured.
So, intrepid explorer that you are, you measure the legs of this triangle (using a ruler
incremented in millimeters) and arrive at the following two measurements:
The horizontal leg
is 10.25 cm and the vertical leg 7.35 cm.
(How to deal with the unstated uncertainties
associated with these measurements will be discussed later.)
We can now use the theorem of
Pythagoras to calculate the length of our hypotenuse.
Figure 1
222
CA
B
=+
, or
22
B
.
See figure 1 above.
This results in
(10.25
)
(7.35
)
cc
m
c
m
.
Using our scientific calculator, it returns an
answer for c of 12.6128902 cm.
Now, if we tell someone that the hypotenuse of that triangle
is 12.6128902 cm, they would be justified in thinking that we spent great efforts to measure
that hypotenuse to within a few wavelengths of visible light, because we are telling them that
we know something about the 10
9
meter decimal place with the answer we’ve given.
Clearly, a plastic meter stick (see figure 2) from the local office supply company is not
capable of making measurements with that sort of precision.
What we need, then, is a set of
standards to guide us in how to round our calculated answers so that we are not alleging to
know things that we cannot possibly know.
This is where proper significant figure techniques
come into play.
Ruler
Fig.2
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View Full DocumentWhat defines significant figures?
Ultimately, the number digits we report in our final answer (in other words, how many digits
we round to) is determined by how many digits are in the inputs to our calculation, and how
large our uncertainty is.
In order to determine the number of significant figures in a number,
we need to understand what a significant figure is, and we need to understand how to count
them.
What is a significant figure?
Informally, it is any digit in a reported number that we have
some degree of certainty about.
For example, if I tell you a length is 7.35 cm, I am telling you
that I have some degree of confidence that I know each digit of that length.
I know that the
length exceeds 7 cm, but not 8 cm.
I know that it exceeds 7 cm by about 0.3.
I also know that
it is half way between 7.3 cm and 7.4 cm.
But how do we decide how many significant figures there are in a number?
We have some
simple rules for determining this.
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 Spring '08
 Staff
 Physics, Decimal, 2 cm, 1 mm, 7.3 cm

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