Significant
Figures
If the error in a measurement can only be estimated to the nearest
order of magnitude, it may be most convenient to include the error
implicitly, by writing only the appropriate number of significant
figures in the reported result. The rightmost significant figure
in a result should be the only one containing any uncertainty. Or,
if you want to be quite conservative, it should instead be the last
digit known with certainty. For example, if the raw measurement
(or calculator display) is 1.70784, and the four digits on the right
(0784) are not known exactly, you should either write your result
to three significant figures, as 1.70, or to two significant figures,
as 1.7.
The number of significant figures gives a rough approximation of
the fractional error in the measurement. The unit size of the least
significant digit is an indication of the absolute error. In the
first example above, the unit size of the least significant digit
is 0.01; this corresponds to the reported absolute error. The reported
fractional error, then, is 0.01/1.70 = 0.6%. When collecting data
and performing calculations, it's a good idea to use a few more
digits than are actually significant, to avoid losing information
unnecessarily. All reported results, however, should either show
explicit error estimates, or be rounded to the proper number of
significant figures.
As discussed in the lab manual introduction, when multiplying two
numbers together, the general rule of thumb is to write the answer
using the same number of significant figures as the multipliers.
When the multipliers have different numbers of significant figures,
the smallest is used. Put another way, the fractional error in the
product will be of the same order of magnitude as the largest of
the fractional errors in the numbers you started with. Thus 0.3526
x 1.2 = 0.42 (not 0.42312). This same method should be used for
division.
Addition is different. Consider the example: 0.2056 + 14.25 + 576.1
= 593.1. An answer of 593.1276 is not appropriate because the last
three digits (.0276) add nothing to the accuracy of the results,
since one of the numbers being added (576.1) is accurate only to
tenths. Here the absolute error in the sum will be of the same order
of magnitude as the largest of the absolute errors in the original
numbers. Subtraction works similarly.
Rules for counting significant figures in a number:
 The leftmost nonzero digit is the first significant figure.
 If there is no decimal point, the rightmost nonzero digit is
the last significant figure.
 If there is a decimal point, the rightmost digit is significant,
zero or not.
 Any digits between significant figures are also significant.
