Suppose that during the process of an experiment you are confronted with the
task of determining the circumference of a pulley. You measure the diameter
of the pulley with a good ruler whose smallest division is 1/10 inch and
record the diameter as 4.15 inches. Taking
and multiplying out
,
you write down: Circumference equals 13.037640 inches.
Arithmetically the result is correct. Physically it is misleading.
Confronted with the ruler, you would readily admit that you could not
estimate with certainty to one-hundredth of an inch, and that the diameter
of the pulley might well be 4.14 inches instead of 4.15 inches. If, however,
the diameter of the pulley is taken as 4.14 inches, the circumference would
come out according to this calculation as 13.006229 inches. Comparing this
result with the former, you find that they differ in the fourth figure.
Which answer is correct? You really do not know, and in fact using the apparatus mentioned no one would be willing to say. Another observer might estimate the diameter to be 4.16 inches and a third ``circumference" would be obtained. Because of this uncertainty the physicist says that all the answers are wrong since they explicitly express an accuracy in the determination of the circumference which is not warranted by the precision in the measurement of the diameter. The physicist says that some of the numbers written down are significant figures and that the rest are meaningless.
How, then, are we to determine the number of significant figures in a result?
For use in the analysis of the results of carefully performed experiments
there are statistical methods available which allow a prediction to be made
of the precision. One such system is outlined in Experiment 1. We shall not
use such a system here but content ourselves instead with a less exact method.
Returning to our example, we assume that after a number of trial measurements
we would be willing to state that the diameter of the pulley is
inches (to be read ``4.15 plus or minus 0.01 inches"). In other words we are
willing to state that the measured diameter is correct to about
%.
As will be shown later, the determination of the circumference
is then only accurate to
%
and should be written as
inches.
[ Another convention, not so universally respected, is to always place a leading zero before the decimal point in expressing a number less than one. For instance, we try to avoid writing ``.1" rather than ``0.1" because the leading decimal point can too easily be overlooked in a cursory glance. ]
The experimenter should recognize that the accuracy of the result is
independent of the position of the decimal point in the result. Changing
the units in which the result is expressed can in no way alter the accuracy
of the result. The term ``significant figures" is often used to give this
idea: a result accurate to four significant figures would be one in which
the observer relies on the first four figures counting always from the first
number at the left, other than zero. As an illustration, if a certain
distance is known to be 15.1 meters, accurate to about 1 percent -- that is,
to three significant figures -- then the same result would be 0.0151
kilometers or
millimeters,
using as before three significant figures.
A frequent misunderstanding arises in the case where the last significant figure is zero. A measurement made with a centimeter rule of the diameter of a cylinder might be 7.10 cm. Here the 0 is a significant figure; the observer could have seen if the measurement had been 7.09 or 7.11. A record of 7.1 cm would indicate that the observer had failed to read as closely as possible, but on the other hand a record of 7.10000 cm would not truthfully represent the observation, since it could not have been possible to observe with the ruler what figures occupied the fourth, fifth and sixth places.
In arithmetical computation, insignificant figures may be dropped at each stage of the computation. The computation of the volume of the cylinder whose dimensions are given below will serve as an illustration.
The number 7.138 is accurate to 1/9 percent, therefore every time this number is used as a factor it may introduce an error of 1/9 percent, or the fourth figure will be uncertain. As a consequence, we may omit all figures beyond the fourth which come in as a result of multiplication. Thus we continue the calculation:
[ Several other conventions are implicit here: first, units are suppressed until the end. This is a time-saving convention which sometimes backfires - in more dimensionally elaborate calculations it is often worth carrying along the units at every step to ensure that no step is ``dimensionally absurd" and to aid in locating errors when the final answer doesn't make sense. Second, integers (like 4) are exempt from the significant figures convention, for obvious reasons. We could write 4.000 instead, but that would be silly. ]
The procedure for assigning uncertainty to the result will be considered in Experiment 1.