The simplest way to indicate the precision of a measurement is by the number of digits that you write down when you record it. In the example above, notice that the measurement was recorded as 25.45 cm - not 25.4 cm, or 25.5 cm, or 25.52 cm. The number of digits that you write down for a measurement is called the number of significant digits (or significant figures) in the measurement. Scientists understand that the last digit (and only the last digit) in a measurement is an estimate. By writing 25.45 cm, you indicate that you are sure that the measurement was between 25.4 cm and 25.5 cm, and you estimate that it was about 5/10 of the way between them.
25.45 cm is the best estimate (so far) for the length of a pendulum. We could take a step beyond significant digits and estimate the range of values that we believe the "true length" lies within. In the diagram above, you could certainly tell if the measurement were smaller than about 25.43 cm or larger than 25.47 cm. Therefore, you are confident that the "true value" lies within 0.2 mm (0.02 cm) of 25.45 cm. You can indicate this by writing the measurement as
This is why "measurements are not numbers". A physical measurement consists of the best estimate of the "true value" and the range in which the "true value" probably lies. How did I come up with the 0.2 mm range above? Well, by looking at the diagram above, and thinking about it. It seemed like a reasonable value to me. No, there is no set of rules that will tell you what range to use (although there are some ways to get suggestions...). You have to pick a range that seems justifiable to you, and then be ready to justify it.
Determining the number of significant digits (also called significant figures) in a number is pretty easy once you catch on. There are only three rules to know. The only "tricky" digits are zeros...
(Significant Digits applet temporarily removed)
The basic ideas are:
Suppose you need to add the three lengths 1.6 m (2 significant digits), 14.32 m (4 significant digits), and 8.014 m (4 significant digits). Your calculator will tell you the sum is 23.934 m (5 significant digits). How many digits should you keep in this answer? Certainly we can't keep all 5 digits - the sum of a group of measurements can't be more precise than the measurements that were added to produce it!
This problem is illustrated at left as it would be done "by hand".
The estimated digit in each measurement is circled in red. Notice
that all three digits to the right of the decimal point are
estimates, since they were found by adding at least one estimated
digit. In order to keep just one estimated digit, the answer must be
rounded to the tenths place. The sum is 23.9 m.
subtraction problem "86.34 cm - 9.1 cm" is shown at left. Notice that
the calculator's answer - 79.24 cm - carries 2 estimated digits. The
answer must be rounded to 79.2 cm.
The General Rule for Adding And Subtracting Measurements:
When adding or subtracting measurements, find the left-most decimal place that contains an estimated digit. Round your answer to this decimal place.
In each example above one of the measurements had an estimate in the tenths place. Therefore, the answer needed to be rounded to the tenths place.
a rectangle is 24.3 m long and 2.3 m wide. What is the area of the
rectangle? Your calculator says 55.89 m2, but you can't
get a four-significant-digit result from a two-significant digit and
a three-significant-digit measurement. Look at the problem as it is
displayed at the left. All three digits of the intermediate result
"729" must be estimates, since they are produced by multiplying by an
estimated digit. All other estimated digits in the problem are
circled in red. In order to keep just one estimated digit, the
product must be rounded to 56 m2.
The General Rule for Multiplying and Dividing Measurements:
When multiplying or dividing measurements, the answer should contain the same number of significant digits as the measurement with the least number of significant digits.
In the example above, one of the measurements contained 2 significant digits, and the other measurement contained 3 significant digits. The answer must be rounded to 2 significant digits.