Measurements and Significant Figures/Digits
Note: There are many references on the important subject of significant figures (or digits) and
physical measurements. The following was adapted from a Wellesley College Internet site
(www.astro.wellesley.edu/kmcleod/toolkit/sigfigs) that is no longer in operation.
Introduction
If you take your calculator and multiply 1.378 times 2.3 you will get 3.1694 as a result. If you
divide 3.7 by 1.336 you will get 2.769461078. These results are "correct" in a pure mathematical
sense that assumes you know the values of the initial numbers exactly. That is, that 3.7 is
actually 3.7000000000... and that 1.378 is actually 1.378000000000... and so on. In the real
world when we make measurements of anything, the value we get is not known exactly, but
rather has some uncertainty associated with it. How large this uncertainty is depends to a high
degree on the type of measuring device used as well as how it is used.
For example, suppose that three people were told to determine the length of a piece of wood and
were given a tape measure whose smallest markings were at 0.1-centimeter intervals:
They report the following values:
Person
Value measured for length
1
11.6 cm
2
11.6283476 cm
3
11.63 cm
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Who is right? The correct question is "who has quoted a value of the proper accuracy?" Person 1
has been overly conservative because it is possible to estimate how far between the 0.1 cm marks
the edge of the wood lies. Person 2 is being silly, because it is impossible to estimate such small
distances by eye. Person 3 has made the best estimate of where the edge lies. No precision is lost
as in case 1, and no unrealistic precision is claimed as in case 2. We’ve assumed here that one
can measure precisely to one-tenth of the smallest markings on the ruler—in this case, that
means to the nearest 0.01 cm. This example illustrates the general concept of significant figures
(hereafter called SF) and the precision of the least significant digit.
So Person 3’s length measurement of 11.63 cm has 4 SF, with the least significant digit in the
hundredths place. Now Person 3 measures the width of the same piece of wood and finds it to be
5.74 cm. This result has the same precision as the previous measurement (both are given to the
hundredths place), but only has three SF. Similarly, her thickness measurement of 0.42 cm has
the same precision but only two SF.
The next sections will explain rules for determining how many SF are in a number and how
many SF to keep when you do calculations on numbers derived from measurements.
Determining the Number of Significant Figures
Here are a few rules to help you determine how many SF are in a given number, and which of
the digits is the least significant one:
1. Integers are exact and are considered to have an infinite number of SF; i.e. 2 is
understood to be 2.0000000…
2. Numerical constants such as pi can have as many SF as you need, e.g. 3.14, 3.14159,
3.14159265…
3. If the number is neither an integer nor a numerical constant, then the number of SF is
equal to the number of digits excluding certain zeros that act only as "placeholders." In a
number with a decimal point, any leading zeros are placeholders and are not significant,
but trailing zeros in this case are significant. In a number without a decimal point, such as
2100, trailing zeros might or might not be significant; you can only tell from the context.
To make such cases clear, many scientists use scientific notation:
2.1 x 102
2.10 x 102
2.100 x 102
implies 2 SF
implies 3 SF
implies 4 SF
In any case, the least significant digit is then the significant digit that is farthest to the right. For
our purposes, the precision of any quoted value can be assumed to be ± 1 of the least significant
digit, unless stated otherwise.
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