# Significant Figures

The significant figures of a (measured or calculated) quantity are the meaningful digits in it. There are conventions which you should learn and follow for how to express numbers so as to properly indicate their significant figures.

- Any digit that is not zero is significant. Thus 642 has three significant figures and 6.792 has four significant figures.
- Zeros between non zero digits are significant. Thus 4023 has four significant figures.
- Zeros to the left of the first non zero digit are not significant. Thus 0.000072 has only two significant figures. This is more easily seen if it is written as 7.2 x10
^{-5}. - For numbers with decimal points, zeros to the right of a non zero digit are significant. Thus 6.00 has three significant figures and 0.050 has two significant figures. For this reason it is important to keep the trailing zeros to indicate the actual number of significant figures.
- For numbers without decimal points, trailing zeros may or may not be significant. Thus, 400 indicates only one significant figure. To indicate that the trailing zeros are significant a decimal point must be added. For example, 400. has three significant figures, and has one significant figure.
- Exact numbers have an infinite number of significant digits. For example, if there are two oranges on a table, then the number of oranges is 2.000... . Defined numbers are also like this. For example, the number of centimeters per inch (2.54) has an infinite number of significant digits, as does the speed of light (299792458 m/s).
- There are also specific rules for how to consistently express the uncertainty associated with a number. In general, the last significant figure in any result should be of the same order of magnitude (i.e.. in the same decimal position) as the uncertainty. Also, the uncertainty should be rounded to one or two significant figures. Always work out the uncertainty after finding the number of significant figures for the actual measurement.

For example:

9.82 ± 0.02

10.0 ± 1.5

4 ± 1

The following numbers are all **incorrect**.

9.82 ± 0.02385 is wrong but 9.82 ± 0.02 is fine

10.0 ± 2 is wrong but 10.0 ± 2.0 is fine

4 ± 0.5 is wrong but 4.0 ± 0.5 is fine

In practice, when doing mathematical calculations, it is a good idea to keep one more digit than is significant to reduce rounding errors. But in the end, the answer must be expressed with only the proper number of significant figures. After addition or subtraction, the result is significant only to the place determined by the largest last significant place in the original numbers. For example,

89.332 + 1.1 = 90.432

should be rounded to get 90.4 (the tenths place is the last significant place in 1.1). After multiplication or division, the number of significant figures in the result is determined by the original number with the smallest number of significant figures. For example,

(2.80) (4.5039) = 12.61092

should be rounded off to 12.6 (three significant figures like 2.80).