When are Digits Significant?
Non-zero digits are always significant. Thus, 22 has two significant digits, and 22.3 has three significant digits.
With zeroes, the situation is more complicated: a. Zeroes placed before other digits are not significant; 0.046 has two significant digits. b. Zeroes placed between other digits are always significant; 4009 kg has four significant digits. c. Zeroes placed after other digits but behind a decimal point are significant; 7.90 has three significant digits. d. Zeroes at the end of a number are significant only if they are behind a decimal point as in (c). Otherwise, it is impossible to tell if they are significant. For example, in the number 8200, it is not clear if the zeroes are significant or not. The number of significant digits in 8200 is at least two, but could be three or four. To avoid uncertainty, use scientific notation to place significant zeroes behind a decimal point:
8.200  103 has four significant digits
8.20  103 has three significant digits
8.2  103 has two significant digits
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)