Significant Figures

The significant figures (also called significant digits and abbreviated sig figs, sign.figs, sig digs or s.f.) of a number are those digits that carry meaning contributing to its precision (see entry for Accuracy and precision). This includes all digits except:

leading and trailing zeros where they serve merely as placeholders to indicate the scale of the number.

spurious digits introduced, for example, by calculations carried out to greater accuracy than that of the original data, or measurements reported to a greater precision than the equipment supports.

The concept of significant figures is often used in connection with rounding. Rounding to n significant figures is a more general-purpose technique than rounding to n decimal places, since it handles numbers of different scales in a uniform way. For example, the population of a city might only be known to the nearest thousand and be stated as 52,000, while the population of a country might only be known to the nearest million and be stated as 52,000,000. The former might be in error by hundreds, and the latter might be in error by hundreds of thousands, but both have two significant figures (5 and 2). This reflects the fact that the significance of the error (its likely size relative to the size of the quantity being measured) is the same in both cases.

Computer representations of floating point numbers typically use a form of rounding to significant figures, but with binary numbers.

The term "significant figures" can also refer to a crude form of error representation based around significant figure rounding; for this use, see Significance arithmetic.

The rules for identifying significant digits when writing or interpreting numbers are as follows:

All non-zero digits are considered significant. For example, 91 has two significant figures (9 and 1), while 123.45 has five significant figures (1, 2, 3, 4 and 5).

Zeros appearing anywhere between two non-zero digits are significant. Example: 101.12 has five significant figures: 1, 0, 1, 1 and 2.

Leading zeros are not significant. For example, 0.00052 has two significant figures: 5 and 2.

Trailing zeros in a number containing a decimal point are significant. For example, 12.2300 has six significant figures: 1, 2, 2, 3, 0 and 0. The number 0.000122300 still has only six significant figures (the zeros before the 1 are not significant). In addition, 120.00 has five significant figures. This convention clarifies the precision of such numbers; for example, if a result accurate to four decimal places is given as 12.23 then it might be understood that only two decimal places of accuracy are available. Stating the result as 12.2300 makes clear that it is accurate to four decimal places.