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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.

A practical calculation that uses any irrational number necessitates rounding the number, and hence the answer, to a finite number of significant figures. 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.

Identifying significant digits

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.
• The significance of trailing zeros in a number not containing a decimal point can be ambiguous. For example, it may not always be clear if a number like 1300 is accurate to the nearest unit (and just happens coincidentally to be an exact multiple of a hundred) or if it is only shown to the nearest hundred due to rounding or uncertainty. Various conventions exist to address this issue:
• A bar may be placed over the last significant digit; any trailing zeros following this are insignificant. For example, $13 \bar{0} 0$ has three significant figures (and hence indicates that the number is accurate to the nearest ten).
• The last significant figure of a number may be underlined; for example, "20000" has two significant figures.
• A decimal point may be placed after the number; for example "100." indicates specifically that three significant figures are meant.[1]

However, these conventions are not universally used, and it is often necessary to determine from context whether such trailing zeros are intended to be significant. If all else fails, the level of rounding can be specified explicitly. The abbreviation s.f. is sometimes used, for example "20 000 to 2 s.f." or "20 000 (2 sf)". Alternatively, the uncertainty can be stated separately and explicitly, as in 20 000 ± 1%, so that significant-figures rules do not apply.

Scientific notation

Generally, the same rules apply to numbers expressed in scientific notation. However, in the normalized form of that notation, placeholder leading and trailing digits do not occur, so all digits are significant. For example, 0.00012 (two significant figures) becomes 1.2×10−4, and 0.000122300 (six significant figures) becomes 1.22300×10−4. In particular, the potential ambiguity about the significance of trailing zeros is eliminated. For example, 1300 to four significant figures is written as 1.300×103, while 1300 to two significant figures is written as 1.3×103.

Engineering notation

See main article on Engineering notation

Rounding

To round to n significant figures:

• Start with the leftmost non-zero digit (e.g. the '1' in 1200, or the '2' in 0.0256).
• Keep n digits. Replace the rest with zeros.
• Round up by one if appropriate. For example, if rounding 0.039 to 1 significant figure, the result would be 0.04. There are several different rules for handling borderline cases — see rounding for more details.
• Officially[2] if the first not significant figure is a 5 not followed by any other digits or followed only by zeros, the last significant figure should be rounded down or up to the nearest even number. So to round 1.25 to 2 significant figures should be 1.2 and 1.35 should be 1.4. If the first non-significant digit is a 5 followed by other non-zero digits, the last significant digit should be rounded up. For example, 1.2459 as the result of a calculation or measurement that only allows for 3 significant figures should be written 1.25.

Arithmetic

For multiplication and division, the result should have as many significant figures as the measured number with the smallest number of significant figures.

For addition and subtraction, the result should have as many decimal places as the measured number with the smallest number of decimal places.

Importance

Superfluous precision

If a sprinter is measured to have completed a 100-metre race in 11.71 seconds, what is the sprinter's average speed? By dividing the distance by the time using a calculator, we get a speed of 8.53970965 m/s.

The most straightforward way to indicate the precision of this result (or any result) is to state the uncertainty separately and explicitly, for example in the above case as 8.5397±0.0037 m/s or equivalently 8.5397(37) m/s. This is particularly appropriate when the uncertainty itself is important and precisely known (here, 100 m is presumed to be precise, and the time is 11.71±0.005 s, or an uncertainty of nearly 430 ppm). In this case, it is safe and indeed advantageous to provide more digits than would be called for by the significant-figures rules.

If the degree of precision in the answer is not important, it is again safe to express trailing digits that are not known exactly, for example 8.5397 m/s.

If, however, we are forced to apply significant-figures rules, expressing the result as 8.53970965 m/s would seem to imply that the speed is known to the nearest 10 nm/s or thereabouts, which would improperly overstate the precision of the measurement. Reporting the result using three significant figures (8.54 m/s) might be interpreted as implying that the speed is somewhere between 8.535 and 8.545 m/s. This is actually very close to the true precision, the actual speed being somewhere between 8.5360 and 8.5434 m/s. Reporting the result using two significant figures (8.5 m/s) would introduce considerable roundoff error and degrade the precision of the result.

(Note: in actual practice, 100 m is not this precise! For example, a pair of 0.05-metre-wide (2-inch) lines at the start and end would introduce a separate uncertainty of ±0.05 m (2 in) or 500 ppm to the above calculation. Now the total uncertainty has risen to 500 + 430 = 930 ppm, since both sources must be added together. Applied to the speed, that now becomes 8.5397±0.0080 or 8.5397(80) m/s, the actual speed being somewhere between 8.5317 and 8.5477 m/s.)

Comprehension

Numbers are often rounded off to make them easier to read. It's easier for someone to compare (say) 18% to 36% than to compare 18.148% to 35.922%. Similarly, when reviewing a budget, a series of figures like:

Division A: $185 000 Division B:$ 45 000
Division C: $67 000  is easier to understand and compare than a series like: Division A:$184 982
Division B: $44 689 Division C:$ 67 422


To reduce ambiguity, such data are sometimes represented to the nearest order of magnitude, like:

Revenue (in thousands of dollars):
Division A: 185
Division B:  45
Division C:  67


Least Significant

The last digit is the least significant digit. This is sometimes referred to in product manuals specifying accuracy. For example, an accuracy of +-(.3%+2) for 73.6 volts means there is an uncertainty of +-.42 volts in the nominal value. For the same accuracy, an uncertainty of .2 volts would exist for a reading of zero, and an uncertainty of .5 for a reading of 100 volts.

References

1. ^ Myers, R. Thomas; Oldham, Keith B.; Tocci, Salvatore.. "2" (Textbook). Chemistry. Austin, Texas: Holt Rinehart Winston. p. 59. ISBN 0-03-052002-9.
2. ^ http://www.chem.sc.edu/faculty/morgan/resources/sigfigs/sigfigs6.html