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:
The concept of significant figures is often used in connection with rounding. Rounding to n significant figures is a more generalpurpose 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.
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The rules for identifying significant digits when writing or interpreting numbers are as follows:
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 significantfigures rules do not apply.
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 1.300×10^{3}, while 1300 to two significant figures is written as 1.3×10^{3}.
See main article on Engineering notation
To round to n significant figures:
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.
If a sprinter is measured to have completed a 100metre 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 significantfigures 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 significantfigures 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 0.05metrewide (2inch) 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.)
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
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.
