Significance arithmetic is a set of rules (sometimes called significant figure rules) for approximating the propagation of uncertainty in scientific or statistical calculations. These rules can be used to find the appropriate number of significant figures to use to represent the result of a calculation. If a calculation is done without analysis of the uncertainty involved, a result that is written with too many significant figures can be taken to imply a higher precision than is known, and a result that is written with too few significant figures results in an avoidable loss of precision. Understanding these rules requires a good understanding of the concept of significant and insignificant figures.
The rules of significance arithmetic are an approximation based on statistical rules for dealing with probability distributions. See the article on propagation of uncertainty for these more advanced and precise rules. Significance arithmetic rules rely on the assumption that the number of significant figures in the operands gives accurate information about the uncertainty of the operands and hence the uncertainty of the result. For alternatives see Interval arithmetic and Floating-point error mitigation.
An important caveat is that significant figures apply only to measured values. Values known to be exact should be ignored for determining the number of significant figures that belong in the result. Examples of such values include: integer counts (e.g. the number of oranges in a bag) definitions of one unit in terms of another (e.g. a minute is 60 seconds) actual prices asked or offered, and quantities given in requirement specifications legally defined conversions, such as international currency exchange scalar operations, such as "tripling" or "halving" mathematical constants, such as π and e
Physical constants such as the gravitational constant, however, have a limited number of significant digits, because these constants are known to us only by measurement. On the other hand, c (the speed of light) is exactly 299,792,458 m/s by definition.
Multiplication and division using significance arithmetic
When multiplying or dividing numbers, the result is rounded to the number of significant figures in the factor with the least significant figures. Here, the quantity of significant figures in each of the factors is important—not the position of the significant figures. For instance, using significance arithmetic rules:
8 × 8 ≈ 6 × 101
8 × 8.0 ≈ 6 × 101
8.0 × 8.0 ≈ 64
8.02 × 8.02 ≈ 64.3
8 / 2.0 ≈ 4
8.6 / 2.0012 ≈ 4.3
2 × 0.8 ≈ 2
If, in the above, the numbers are assumed to be measurements (and therefore probably inexact) then "8" above represents an inexact measurement with only one significant digit. Therefore, the result of is rounded to a result with only one significant digit, i.e., instead of the unrounded "64" that one might expect. In many cases, the rounded result is less accurate than the non-rounded result; a measurement of "8" has an actual underlying quantity between 7.5 and 8.5. The true square would be in the range between 56.25 and 72.25. So is the best one can give, as other possible answers give a false sense of accuracy. Further, the is itself confusing (as it might be considered to imply , which is over-optimistic; more accurate would be ).
Addition and subtraction using significance arithmetic
When adding or subtracting using significant figures rules, results are rounded to the position of the least significant digit in the most uncertain of the numbers being added (or subtracted). That is, the result is rounded to the last digit that is significant in each of the numbers being summed. Here the position of the significant figures is important, but the quantity of significant figures is irrelevant. Some examples using these rules are: