# Arithmetic precision

measure of random errors in a system: the variation among measurements of the same value

The precision of a numeric value describes the number of digits that are used to show that value. In a scientific setting, this would be the total number of digits (sometimes called the significant figures or significant digits) or, less commonly, the number of fractional digits or decimal places (the number of digits following the decimal point). This second definition is useful in financial and engineering applications where the count of digits in the fractional part has particular importance.

In both cases, the term "precision" can be used to describe the position at which an inexact result will be rounded. For example, in floating point arithmetic, a result is rounded to a given or fixed precision, which is the length of the resulting significand. In financial calculations, a number is often rounded to a given number of places (for example, to two places after the decimal separator for many world currencies).

As an example, the decimal quantity 12.345 can be expressed with various numbers of significant digits or decimal places. If insufficient precision is available then the number is rounded in some manner to fit the available precision. The following table shows the results for various total precisions and decimal places rounded to the nearest value using the round-to-even method.

Precision
Rounded to
significant digits
Rounded to
decimal places
Five 12.345 12.34500
Four 12.34 12.3450
Three 12.3 12.345
Two 12 12.34
One 1 × 101 12.3
Zero n/a 12

Note that it is often not appropriate to display a figure with more digits than that which can be measured. For instance, if a device measures to the nearest gram and gives a reading of 12.345 kg, it would create false precision if the measurement were expressed "12.34500 kg" with 2 extra zeroes ("00") at the end.

The representation of a positive number x to a precision of p significant digits has a numerical value that is given by the formula

round(10−n·x)·10n, where n = floor(log10 x) + 1 − p.

For a negative number, the numerical value is minus that of the absolute value. The number 0, to any precision, can be taken to be 0.