Approximation is the setting of an approximate value Va in place of a true but imprecisely known value V where V - Va lies within known limits. There are several different ways of expressing approximations and their limits of accuracy.


If we approximate 2.3654202 by 2.365420 we say that this is correct to 6 decimal places, in that the six figures after the decimal point are correct. It is common practice, when approximating to (n - 1) decimal places a number that has n decimal places, to round up if the nth figure after the point is 5 or greater, down if it is less than 5. Thus 3.65 can be written as 3.7 to 1 decimal place; 3.64 as 3.6.


Similarly, 2.3654202 can be expressed as 2.365420 to 7 significant figures, since the first 7 figures are correct, rounding down.


An alternative way of writing approximations is by use of the sign ± ("plus or minus"). Thus 2.365420±0.0000004 is an approximation stating that the correct value lies between 2.3654196 and 2.3654204.


Approximation is required in almost all computations, due either to inherent inaccuracies in the calculating device or to the technique of calculation, or because greater accuracy is unnecessary. The techniques of approximation to a function are of paramount importance in calculus.