The absolute difference of two
real numbers and is given by , the
absolute value of their
difference. It describes the distance on the
real line between the points corresponding to and . It is a special case of the
Lp distance for all and is the standard
metric used for both the set of
rational numbers and their completion, the set of real numbers .
(triangle inequality); in the case of the absolute difference, equality holds if and only if or .
By contrast, simple
subtraction is not non-negative or commutative, but it does obey the second and fourth properties above, since if and only if , and .
When it is desirable to avoid the absolute value function – for example because it is expensive to compute, or because its derivative is not continuous – it can sometimes be eliminated by the identity
if and only if .
This follows since and squaring is
monotonic on the nonnegative reals.