Absolute difference

The absolute difference of two real numbers

It describes the distance on the real line between the points corresponding to

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

As with any metric, the metric properties hold: By contrast, simple subtraction is not non-negative or commutative, but it does obey the second and fourth properties above, since

The absolute difference is used to define other quantities including the relative difference, the L1 norm used in taxicab geometry, and graceful labelings in graph theory.

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 This follows since

and squaring is monotonic on the nonnegative reals.

This algebra-related article is a stub.