# Absolute difference

The **absolute difference** of two real numbers *x*, *y* is given by |*x* − *y*|, the absolute value of their difference. It describes the distance on the real line between the points corresponding to *x* and *y*. It is a special case of the L^{p} distance for all 1 ≤ *p* ≤ ∞ and is the standard metric used for both the set of rational numbers **Q** and their completion, the set of real numbers **R**.

As with any metric, the metric properties hold:

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*x*−*y*| ≥ 0, since absolute value is always non-negative. - |
*x*−*y*| = 0 if and only if*x*=*y*. - |
*x*−*y*| = |*y*−*x*| (*symmetry*or*commutativity*). - |
*x*−*z*| ≤ |*x*−*y*| + |*y*−*z*| (*triangle inequality*); in the case of the absolute difference, equality holds if and only if*x*≤*y*≤*z*.

By contrast, simple subtraction is not non-negative or commutative, but it does obey the second and fourth properties above, since *x* − *y* = 0 if and only if *x* = *y*, and *x* − *z* = (*x* − *y*) + (*y* − *z*).

The absolute difference is used to define other quantities including the relative difference, the L^{1} 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

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*x*−*y*| < |*z*−*w*| if and only if (*x*−*y*)^{2}< (*z*−*w*)^{2}.

This follows since |*x* − *y*|^{2} = (*x* − *y*)^{2} and squaring is monotonic on the nonnegative reals.