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