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 Lp 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:
- |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).
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
- |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.
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