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