Showing the absolute difference of real numbers
x and
y as the distance between them on the
real line.
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:

x − y ≥ 0, since absolute value is always nonnegative.

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 nonnegative 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

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.
References
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