Absolute difference

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 $x$ and $y$ 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 $1\leq p\leq \infty$ and is the standard metric used for both the set of rational numbers $\mathbb {Q}$ and their completion, the set of real numbers $\mathbb {R}$.

As with any metric, the metric properties hold:

• $|x-y|\geq 0$, since absolute value is always non-negative.
• $|x-y|=0$   if and only if   $x=y$.
• $|x-y|=|y-x|$     (symmetry or commutativity).
• $|x-z|\leq |x-y|+|y-z|$     (triangle inequality); in the case of the absolute difference, equality holds if and only if $x\leq y\leq z$ or $x\geq y\geq z$.

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 L¹ 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.

Additional Properties

• In any subset S of the real numbers which has an Infimum and a Supremum, the absolute difference between any two numbers in S is less or equal then the absolute difference of the Infimum and Supremum of S.

See also

• Absolute deviation

References

• Weisstein, Eric W. "Absolute Difference". MathWorld.