Absolute difference

Showing the absolute difference of real numbers ${\displaystyle x}$ and ${\displaystyle y}$ as the distance between them on the real line.

The absolute difference of two real numbers ${\displaystyle x}$ and ${\displaystyle y}$ is given by ${\displaystyle |x-y|}$, the absolute value of their difference. It describes the distance on the real line between the points corresponding to ${\displaystyle x}$ and ${\displaystyle y}$. It is a special case of the L^p distance for all ${\displaystyle 1\leq p\leq \infty }$ and is the standard metric used for both the set of rational numbers ${\displaystyle \mathbb {Q} }$ and their completion, the set of real numbers ${\displaystyle \mathbb {R} }$.

As with any metric, the metric properties hold:

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

By contrast, simple subtraction is not non-negative or commutative, but it does obey the second and fourth properties above, since ${\displaystyle x-y=0}$ if and only if ${\displaystyle x=y}$, and ${\displaystyle x-z=(x-y)+(y-z)}$.

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

${\displaystyle |x-y|<|z-w|}$ if and only if ${\displaystyle (x-y)^{2}<(z-w)^{2}}$.

This follows since ${\displaystyle |x-y|^{2}=(x-y)^{2}}$ 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.