Absolute difference

Absolute difference has the following properties:

• For ${\displaystyle x\geq 0}$, ${\displaystyle |x-0|=x}$ (zero is the identity element on non-negative numbers)^[1]
• For all ${\displaystyle x}$, ${\displaystyle |x-x|=0}$ (every element is its own inverse element)^[1]
• ${\displaystyle |x-y|\geq 0}$ (non-negativity)^[2]
• ${\displaystyle |x-y|=0}$ if and only if ${\displaystyle x=y}$ (nonzero for distinct arguments).^[2]
• ${\displaystyle |x-y|=|y-x|}$ (symmetry or commutativity).^[1][2]
• ${\displaystyle |x-z|\leq |x-y|+|y-z|}$ (the triangle inequality);^[2][3] equality holds if and only if ${\displaystyle x\leq y\leq z}$ or ${\displaystyle x\geq y\geq z}$.

Because it is non-negative, nonzero for distinct arguments, symmetric, and obeys the triangle inequality, the real numbers form a metric space with the absolute difference as its distance, the familiar measure of distance along a line.^[4] It has been called "the most natural metric space",^[5] and "the most important concrete metric space".^[2] This distance generalizes in many different ways to higher dimensions, as a special case of the L^p distances for all ${\displaystyle 1\leq p\leq \infty }$, including the ${\displaystyle p=1}$ and ${\displaystyle p=2}$ cases (taxicab geometry and Euclidean distance, respectively). It is also the one-dimensional special case of hyperbolic distance.

Instead of ${\displaystyle |x-y|}$, the absolute difference may also be expressed as $${\displaystyle \max(x,y)-\min(x,y).}$$ Generalizing this to more than two values, in any subset ${\displaystyle S}$ of the real numbers which has an infimum and a supremum, the absolute difference between any two numbers in ${\displaystyle S}$ is less or equal than the absolute difference of the infimum and supremum of ${\displaystyle S}$.

The absolute difference takes non-negative integers to non-negative integers. As a binary operation that is commutative but not associative, with an identity element on the non-negative numbers, the absolute difference gives the non-negative numbers (whether real or integer) the algebraic structure of a commutative magma with identity.^[1]