Absolute difference

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 $1\le p\le\infty$ 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:

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

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

This follows since $, x-y, ^2=(x-y)^2$ and squaring is monotonic on the nonnegative reals.

See also

* Absolute deviation

References

*

Real numbers
Distance

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