Absolute difference

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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 ${\displaystyle y}$ , and is a special case of the L^p distance for all ${\displaystyle 1\leq p\leq \infty }$ . Its applications in statistics include the absolute deviation from a central tendency.

Properties#edit

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]

Applications#edit

The absolute difference is used to define the relative difference, the absolute difference between a given value and a reference value divided by the reference value itself.^[6]

In the theory of graceful labelings in graph theory, vertices are labeled by natural numbers and edges are labeled by the absolute difference of the numbers at their two vertices. A labeling of this type is graceful when the edge labels are distinct and consecutive from 1 to the number of edges.^[7]

As well as being a special case of the L^p distances, absolute difference can be used to define Chebyshev distance (L^∞), in which the distance between points is the maximum or supremum of the absolute differences of their coordinates.^[8]

In statistics, the absolute deviation of a sampled number from a central tendency is its absolute difference from the center, the average absolute deviation is the average of the absolute deviations of a collection of samples, and least absolute deviations is a method for robust statistics based on minimizing the average absolute deviation.

References#edit

^ a b c d Talukdar, D.; Das, N. R. (July 1996). "80.33 Measuring associativity in a groupoid of natural numbers". The Mathematical Gazette. 80 (488): 401–404. doi:10.2307/3619592. JSTOR 3619592.
^ a b c d e Kubrusly, Carlos S. (2001). Elements of Operator Theory. Boston: Birkhäuser. p. 86. doi:10.1007/978-1-4757-3328-0. ISBN 9781475733280.
^ Khamsi, Mohamed A.; Kirk, William A. (2011). "1.3 The triangle inequality in ${\displaystyle \mathbb {R} }$ ". An Introduction to Metric Spaces and Fixed Point Theory. John Wiley & Sons. pp. 7–8. ISBN 9781118031322.
^ Georgiev, Svetlin G.; Zennir, Khaled (2019). Functional Analysis with Applications. Walter de Gruyter GmbH. p. 25. ISBN 9783110657722.
^ Khamsi & Kirk (2011), p. 14.
^ Reba, Marilyn A.; Shier, Douglas R. (2014). Puzzles, Paradoxes, and Problem Solving: An Introduction to Mathematical Thinking. CRC Press. p. 463. ISBN 9781482297935.
^ Golomb, Solomon W. (1972). "How to number a graph". In Read, Ronald C. (ed.). Graph Theory and Computing. Academic Press. pp. 23–37. doi:10.1016/B978-1-4832-3187-7.50008-8. MR 0340107.
^ Webb, Andrew R. (2003). Statistical Pattern Recognition (2nd ed.). John Wiley & Sons. p. 421. ISBN 9780470854785.

External links#edit

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

Categories Real numbers Distance

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