absolute difference

Absolute difference has the following properties:

• For $x\backslash ge\; 0$, $|x-0|=x$ (zero is the identity element on non-negative numbers)
• For all $x$, $|x-x|=0$ (every element is its own inverse element)
• $|x-y|\backslash ge\; 0$ (non-negativity){{cite book

| last = Kubrusly | first = Carlos S.

| doi = 10.1007/978-1-4757-3328-0

| isbn = 9781475733280

| location = Boston

| page = 86

| publisher = Birkhäuser

| title = Elements of Operator Theory

| url = https://books.google.com/books?id=0ijlBwAAQBAJ&pg=PA86

| year = 2001}}

• $|x-y|=\; 0$ if and only if $x=y$ (nonzero for distinct arguments).
• $|x-y|=|y-x|$ (symmetry or commutativity).
• $|x-z|\backslash le|x-y|+|y-z|$ (the triangle inequality);{{cite book|title=An Introduction to Metric Spaces and Fixed Point Theory|first1=Mohamed A.|last1=Khamsi|first2=William A.|last2=Kirk|publisher=John Wiley & Sons|year=2011|isbn=9781118031322|contribution=1.3 The triangle inequality in $\backslash R$|pages=7–8|contribution-url=https://books.google.com/books?id=3ZlpXpedkasC&pg=PA7}} equality holds if and only if $x\backslash le\; y\backslash le\; z$ or $x\backslash ge\; y\backslash ge\; z$.

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.{{cite book|title=Functional Analysis with Applications|first1=Svetlin G.|last1=Georgiev|first2=Khaled|last2=Zennir|publisher=Walter de Gruyter GmbH|year=2019|isbn=9783110657722|page=25|url=https://books.google.com/books?id=1XicDwAAQBAJ&pg=PA25}} It has been called "the most natural metric space",{{sfnp|Khamsi|Kirk|2011|p=14}} and "the most important concrete metric space". This distance generalizes in many different ways to higher dimensions, as a special case of the L^p distances for all $1\backslash le\; p\backslash le\backslash infty$, including the $p=1$ and $p=2$ cases (taxicab geometry and Euclidean distance, respectively). It is also the one-dimensional special case of hyperbolic distance.

Instead of $|x-y|$, the absolute difference may also be expressed as $$\backslash max(x,y)-\backslash min(x,y).$$ Generalizing this to more than two values, 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 {{nowrap|of $S$.}}

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.{{cite journal

| last1 = Talukdar | first1 = D.

| last2 = Das | first2 = N. R.

| date = July 1996

| doi = 10.2307/3619592

| issue = 488

| journal = The Mathematical Gazette

| jstor = 3619592

| pages = 401–404

| title = 80.33 Measuring associativity in a groupoid of natural numbers

| volume = 80}}

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.{{cite book|title=Puzzles, Paradoxes, and Problem Solving: An Introduction to Mathematical Thinking|first1=Marilyn A.|last1=Reba|first2=Douglas R.|last2=Shier|publisher=CRC Press|year=2014|isbn=9781482297935|page=463|url=https://books.google.com/books?id=4rDNBQAAQBAJ&pg=PA463}}

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.{{cite book

| last = Golomb | first = Solomon W. | author-link = Solomon W. Golomb

| editor-last = Read | editor-first = Ronald C. | editor-link = Ronald C. Read

| contribution = How to number a graph

| doi = 10.1016/B978-1-4832-3187-7.50008-8

| mr = 340107

| pages = 23–37

| publisher = Academic Press

| title = Graph Theory and Computing

| url = https://books.google.com/books?id=ja7iBQAAQBAJ&pg=PA23

| year = 1972}}

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.{{cite book|title=Statistical Pattern Recognition|first=Andrew R.|last=Webb|edition=2nd|publisher=John Wiley & Sons|year=2003|isbn=9780470854785|page=421|url=https://books.google.com/books?id=ivMBWCe_f0gC&pg=PA421}}

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.

{{reflist}}

• {{MathWorld | urlname=AbsoluteDifference | title=Absolute Difference}}

{{Real numbers}}

Category:Real numbers

Category:Distance