Absolute value: Difference between revisions

{{about|the [[Magnitude (mathematics)#Numbers|magnitude]] of real and complex numbers|a generalization of the concept|Absolute value (algebra)|other uses}}

In [[mathematics]], the '''absolute value''' or '''modulus''' of a [[real number]] <math>x</math>, {{nowrap|denoted <math>|x|</math>,}} is the ([[non-negative]]) [[Magnitude (mathematics)#Numbers|magnitude]] {{nowrap|of <math>x</math>}} [[Measurement|measured]] without regard to its [[sign (mathematics)|sign]]. Namely, <math>|x|=x</math> if <math>x</math> is a [[positive number]], and <math>|x|=-x</math> if <math>x</math> is [[negative number|negative]] (in which case <math>-x</math> is positive), and {{nowrap|<math>|0|=0</math>.}} For example, the absolute value of 3 {{nowrap|is 3,}} and the absolute value of −3 is {{nowrap|also 3.}} The absolute value of a number may be thought of as its [[distance]] from zero.

In 1806, [[Jean-Robert Argand]] introduced the term ''module'', meaning ''unit of measure'' in French, specifically for the ''complex'' absolute value,<ref name=oed>[[Oxford English Dictionary]], Draft Revision, June 2008</ref><ref>Nahin, [http://www-history.mcs.st-andrews.ac.uk/Mathematicians/Argand.html O'Connor and Robertson], and [http://functions.wolfram.com/ComplexComponents/Abs/35/ functions.Wolfram.com.]; for the French sense, see [[Dictionnaire de la langue française (Littré)|Littré]], 1877</ref> and it was borrowed into English in 1866 as the Latin equivalent ''modulus''.<ref name=oed /> The term ''absolute value'' has been used in this sense from at least 1806 in French<ref>[[Lazare Nicolas Marguerite Carnot|Lazare Nicolas M. Carnot]], ''Mémoire sur la relation qui existe entre les distances respectives de cinq point quelconques pris dans l'espace'', p.&nbsp;105 [https://books.google.com/books?id=YyIOAAAAQAAJ&pg=PA105 at Google Books]</ref> and 1857 in English.{{efn|The oldest citation in the 2nd edition of the Oxford English Dictionary is from 1907. The term ''absolute value'' is also used in contrast to ''relative value''.<ref>{{cite book|first=James Mill|last=Peirce|title=A Text-book of Analytic Geometry|url=https://archive.org/details/atextbookanalyt00peirgoog/page/n60|page=42|via=Internet Archive}}</ref>}} The notation {{math|{{abs|{{mvar|x}}}}}}, with a [[vertical bar]] on each side, was introduced by [[Karl Weierstrass]] in 1841.<ref>Nicholas J. Higham, ''Handbook of writing for the mathematical sciences'', SIAM. {{ISBN|0-89871-420-6}}, p.&nbsp;25</ref> Other names for ''absolute value'' include ''numerical value''<ref name=oed /> and ''magnitude''.<ref name=oed /> The absolute value of <math>x</math> has also been denoted <math>\operatorname{abs} x</math> in some mathematical publications,{{sfnp|Siegel|1942}} and in [[spreadsheet]]s, programming languages, and computational software packages, the absolute value of <math display="inline">x</math> is generally represented by <code>abs(''x'')</code>, or a similar expression,{{sfnp|Bluttman|2015|p=[https://books.google.com/books?id=3pVxBgAAQBAJ&pg=PA135 135]}} as it has been since the earliest days of [[high-level programming language]]s.{{sfnp|Knuth|1962|p=43, 126}}

The vertical bar notation also appears in a number of other mathematical contexts: for example, when applied to a set, it denotes its [[cardinality]]; when applied to a [[Matrix (math)|matrix]], it denotes its [[determinant]].{{sfnp|Sargent|2025|p=10}} Vertical bars denote the absolute value only for algebraic objects for which the notion of an absolute value is defined, notably an [[Element (mathematics)|element]] of a [[normed division algebra]], for example, a real number, a complex number, or a quaternion. A closely related but distinct notation is the use of vertical bars for either the [[Euclidean norm]]{{sfnp|Spivak|1965|p=1}} or [[sup norm]]{{sfnp|Munkres|1991|p=4}} of a vector {{nowrap|in <math>\R^n</math>,}} although double vertical bars with subscripts {{nowrap|(<math>\|\cdot\|_2</math>}} {{nowrap|and <math>\|\cdot\|_\infty</math>,}} respectively) are a more common and less ambiguous notation.

For any {{nowrap|[[real number]] <math>x</math>,}} the absolute value or modulus {{nowrap|of <math>x</math>}} is denoted {{nowrap|by <math>|x|</math>}}, with a [[vertical bar]] on each side of the quantity, and is defined as{{sfnp|Mendelson|2008|p=[https://books.google.com/books?id=A8hAm38zsCMC&pg=PA2 2]}}

The absolute value {{nowrap|of <math>x</math>}} is thus always either a [[positive number]] or [[0|zero]], but never [[negative number|negative]]. When <math>x</math> itself is negative {{nowrap|(<math>x < 0</math>),}} then its absolute value is necessarily positive {{nowrap|(<math>|x|=-x>0</math>).{{sfnp|Smith|2013|p=[https://books.google.com/books?id=ZUJbVQN37bIC&pg=PA8 8]}}}}

From an [[analytic geometry]] point of view, the absolute value of a real number is that number's [[distance]] from zero along the [[real number line]], and more generally, the absolute value of the difference of two real numbers (their [[absolute difference]]) is the distance between them.{{sfnp|Smith|2013|p=[https://books.google.com/books?id=ZUJbVQN37bIC&pg=PA8 8]}} The notion of an abstract [[distance function]] in mathematics can be seen to be a generalisation of the absolute value of the difference.{{sfnp|Tabak|2014|p=[https://books.google.com/books?id=r0HuPiexnYwC&pg=PA150 150]}} See {{slink||Distance}} below.

Since the [[radical symbol|square root symbol]] represents the unique ''positive'' [[square root]], when applied to a positive number, it follows that{{sfnp|Varberg|Purcell|Rigdon|2007|p=[https://archive.org/details/matematika-a-purcell-calculus-9th-ed/page/13 13]}}

This is equivalent to the definition above, and may be used as an alternative definition of the absolute value of real numbers.{{sfnp|Stewart|2001|p=A5}}

The absolute value has the following four fundamental properties (<math display="inline">a</math>, <math display="inline">b</math> are real numbers), that are used for generalization of this notion to other domains:{{sfnp|Shechter|1997|p=[https://books.google.com/books?id=eqUv3Bcd56EC&pg=PA259 259]}}

| Non-negativity{{sfnp|Shechter|1997|p=[https://books.google.com/books?id=eqUv3Bcd56EC&pg=PA259 259]}}

|Positive-definiteness{{sfnp|Shechter|1997|p=[https://books.google.com/books?id=eqUv3Bcd56EC&pg=PA259 259]}}

|[[Multiplicativeness|Multiplicativity]]{{sfnp|Shechter|1997|p=[https://books.google.com/books?id=eqUv3Bcd56EC&pg=PA259 259]}}

| [[Subadditivity]], specifically the [[triangle inequality]]{{sfnp|Shechter|1997|p=[https://books.google.com/books?id=eqUv3Bcd56EC&pg=PA259 259]}}

Non-negativity, positive definiteness, and multiplicativity are readily apparent from the definition. To see that subadditivity holds, first note that <math>|a+b|=s(a+b)</math> {{nowrap|where <math>s=\pm 1</math>,}} with its sign chosen to make the result positive. Now, since <math>-1 \cdot x \le |x|</math> {{nowrap|and <math>+1 \cdot x \le |x|</math>,}} it follows that, whichever of <math>\pm1</math> is the value {{nowrap|of <math>s</math>,}} one has <math>s \cdot x\leq |x|</math> for all {{nowrap|real <math>x</math>.}} Consequently, <math>|a+b|=s \cdot (a+b) = s \cdot a + s \cdot b \leq |a| + |b|</math>, as desired.

Some additional useful properties are given below. These are either immediate consequences of the definition or implied by the four fundamental properties above.

|[[even function|Evenness]] ([[reflection symmetry]] of the graph){{sfnp|Varberg|Purcell|Rigdon|2007|p=[https://archive.org/details/matematika-a-purcell-calculus-9th-ed/page/32 32]}}

|Preservation of division &ndash; equivalent to multiplicativity{{sfnp|Varberg|Purcell|Rigdon|2007|p=[https://archive.org/details/matematika-a-purcell-calculus-9th-ed/page/11 11]}}

|[[Reverse triangle inequality]] &ndash; equivalent to subadditivity{{sfnp|Varberg|Purcell|Rigdon|2007|p=[https://archive.org/details/matematika-a-purcell-calculus-9th-ed/page/11 11]}}

Two other useful properties concerning inequalities are:{{sfnp|Varberg|Purcell|Rigdon|2007|p=[https://archive.org/details/matematika-a-purcell-calculus-9th-ed/page/11 11]}}

where <math>x</math> and <math>y</math> are real numbers, the absolute value or modulus {{nowrap|of <math>z</math>}} is {{nowrap|denoted <math>|z|</math>}} and is defined by{{sfnp|González|1992|p=[https://books.google.com/books?id=ncxL7EFr7GsC&pg=PA19 19]}}

the [[Pythagorean addition]] of <math>x</math> and <math>y</math>, where <math>\operatorname{Re}(z)=x</math> and <math>\operatorname{Im}(z)=y</math> denote the real and imaginary parts {{nowrap|of <math>z</math>,}} respectively. When the {{nowrap|imaginary part <math>y</math>}} is zero, this coincides with the definition of the absolute value of the {{nowrap|real number <math>x</math>.}}{{sfnp|González|1992|p=[https://books.google.com/books?id=ncxL7EFr7GsC&pg=PA19 19]}}

Since the product of any complex number <math>z</math> and its {{nowrap|[[complex conjugate]] <math>\bar z = x - iy</math>,}} with the same absolute value, is always the non-negative real number {{nowrap|<math>\left(x^2 + y^2\right)</math>,}} the absolute value of a complex number <math>z</math> is the square root {{nowrap|of  <math>z \cdot \overline{z},</math>}} which is therefore called the [[absolute square]] or ''squared modulus'' {{nowrap|of <math>z</math>:}}{{sfnp|González|1992|p=[https://books.google.com/books?id=ncxL7EFr7GsC&pg=PA19 19]}}

The complex absolute value shares the four fundamental properties given above for the real absolute value. The identity <math>|z|^n = |z^n|</math> is a special case of multiplicativity that is often useful by itself.{{sfnp|González|1992|p=[https://books.google.com/books?id=ncxL7EFr7GsC&pg=PA19 19]}}

[[Image:Absolute value.svg|thumb|The [[graph of a function|graph]] of the absolute value function for real numbers]]

[[Image:Absolute value composition.svg|thumb|[[composition of functions|Composition]] of absolute value with a [[cubic function]] in different orders]]

The real absolute value function is [[continuous function|continuous]] everywhere. It is [[differentiable]] everywhere except for {{math|1=''x'' = 0}}.  It is [[monotonic function|monotonically decreasing]] on the [[Interval (mathematics)|interval]] {{open-closed|−∞, 0}} and monotonically increasing on the interval {{closed-open|0, +∞}}.{{sfnp|Varberg|Purcell|Rigdon|2007|p=[https://archive.org/details/matematika-a-purcell-calculus-9th-ed/page/84 84]}} Since a real number and its [[additive inverse|opposite]] have the same absolute value, it is an [[even function]], and is hence not [[Inverse function|invertible]].{{sfnp|Baronti et al.|2016|p=[http://books.google.com/books?id=dBFuDQAAQBAJ&pg=PA37 37]}} The real absolute value function is a [[piecewise linear function|piecewise linear]], [[convex function]].{{sfnp|Varberg|Purcell|Rigdon|2007|p=[https://archive.org/details/matematika-a-purcell-calculus-9th-ed/page/32 32]}}

For both real and complex numbers, the absolute value function is [[idempotent]] (meaning that the absolute value of any absolute value is itself).

The real absolute value function has a [[derivative]] for every {{math|''x'' ≠ 0}}, given by a [[step function]] equal to the [[sign function]] except at {{math|1=''x'' = 0}} where the absolute value function is not [[differentiable]]:<ref name="MathWorld">{{cite web| url = http://mathworld.wolfram.com/AbsoluteValue.html| title = Weisstein, Eric W. ''Absolute Value.'' From MathWorld – A Wolfram Web Resource.}}</ref>{{sfnp|Bartle|2011|p=163}}

<math display="block">\begin{align}

The [[subderivative|subdifferential]] of&nbsp;{{math|{{abs|{{mvar|x}}}}}} at&nbsp;{{math|1=''x'' = 0}} is the interval&nbsp;{{closed-closed|−1, 1}}.{{sfnp|Curnier|1999|p=[https://books.google.com/books?id=tiBtC4GmuKcC&pg=PA31 31&ndash;32]}}

=== Derivatives and antiderivatives of compositions ===

The following three formulae are special cases of the [[chain rule]]:

<math>{\text{d}^n \over \text{d}x^n} f(|x|)= (\sgn x)^n f^{(n)}(|x|)\quad \text{for } x \ne 0\,,</math>

<math>{\text{d}^n \over \text{d}x^n} |f(x)|=\sgn(f(x)) f^{(n)}(x)\quad \text{for } f(x) \ne 0\,,</math>

 

 

<math>{\text{d}^n \over \text{d}x^n} |f(|x|)|=(\sgn x)^n \sgn(f(|x|)) f^{(n)}(|x|)\quad \text{for } x \ne 0, f(|x|) \ne 0\,.</math>

 

 

 

<math>\int {|f(x)| \text{d}x} = \int {{|f(x)| \over f(x)} f(x) \text{d}x} = \sgn(f(x)) F(x)\quad \text{for } f(x) \ne 0\,,</math>

 

<math>\int {|f(|x|)| \text{d}x} = \int {{|f(|x|)| \over f(|x|)} f(|x|) \text{d}x} = \sgn (x) \sgn(f(|x|)) F(|x|)\quad \text{for } x \ne 0, f(|x|) \ne 0\,,</math>

supposing the derivative of the [[Sign function#Differentiation|sign function]] is 0.

 

=== Power rule for expressions with absolute values ===

Using [[Chain rule|chain]] and [[Product rule|product]] rules, the [[power rule]] for expressions of the type <math>x^n |x|^m</math> can be written as:

 

<math>{\text{d} \over \text{d}x} x^n |x|^m = (n+m)x^{n-1}|x|^m\,.</math>

 

This holds true even for <math>n=0</math>:

 

<math>{\text{d} \over \text{d}x} |x|^m = mx^{-1}|x|^m\,.</math>

<math display="block">a = (a_1, a_2, \dots , a_n) </math>

<math display="block">b = (b_1, b_2, \dots , b_n) </math>

in [[Euclidean space|Euclidean {{mvar|n}}-space]] is defined as:{{sfnp|Tabak|2014|p=[https://books.google.com/books?id=r0HuPiexnYwC&pg=PA150 150]}}

<math display="block">\sqrt{\textstyle\sum_{i=1}^n(a_i-b_i)^2}. </math>

A real valued function {{mvar|d}} on a set {{math|''X'' × ''X''}} is called a [[Metric (mathematics)|metric]] (or a ''distance function'') on&nbsp;{{mvar|X}}, if it satisfies the following four axioms:{{efn|These axioms are not minimal; for instance, non-negativity can be derived from the other three: {{math|1=0 = ''d''(''a'', ''a'') ≤ ''d''(''a'', ''b'') + ''d''(''b'', ''a'') = 2''d''(''a'', ''b'')}}.}}

The definition of absolute value given for real numbers above can be extended to any [[ordered ring]]. That is, if&nbsp;{{mvar|a}} is an element of an ordered ring&nbsp;''R'', then the '''absolute value''' of&nbsp;{{mvar|a}}, denoted by {{math|{{abs|''a''}}}}, is defined to be:

<math display="block">|a| = \left\{

where {{math|−''a''}} is the [[additive inverse]] of&nbsp;{{mvar|a}}, 0 is the [[additive identity]], and < and ≥ have the usual meaning with respect to the ordering in the ring.{{sfnp|Mac Lane|Birkhoff|1999|p=[https://books.google.com/books?id=L6FENd8GHIUC&pg=PA264 264]}}

A real-valued function&nbsp;{{mvar|v}} on a [[field (mathematics)|field]]&nbsp;{{mvar|F}} is called an ''absolute value'' (also a ''modulus'', ''magnitude'', ''value'', or ''valuation''){{sfnp|Shechter|1997|p=[https://books.google.com/books?id=eqUv3Bcd56EC&pg=PA260 260]}}{{efn|1=This meaning of ''valuation'' is rare. Usually, a [[valuation (algebra)|valuation]] is the logarithm of the inverse of an absolute value.}} if it satisfies the following four axioms:

An absolute value which satisfies any (hence all) of the above conditions is said to be '''non-Archimedean''', otherwise it is said to be [[Archimedean field|Archimedean]].{{sfnp|Shechter|1997|pp=[https://books.google.com/books?id=eqUv3Bcd56EC&pg=PA260 260–261]}}

In general, the norm of a composition algebra may be a [[quadratic form]] that is not definite and has [[null vector]]s. However, as in the case of division algebras, when an element ''x'' has a non-zero norm, then ''x'' has a [[multiplicative inverse]] given by ''x''*/''N''(''x'').

== Notes ==

{{notelist}}

 

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*{{cite web|last1=O'Connor|first1=J.J.|last2=Robertson|first2=E.F.|archive-url=https://web.archive.org/web/20190402101735/http://www-history.mcs.st-andrews.ac.uk/Biographies/Argand.html|archive-date=2 April 2019|url=http://www-history.mcs.st-andrews.ac.uk/Biographies/Argand.html|title=Jean Robert Argand|publisher=School of Mathematics and Statistics, University of St Andrews|location=[[Scotland]]}}

| url = https://www.unicode.org/notes/tn28/UTN28-PlainTextMath-v3.3.pdf

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