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Absolute value

From Wikipedia, the free encyclopedia
Distance from zero to a number
This article is about the absolute value of real and complex numbers. For a generalization of the concept, seeAbsolute value (algebra). For other uses, seeAbsolute value (disambiguation).

Thegraph of the absolute value function for real numbers
The absolute value of a number may be thought of as its distance from zero.

Inmathematics, theabsolute value ormodulus of areal numberx{\displaystyle x},denoted|x|{\displaystyle |x|}, is thenon-negative valueofx{\displaystyle x} without regard to itssign. Namely,|x|=x{\displaystyle |x|=x} ifx{\displaystyle x} is apositive number, and|x|=−x{\displaystyle |x|=-x} ifx{\displaystyle x} isnegative (in which case negatingx{\displaystyle x} makes−x{\displaystyle -x} positive), and|0|=0{\displaystyle |0|=0}. For example, the absolute value of 3is 3, and the absolute value of −3 isalso 3. The absolute value of a number may be thought of as itsdistance from zero.

Generalisations of the absolute value for real numbers occur in a wide variety of mathematical settings. For example, an absolute value is also defined for thecomplex numbers, thequaternions,ordered rings,fields andvector spaces. The absolute value is closely related to the notions ofmagnitude,distance, andnorm in various mathematical and physical contexts.

Terminology and notation

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In 1806,Jean-Robert Argand introduced the termmodule, meaningunit of measure in French, specifically for thecomplex absolute value,[1][2] and it was borrowed into English in 1866 as the Latin equivalentmodulus.[1] The termabsolute value has been used in this sense from at least 1806 in French[3] and 1857 in English.[4] The notation|x|, with avertical bar on each side, was introduced byKarl Weierstrass in 1841.[5] Other names forabsolute value includenumerical value[1] andmagnitude.[1] The absolute value ofx{\displaystyle x} has also been denotedabs⁡x{\displaystyle \operatorname {abs} x} in some mathematical publications,[6] and inspreadsheets, programming languages, and computational software packages, the absolute value ofx{\textstyle x} is generally represented byabs(x), or a similar expression,[7] as it has been since the earliest days ofhigh-level programming languages.[8]

The vertical bar notation also appears in a number of other mathematical contexts: for example, when applied to a set, it denotes itscardinality; when applied to amatrix, it denotes itsdeterminant.[9] Vertical bars denote the absolute value only for algebraic objects for which the notion of an absolute value is defined, notably anelement of anormed 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 theEuclidean norm[10] orsup norm[11] of a vectorinRn{\displaystyle \mathbb {R} ^{n}}, although double vertical bars with subscripts(‖⋅‖2{\displaystyle \|\cdot \|_{2}}and‖⋅‖∞{\displaystyle \|\cdot \|_{\infty }}, respectively) are a more common and less ambiguous notation.

Definition and properties

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Real numbers

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For anyreal numberx{\displaystyle x}, theabsolute value ormodulusofx{\displaystyle x} is denotedby|x|{\displaystyle |x|}, with avertical bar on each side of the quantity, and is defined as[12]|x|={x,if x≥0−x,if x<0.{\displaystyle |x|={\begin{cases}x,&{\text{if }}x\geq 0\\-x,&{\text{if }}x<0.\end{cases}}}

The absolute valueofx{\displaystyle x} is thus always either apositive number orzero, but nevernegative. Whenx{\displaystyle x} itself is negative(x<0{\displaystyle x<0}), then its absolute value is necessarily positive(|x|=−x>0{\displaystyle |x|=-x>0}).[13]

From ananalytic geometry point of view, the absolute value of a real number is that number'sdistance from zero along thereal number line, and more generally, the absolute value of the difference of two real numbers (theirabsolute difference) is the distance between them.[13] The notion of an abstractdistance function in mathematics can be seen to be a generalisation of the absolute value of the difference.[14] See§ Distance below.

Since thesquare root symbol represents the uniquepositivesquare root, when applied to a positive number, it follows that[15]}|x|=x2.{\displaystyle |x|={\sqrt {x^{2}}}.}This is equivalent to the definition above, and may be used as an alternative definition of the absolute value of real numbers.[16]

The absolute value has the following four fundamental properties (a{\textstyle a},b{\textstyle b} are real numbers), that are used for generalization of this notion to other domains:[17]

|a|≥0{\displaystyle |a|\geq 0}Non-negativity[17]
|a|=0⟺a=0{\displaystyle |a|=0\iff a=0}Positive-definiteness[17]
|ab|=|a||b|{\displaystyle |ab|=\left|a\right|\left|b\right|}Multiplicativity[17]
|a+b|≤|a|+|b|{\displaystyle |a+b|\leq |a|+|b|}Subadditivity, specifically thetriangle inequality[17]

Non-negativity, positive definiteness, and multiplicativity are readily apparent from the definition. To see that subadditivity holds, first note that|a+b|=s(a+b){\displaystyle |a+b|=s(a+b)}wheres=±1{\displaystyle s=\pm 1}, with its sign chosen to make the result positive. Now, since−1⋅x≤|x|{\displaystyle -1\cdot x\leq |x|}and+1⋅x≤|x|{\displaystyle +1\cdot x\leq |x|}, it follows that, whichever of±1{\displaystyle \pm 1} is the valueofs{\displaystyle s}, one hass⋅x≤|x|{\displaystyle s\cdot x\leq |x|} for allrealx{\displaystyle x}. Consequently,|a+b|=s⋅(a+b)=s⋅a+s⋅b≤|a|+|b|{\displaystyle |a+b|=s\cdot (a+b)=s\cdot a+s\cdot b\leq |a|+|b|}, 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.

||a||=|a|{\displaystyle {\bigl |}\left|a\right|{\bigr |}=|a|}Idempotence (the absolute value of the absolute value is the absolute value)
|−a|=|a|{\displaystyle \left|-a\right|=|a|}Evenness (reflection symmetry of the graph)[18]
|a−b|=0⟺a=b{\displaystyle |a-b|=0\iff a=b}Identity of indiscernibles (equivalent to positive-definiteness)
|a−b|≤|a−c|+|c−b|{\displaystyle |a-b|\leq |a-c|+|c-b|}Triangle inequality (equivalent to subadditivity)
|ab|=|a||b| {\displaystyle \left|{\frac {a}{b}}\right|={\frac {|a|}{|b|}}\ } (ifb≠0{\displaystyle b\neq 0})Preservation of division – equivalent to multiplicativity[19]
|a−b|≥||a|−|b||{\displaystyle |a-b|\geq {\bigl |}\left|a\right|-\left|b\right|{\bigr |}}Reverse triangle inequality – equivalent to subadditivity[19]

Two other useful properties concerning inequalities are:[19]

|a|≤b⟺−b≤a≤b{\displaystyle |a|\leq b\iff -b\leq a\leq b}
|a|≥b⟺a≤−b {\displaystyle |a|\geq b\iff a\leq -b\ } ora≥b{\displaystyle a\geq b}

These relations may be used to solve inequalities involving absolute values. For example:

|x−3|≤9{\displaystyle |x-3|\leq 9}⟺−9≤x−3≤9{\displaystyle \iff -9\leq x-3\leq 9}
⟺−6≤x≤12{\displaystyle \iff -6\leq x\leq 12}

The absolute value, as "distance from zero", is used to define the absolute difference between arbitrary real numbers, the standardmetric on the real numbers.

Complex numbers

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The absolute value of acomplex numberz{\displaystyle z} is thedistancer{\displaystyle r}ofz{\displaystyle z} from the origin. It is also seen in the picture thatz{\displaystyle z} and itscomplex conjugatez¯{\displaystyle {\bar {z}}} have the same absolute value.

Since thecomplex numbers are notordered, the definition given at the top for the real absolute value cannot be directly applied to complex numbers. However, the geometric interpretation of the absolute value of a real number as its distance from 0 can be generalised. The absolute value of a complex number is defined by the Euclidean distance of its corresponding point in thecomplex plane from theorigin. This can be computed using thePythagorean theorem: for any complex numberz=x+iy,{\displaystyle z=x+iy,}wherex{\displaystyle x} andy{\displaystyle y} are real numbers, theabsolute value ormodulusofz{\displaystyle z} isdenoted|z|{\displaystyle |z|} and is defined by[20]|z|=Re⁡(z)2+Im⁡(z)2=x2+y2,{\displaystyle |z|={\sqrt {\operatorname {Re} (z)^{2}+\operatorname {Im} (z)^{2}}}={\sqrt {x^{2}+y^{2}}},}thePythagorean addition ofx{\displaystyle x} andy{\displaystyle y}, whereRe⁡(z)=x{\displaystyle \operatorname {Re} (z)=x} andIm⁡(z)=y{\displaystyle \operatorname {Im} (z)=y} denote the real and imaginary partsofz{\displaystyle z}, respectively. When theimaginary party{\displaystyle y} is zero, this coincides with the definition of the absolute value of thereal numberx{\displaystyle x}.[20]

When a complex numberz{\displaystyle z} is expressed in itspolar formasz=reiθ,{\displaystyle z=re^{i\theta },} its absolute valueis|z|=r.{\displaystyle |z|=r.}

Since the product of any complex numberz{\displaystyle z} and itscomplex conjugatez¯=x−iy{\displaystyle {\bar {z}}=x-iy}, with the same absolute value, is always the non-negative real number(x2+y2){\displaystyle \left(x^{2}+y^{2}\right)}, the absolute value of a complex numberz{\displaystyle z} is the square rootofz⋅z¯,{\displaystyle z\cdot {\overline {z}},} which is therefore called theabsolute square orsquared modulusofz{\displaystyle z}:[20]|z|=z⋅z¯.{\displaystyle |z|={\sqrt {z\cdot {\overline {z}}}}.}This generalizes the alternative definition for reals:|x|=x⋅x{\textstyle |x|={\sqrt {x\cdot x}}}.

The complex absolute value shares the four fundamental properties given above for the real absolute value. The identity|z|n=|zn|{\displaystyle |z|^{n}=|z^{n}|} is a special case of multiplicativity that is often useful by itself.[20]

Absolute value function

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Thegraph of the absolute value function for real numbers
Composition of absolute value with acubic function in different orders

The real absolute value function iscontinuous everywhere. It isdifferentiable everywhere except forx = 0. It ismonotonically decreasing on theinterval(−∞, 0] and monotonically increasing on the interval[0, +∞).[21] Since a real number and itsopposite have the same absolute value, it is aneven function, and is hence notinvertible.[22] The real absolute value function is apiecewise linear,convex function.[18]

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

Relationship to the sign function

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The absolute value function of a real number returns its value irrespective of its sign, whereas thesign (or signum) function returns a number's sign irrespective of its value. The following equations show the relationship between these two functions:

|x|=xsgn⁡(x),{\displaystyle |x|=x\operatorname {sgn}(x),}

or

|x|sgn⁡(x)=x,{\displaystyle |x|\operatorname {sgn}(x)=x,}

and forx ≠ 0,

sgn⁡(x)=|x|x=x|x|.{\displaystyle \operatorname {sgn}(x)={\frac {|x|}{x}}={\frac {x}{|x|}}.}

Relationship to the max and min functions

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Lets,t∈R{\displaystyle s,t\in \mathbb {R} }, then the following relationship to theminimum andmaximum functions hold:

|t−s|=−2min(s,t)+s+t{\displaystyle |t-s|=-2\min(s,t)+s+t}

and

|t−s|=2max(s,t)−s−t.{\displaystyle |t-s|=2\max(s,t)-s-t.}

The formulas can be derived by considering each cases>t{\displaystyle s>t} andt>s{\displaystyle t>s} separately.

From the last formula one can derive also|t|=max(t,−t){\displaystyle |t|=\max(t,-t)}.

Derivative

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The real absolute value function has aderivative for everyx ≠ 0, given by astep function equal to thesign function except atx = 0 where the absolute value function is notdifferentiable:[23][24]d|x|dx=x|x|={−1x<01x>0=sgn⁡xfor x≠0.{\displaystyle {\begin{aligned}{\frac {d\left|x\right|}{dx}}&={\frac {x}{|x|}}={\begin{cases}-1&x<0\\1&x>0\end{cases}}\\[7mu]&=\operatorname {sgn} x\quad {\text{for }}x\neq 0.\end{aligned}}}

The real absolute value function is an example of a continuous function that achieves aglobal minimum where the derivative does not exist.

Thesubdifferential of |x| at x = 0 is the interval [−1, 1].[25]

Thecomplex absolute value function is continuous everywhere butcomplex differentiablenowhere because it violates theCauchy–Riemann equations.[23]

The second derivative of |x| with respect to x is zero everywhere except zero, where it does not exist. As ageneralised function, the second derivative may be taken as two times theDirac delta function.

Antiderivative

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Theantiderivative (indefiniteintegral) of the real absolute value function is

∫|x|dx=x|x|2+C,{\displaystyle \int \left|x\right|dx={\frac {x\left|x\right|}{2}}+C,}

whereC is an arbitraryconstant of integration. This is not acomplex antiderivative because complex antiderivatives can only exist for complex-differentiable (holomorphic) functions, which the complex absolute value function is not.

Derivatives of compositions

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The following two formulae are special cases of thechain rule:

ddxf(|x|)=x|x|(f′(|x|)){\displaystyle {d \over dx}f(|x|)={x \over |x|}(f'(|x|))}

if the absolute value is inside a function, and

ddx|f(x)|=f(x)|f(x)|f′(x){\displaystyle {d \over dx}|f(x)|={f(x) \over |f(x)|}f'(x)}

if another function is inside the absolute value. In the first case, the derivative is always discontinuous atx=0{\textstyle x=0} in the first case and wheref(x)=0{\textstyle f(x)=0} in the second case.

Distance

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See also:Metric space

The absolute value is closely related to the idea ofdistance. As noted above, the absolute value of a real or complex number is the distance from that number to the origin, along the real number line, for real numbers, or in the complex plane, for complex numbers, and more generally, the absolute value of the difference of two real or complex numbers is the distance between them.

The standardEuclidean distance between two pointsa=(a1,a2,…,an){\displaystyle a=(a_{1},a_{2},\dots ,a_{n})}andb=(b1,b2,…,bn){\displaystyle b=(b_{1},b_{2},\dots ,b_{n})}inEuclideann-space is defined as:[14]∑i=1n(ai−bi)2.{\displaystyle {\sqrt {\textstyle \sum _{i=1}^{n}(a_{i}-b_{i})^{2}}}.}

This can be seen as a generalisation, since fora1{\displaystyle a_{1}} andb1{\displaystyle b_{1}} real, i.e. in a 1-space, according to the alternative definition of the absolute value,

|a1−b1|=(a1−b1)2=∑i=11(ai−bi)2,{\displaystyle |a_{1}-b_{1}|={\sqrt {(a_{1}-b_{1})^{2}}}={\sqrt {\textstyle \sum _{i=1}^{1}(a_{i}-b_{i})^{2}}},}

and fora=a1+ia2{\displaystyle a=a_{1}+ia_{2}} andb=b1+ib2{\displaystyle b=b_{1}+ib_{2}} complex numbers, i.e. in a 2-space,

|a−b|{\displaystyle |a-b|}=|(a1+ia2)−(b1+ib2)|{\displaystyle =|(a_{1}+ia_{2})-(b_{1}+ib_{2})|}
=|(a1−b1)+i(a2−b2)|{\displaystyle =|(a_{1}-b_{1})+i(a_{2}-b_{2})|}
=(a1−b1)2+(a2−b2)2=∑i=12(ai−bi)2.{\displaystyle ={\sqrt {(a_{1}-b_{1})^{2}+(a_{2}-b_{2})^{2}}}={\sqrt {\textstyle \sum _{i=1}^{2}(a_{i}-b_{i})^{2}}}.}

The above shows that the "absolute value"-distance, for real and complex numbers, agrees with the standard Euclidean distance, which they inherit as a result of considering them as one and two-dimensional Euclidean spaces, respectively.

The properties of the absolute value of the difference of two real or complex numbers: non-negativity, identity of indiscernibles, symmetry and the triangle inequality given above, can be seen to motivate the more general notion of adistance function as follows:

A real valued functiond on a setX × X is called ametric (or adistance function) on X, if it satisfies the following four axioms:[26]

d(a,b)≥0{\displaystyle d(a,b)\geq 0}Non-negativity
d(a,b)=0⟺a=b{\displaystyle d(a,b)=0\iff a=b}Identity of indiscernibles
d(a,b)=d(b,a){\displaystyle d(a,b)=d(b,a)}Symmetry
d(a,b)≤d(a,c)+d(c,b){\displaystyle d(a,b)\leq d(a,c)+d(c,b)}Triangle inequality

Generalizations

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Ordered rings

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The definition of absolute value given for real numbers above can be extended to anyordered ring. That is, if a is an element of an ordered ring R, then theabsolute value of a, denoted by|a|, is defined to be:|a|={a,if a≥0−a,if a<0.{\displaystyle |a|=\left\{{\begin{array}{rl}a,&{\text{if }}a\geq 0\\-a,&{\text{if }}a<0.\end{array}}\right.}where−a is theadditive inverse of a, 0 is theadditive identity, and < and ≥ have the usual meaning with respect to the ordering in the ring.[27]

Fields

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Main article:Absolute value (algebra)

The four fundamental properties of the absolute value for real numbers can be used to generalise the notion of absolute value to an arbitrary field, as follows.

A real-valued function v on afield F is called anabsolute value (also amodulus,magnitude,value, orvaluation)[28][a] if it satisfies the following four axioms:

v(a)≥0{\displaystyle v(a)\geq 0}Non-negativity
v(a)=0⟺a=0{\displaystyle v(a)=0\iff a=\mathbf {0} }Positive-definiteness
v(ab)=v(a)v(b){\displaystyle v(ab)=v(a)v(b)}Multiplicativity
v(a+b)≤v(a)+v(b){\displaystyle v(a+b)\leq v(a)+v(b)}Subadditivity or the triangle inequality

Where0 denotes theadditive identity of F. It follows from positive-definiteness and multiplicativity thatv(1) = 1, where1 denotes themultiplicative identity of F. The real and complex absolute values defined above are examples of absolute values for an arbitrary field.

Ifv is an absolute value on F, then the function d onF × F, defined byd(a, b) =v(a −b), is a metric and the following are equivalent:

An absolute value which satisfies any (hence all) of the above conditions is said to benon-Archimedean, otherwise it is said to beArchimedean.[29]

Vector spaces

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Main article:Norm (mathematics)

Again the fundamental properties of the absolute value for real numbers can be used, with a slight modification, to generalise the notion to an arbitrary vector space.

A real-valued function on avector space V over a field F, represented as‖ · ‖, is called anabsolute value, but more usually anorm, if it satisfies the following axioms:

For all a in F, andv,u in V,

‖v‖≥0{\displaystyle \|\mathbf {v} \|\geq 0}Non-negativity
‖v‖=0⟺v=0{\displaystyle \|\mathbf {v} \|=0\iff \mathbf {v} =0}Positive-definiteness
‖av‖=|a|‖v‖{\displaystyle \|a\mathbf {v} \|=\left|a\right|\left\|\mathbf {v} \right\|}Absolute homogeneity or positive scalability
‖v+u‖≤‖v‖+‖u‖{\displaystyle \|\mathbf {v} +\mathbf {u} \|\leq \|\mathbf {v} \|+\|\mathbf {u} \|}Subadditivity or the triangle inequality

The norm of a vector is also called itslength ormagnitude.

In the case ofEuclidean spaceRn{\displaystyle \mathbb {R} ^{n}}, the function defined by

‖(x1,x2,…,xn)‖=∑i=1nxi2{\displaystyle \|(x_{1},x_{2},\dots ,x_{n})\|={\sqrt {\textstyle \sum _{i=1}^{n}x_{i}^{2}}}}

is a norm called the Euclidean norm. When the real numbersR{\displaystyle \mathbb {R} } are considered as the one-dimensional vector spaceR1{\displaystyle \mathbb {R} ^{1}}, the absolute value is anorm, and is thep-norm (seeLp space) for any p. In fact the absolute value is the "only" norm onR1{\displaystyle \mathbb {R} ^{1}}, in the sense that, for every norm‖ · ‖ onR1{\displaystyle \mathbb {R} ^{1}},‖x‖ = ‖1‖ ⋅ |x|.

The complex absolute value is a special case of the norm in aninner product space, which is identical to the Euclidean norm when the complex plane is identified as theEuclidean plane R2{\displaystyle \mathbb {R} ^{2}}.

Composition algebras

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Main article:Composition algebra

Every composition algebraA has aninvolutionx →x* called itsconjugation. The product inA of an elementx and its conjugatex* is writtenN(x) =x x* and called thenorm of x.

The real numbersR{\displaystyle \mathbb {R} }, complex numbersC{\displaystyle \mathbb {C} }, and quaternionsH{\displaystyle \mathbb {H} } are all composition algebras with norms given bydefinite quadratic forms. The absolute value in thesedivision algebras is given by the square root of the composition algebra norm.

In general, the norm of a composition algebra may be aquadratic form that is not definite and hasnull vectors. However, as in the case of division algebras, when an elementx has a non-zero norm, thenx has amultiplicative inverse given byx*/N(x).

See also

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Notes

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  1. ^This meaning ofvaluation is rare. Usually, avaluation is the logarithm of the inverse of an absolute value.

Footnotes

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  1. ^abcdOxford English Dictionary, Draft Revision, June 2008
  2. ^Nahin,O'Connor and Robertson, andfunctions.Wolfram.com.; for the French sense, seeLittré, 1877
  3. ^Lazare Nicolas M. Carnot,Mémoire sur la relation qui existe entre les distances respectives de cinq point quelconques pris dans l'espace, p. 105at Google Books
  4. ^James Mill Peirce,A Text-book of Analytic Geometryat Internet Archive. The oldest citation in the 2nd edition of the Oxford English Dictionary is from 1907. The termabsolute value is also used in contrast torelative value.
  5. ^Nicholas J. Higham,Handbook of writing for the mathematical sciences, SIAM.ISBN 0-89871-420-6, p. 25
  6. ^Siegel (1942).
  7. ^Bluttman (2015), p. 135.
  8. ^Knuth (1962), p. 43, 126.
  9. ^Sargent (2025), p. 10.
  10. ^Spivak (1965), p. 1.
  11. ^Munkres (1991), p. 4.
  12. ^Mendelson (2008), p. 2.
  13. ^abSmith (2013), p. 8.
  14. ^abTabak (2014), p. 150.
  15. ^Varberg, Purcell & Rigdon (2007), p. 13.
  16. ^Stewart (2001), p. A5.
  17. ^abcdeShechter (1997), p. 259.
  18. ^abVarberg, Purcell & Rigdon (2007), p. 32.
  19. ^abcVarberg, Purcell & Rigdon (2007), p. 11.
  20. ^abcdGonzález (1992), p. 19.
  21. ^Varberg, Purcell & Rigdon (2007), p. 84.
  22. ^Baronti et al. (2016), p. 37.
  23. ^ab"Weisstein, Eric W.Absolute Value. From MathWorld – A Wolfram Web Resource".
  24. ^Bartle (2011), p. 163.
  25. ^Curnier (1999), p. 31–32.
  26. ^These axioms are not minimal; for instance, non-negativity can be derived from the other three:0 =d(a, a) ≤d(a, b) +d(b, a) = 2d(a, b).
  27. ^Mac Lane & Birkhoff (1999), p. 264.
  28. ^Shechter (1997), p. 260.
  29. ^Shechter (1997), pp. 260–261.

References

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External links

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