Absolute_value

In [[mathematics]], the '''absolute value''' (or '''modulus'''<ref name="Argand">[[Jean-Robert Argand]], is credited with introducing the term "modulus" in [[1806]], see: [http://www.amazon.com/gp/reader/0691027951 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.]</ref>) of a [[real number]] is its numerical value without regard to its [[Negative and non-negative numbers|sign]]. So, for example, 3 is the absolute value of both 3 and −3. In [[computer programming]], the [[mathematical function]] used to perform this calculation is usually given the name '''abs()'''.

Generalizations of the absolute value for real numbers occur in a wide variety of mathematical settings. For example an absolute value is also defined for the [[complex number]]s, the [[quaternion]]s, [[ordered ring]]s, [[Field (mathematics)|field]]s and [[vector space]]s.

The absolute value is closely related to the notions of [[magnitude (mathematics)|magnitude]], [[distance]], and [[norm (mathematics)|norm]] in various mathematical and physical contexts.

[[Image:Absolute value.png|frame|The graph of the absolute value function for real numbers.]]

==Real numbers==

For any [[real number]] ''a'' the '''absolute value''' or '''modulus''' of ''a'' is denoted<ref name="Wolfram">[http://functions.wolfram.com/ComplexComponents/Abs/35/ functions.Wolfram.com] credits [[Karl Weierstrass]] with introducing the notation <math>|x|,</math> in [[1841]].</ref> by | ''a'' | and is defined as

:<math>|a| = \begin{cases} a, & \mbox{if } a \ge 0 \\ -a, & \mbox{if } a < 0. \end{cases} </math>

As can be seen from the above definition, the absolute value of ''a'' is always either [[positive number|positive]] or [[0 (number)|zero]], but never [[negative and non-negative numbers|negative]].

From a geometric point of view, the absolute value of a real number is the [[distance]] along the [[real number line]] of that number from zero, and more generally the absolute value of the difference of two real numbers is the distance between them. Indeed the notion of an abstract [[distance function]] in mathematics can be seen to be a generalization of the absolute value of the difference (see [[#Distance|"Distance"]] below).

The following proposition, gives an [[identity (mathematics)|identity]] which is sometimes used as an alternative (and equivalent) definition of the absolute value:

'''PROPOSITION 1''':
:<math>|a| = \sqrt{a^2}</math>

The absolute value has the following four fundamental properties:

'''PROPOSITION 2''':
:{|
|-
| style="width: 250px" |<math>|a| \ge 0 </math>
| Non-negativity
|-
|<math>|a| = 0 \iff a = 0 </math>
|Positive-definiteness
|-
|<math>|ab| = |a||b|\,</math>
|[[Multiplicativeness]]
|-
|<math>|a+b| \le |a| + |b| </math>
|[[Subadditivity]]
|}

Other important properties of the absolute value include:

'''PROPOSITION 3''':
:{|
|-
| style="width:250px" |<math>|-a| = |a|\,</math>
|[[Symmetry]]
|-
|<math>|a - b| = 0 \iff a = b </math>
|[[Identity of indiscernibles]] (equivalent to positive-definiteness)
|-
|<math>|a - b| \le |a - c| +|c - b| </math>
|[[Triangle inequality]] (equivalent to subadditivity)
|-
|<math>|a/b| = |a| / |b| \mbox{ (if } b \ne 0) \,</math>
|Preservation of division (equivalent to multiplicativeness)
|-
|<math>|a-b| \ge ||a| - |b|| </math>
|(equivalent to subadditivity)
|}

Two other useful inequalities are:
:<math>|a| \le b \iff -b \le a \le b </math>
:<math>|a| \ge b \iff a \le -b \mbox{ or } b \le a </math>

The above are often used in solving inequalities; for example:

:{|
|-
|<math>|x-3| \le 9 </math>
|<math>\iff -9 \le x-3 \le 9 </math>
|-
|
|<math>\iff -6 \le x \le 12 </math>
|}

== Complex numbers ==
<div style="float:right; margin-left:3px; margin-right:3px" title="Graphic Representation">
[[Image:complex.png]]
</div>

Since the [[complex number]]s are not [[ordered set|ordered]], the definition given above for the real absolute value cannot be directly generalized for a complex number. However the identity given in Proposition 1:
:<math>|a| = \sqrt{a^2}</math>
can be seen as motivating the following definition.

For any complex number

:<math>z = x + iy\,</math>

where ''x'' and ''y'' are real numbers, the '''absolute value''' or '''modulus''' of <math>z</math> is denoted <math>|z|,</math> and is defined as

:<math>|z| = \sqrt{x^2 + y^2}.</math>

It follows that the absolute value of a real number ''x'' is equal to its absolute value considered as a complex number since:

:<math> |x + i0| = \sqrt{x^2 + 0^2} = \sqrt{x^2} = |x|.</math>

Similar to the geometric interpretation of the absolute value for real numbers, it follows from the [[Pythagorean theorem]] that the absolute value of a complex number is the distance in the [[complex plane]] of that complex number from the [[origin (mathematics)|origin]], and more generally, that the absolute value of the difference of two complex numbers is equal to the distance between those two complex numbers.

The complex absolute value shares all the properties of the real absolute value given in Propositions 2 and 3 above. In addition, If

:<math> z = x + i y = r (\cos \phi + i \sin \phi ) \,</math>

and

:<math>\bar{z} = x - iy</math>

is the [[complex conjugate]] of <math>z</math>, then it is easily seen that

:<math>|z| = r\,</math>

:<math>|z|=|\bar{z}|</math>

:<math>|z| = \sqrt{z\bar{z}}.</math>
The latter formula is the complex analogue of proposition 1 mentioned above in the real case...

Since the positive reals form a subgroup of the complex numbers under multiplication, we may think of absolute value as an [[endomorphism]] of the [[multiplicative group]] of the complex numbers.

== Absolute value functions==
The real absolute value function is [[continuous function|continuous]] everywhere. It is [[derivative|differentiable]] everywhere except for ''x'' = 0. It is [[monotonic function|monotonically decreasing]] on the interval <nowiki>(-∞, 0]</nowiki> and monotonically increasing on the interval <nowiki>[0, ∞)</nowiki>. Since a real number and its negative have the same absolute value, it is an [[even function]], and is hence not [[invertible]].

The [[complex number|complex]] absolute value function is continuous everywhere but (complex) differentiable ''nowhere'' (One way to see this is to show that it does not obey the [[Cauchy-Riemann equations]]).

Both the real and complex functions are [[idempotent]].

It is a [[nonlinear]] function.

==Ordered rings==
The definition of absolute value given for real numbers above can easily be extended to any [[ordered ring]]. That is, if <math>a</math> is an element of an ordered ring <math>R</math>, then the '''absolute value''' of <math>a</math>, denoted by <math>|a| </math>, is defined to be:

:<math>|a| = \begin{cases} a, & \mbox{if } a \ge 0 \\ -a, & \mbox{if } a < 0, \end{cases} </math>

where <math>-a</math> is the [[additive inverse]] of <math>a</math>, and <math>0</math> is the additive [[identity element]].

== Distance==
The absolute value is closely related to the idea of distance. 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 standard [[Euclidean distance]] between two points

:<math>a = (a_1, a_2, \cdots , a_n) </math>

and

:<math>b = (b_1, b_2, \cdots , b_n) </math>

in [[Euclidean space|Euclidean ''n''-space]] is defined as:
:<math>\sqrt{(a_1-b_1)^2 + (a_2-b_2)^2 + \cdots + (a_n-b_n)^2}. </math>

This can be seen to be a generalization of <math>|a - b|,</math> since if <math>a,</math> <math>b </math> are real, then by Proposition 1,
:<math>|a - b| = \sqrt{(a - b)^2}</math>

while if

:<math> a = a_1 + i a_2 \,</math>

and

:<math> b = b_1 + i b_2 \,</math>

are complex numbers, then

:{|
|-
|<math>|a - b| \,</math>
|<math> = |(a_1 + i a_2) - (b_1 + i b_2)|\,</math>
|-
|
|<math> = |(a_1 - b_1) + i(a_2 - b_2)|\,</math>
|-
|
|<math> = \sqrt{(a_1 - b_1)^2 + (a_2 - b_2)^2}</math>
|}

The above shows that the "absolute value" distance for the real numbers or the complex numbers, agrees with the standard Euclidean distance they inherit as a result of considering them as the 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 in Propositions 2 and 3 above, can be seen to motivate the more general notion of a [[distance function]] as follows:

A real valued function <math>d</math> on a set <math>X \times X</math> is called a '''distance function''' (or a '''metric''') for <math>X</math>, if it satisfies the following four axioms:

:{|
|-
|style="width:250px" | <math>d(a, b) \ge 0 </math>
|Non-negativity
|-
|<math>d(a, b) = 0 \iff a = b </math>
|Identity of indiscernibles
|-
|<math>d(a, b) = d(b, a) \,</math>
|Symmetry
|-
|<math>d(a, b) \le d(a, c) + d(c, b) </math>
|Triangle inequality
|}

==Derivatives==
The [[derivative]] of the real absolute value function is the [[signum function]], sgn(''x''), which is defined as

:<math>\sgn (x) = \frac{x}{|x|}</math>

for ''x'' ≠ 0. The absolute value function is not differentiable at ''x'' = 0. Where the absolute value function of a real number returns a value without respect to its sign, the signum function returns a number's sign without respect to its value. Therefore ''x'' = sgn(''x'')abs(''x''). The signum function is a form of the [[Heaviside step function]] used in signal processing, defined as:

:<math> u(x) =
\begin{cases} 0, & x < 0
\\ \frac{1}{2}, & x = 0
\\ 1, & x > 0
\end{cases}
</math>

Where the value of the Heaviside function at zero is conventional. So we have at all nonzero points on the [[real number line]],

:<math>u(x) = \frac{\sgn(x) +1}{2}.\,</math>

The absolute value function has no concavity at any point, the sign function is constant at all points. Therefore the second derivative of |''x''| with respect to ''x'' is zero everywhere except zero, where it is undefined.

The absolute value function is also integrable. Its [[antiderivative]] is

:<math>\int|x|dx=\frac{x|x|}{2}+C</math>.

==Fields==
The fundamental properties of the absolute value for real numbers given in Proposition 2 above, can be used to generalize the notion of absolute value to an arbitrary field, as follows.

A real-valued function <math>v</math> on a [[field (mathematics)|field]] <math>F</math> is called an '''[[absolute value (algebra)|absolute value]]''' (also a ''modulus'', ''magnitude'', ''value'', or ''valuation'') if it satisfies the following four axioms:

:{| cellpadding=10
|-
|<math>v(a) \ge 0 </math>
|Non-negativity
|-
|<math>v(a) = 0 \iff a = \mathbf{0} </math>
|Positive-definiteness
|-
|<math>v(ab) = v(a) v(b) \,</math>
|Multiplicativeness
|-
|<math>v(a+b) \le v(a) + v(b) </math>
|Subadditivity or the triangle inequality
|}

Where <math>\mathbf{0}</math> denotes the additive [[identity element]] of <math>F</math>. It follows from positive-definiteness and multiplicativeness that <math>v(\mathbf{1}) = 1</math>, where <math>\mathbf{1}</math> denotes the multiplicative identity element of <math>F</math>. The real and complex absolute values defined above are examples of absolute values for an arbitrary field.

If <math>v</math> is an absolute value on <math>F</math>, then the function <math>d</math> on <math>F \times F</math>, defined by <math>d(a, b) = v(a - b) </math>, is a metric and the following are equivalent:

* <math>d</math> satisfies the [[ultrametric]] inequality <math> d(x, y) \le \mathrm{max}\{d(x, z), d(y, z)\}.</math>

* <math> \big\{ v\Big(\sum_{k=1}^n \mathbf{1}\Big) : n \in \mathbb{N} \big\} </math> is [[bounded set|bounded]] in '''R'''.

* <math> v\Big(\sum_{k=1}^n \mathbf{1}\Big) \le 1</math> for every <math> n \in \mathbb{N}.</math>

* <math> v(a) \le 1 \Rightarrow v(1+a) \le 1</math> for all <math> a \in F.</math>

* <math> v(a + b) \le \mathrm{max}\{v(a), v(b)\} </math> for all <math> a, b \in F.</math>

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]].<ref name="Shechter">[http://www.amazon.com/gp/reader/0126227608/?keywords=absolute%20value&v=search-inside Schechter, p 260-261].</ref>

== Vector spaces ==
{{main|Norm (mathematics)}}
Again the fundamental properties of the absolute value for real numbers can be used, with a slight modification, to generalize the notion to an arbitrary vector space.

A real valued function ||·|| on a [[vector space]] <math>V</math> over a field <math>F</math>, is called an '''absolute value''' (or more usually a '''norm''') if it satisfies the following axioms:

For all <math>a</math> in <math>F</math>, and <math>\mathbf{v}</math>, <math>\mathbf{u}</math> in <math>V</math>,

:{| cellpadding=10
|-
|<math>\|\mathbf{v}\| \ge 0 </math>
|Non-negativity
|-
|<math>\|\mathbf{v}\| = 0 \iff \mathbf{v} = 0</math>
|Positive-definiteness
|-
|<math>\|a \mathbf{v}\| = |a| \|\mathbf{v}\| </math>
|Positive homogeneity or positive scalability
|-
|<math>\|\mathbf{v} + \mathbf{u}\| \le \|\mathbf{v}\| + \|\mathbf{u}\| </math>
|Subadditivity or triangle inequality
|}

The norm of a vector is also called its ''length'' or ''magnitude''.

In the case of [[Euclidean space]] '''R'''<sup>''n''</sup>, the function defined by

:<math>\|(x_1, x_2, \cdots , x_n) \| = \sqrt{\sum_{i=1}^{n}(x_i)^2}</math>

is a norm called the [[Euclidean norm]]. When the real numbers '''R''' are considered as the one-dimensional vector space [[Euclidean space|'''R'''<sup>1</sup>]], the absolute value is a [[Norm (mathematics)|norm]], and is the [[Norm (mathematics)#Examples|''p''-norm]] for any ''p''. In fact the absolute value is the "only" norm on '''R'''<sup>1</sup>, in the sense that, for every norm ||·|| on '''R'''<sup>1</sup>, ||''x''||=||1||·|''x''|. The complex absolute value is a special case of the norm in an [[inner product space]]. It is identical to the Euclidean norm, if the [[complex plane]] is identified with the [[Euclidean plane]] '''R'''<sup>2</sup>.

== Algorithms ==
In the [[C (programming language)|C programming language]], the <code>abs()</code>, <code>labs()</code>, <code>llabs()</code> (in C99), <code>fabs()</code>, <code>fabsf()</code>, and <code>fabsl()</code> functions compute the absolute value of an operand. Coding the integer version of the function is trivial, ignoring the boundary case where the largest negative integer is input:

int abs (int i)
{
if (i < 0)
return -i;
else
return i;
}

The [[floating-point]] versions are trickier, as they have to contend with special codes for [[infinity]] and [[not-a-number]]s.

The function for absolute value in [[Fortran]], [[Matlab]], and [[GNU Octave]] is <code>abs</code>. It handles integer, real as well as complex numbers.

Using [[assembly language]], it is possible to take the absolute value of a [[processor register|register]] in just three instructions (example shown for a 32-bit register on an [[x86 architecture]], [[Intel]] syntax):

cdq
xor eax, edx
sub eax, edx

<code>cdq</code> extends the sign bit of <code>eax</code> into <code>edx</code>. If <code>eax</code> is nonnegative, then <code>edx</code> becomes zero, and the latter two instructions have no effect, leaving <code>eax</code> unchanged. If <code>eax</code> is negative, then <code>edx</code> becomes 0xFFFFFFFF, or -1. The next two instructions then become a [[two's complement]] inversion, giving the absolute value of the negative value in <code>eax</code>.

==Notes==

{{reflist}}

==References==
* Nahin, Paul J.; [http://www.amazon.com/gp/reader/0691027951 ''An Imaginary Tale'']; Princeton University Press; (hardcover, 1998). ISBN 0-691-02795-1
* O'Connor, J.J. and Robertson, E.F.; [http://www-history.mcs.st-andrews.ac.uk/Mathematicians/Argand.html "Jean Robert Argand"]
* Schechter, Eric; ''Handbook of Analysis and Its Foundations'', pp 259-263, [http://www.amazon.com/gp/reader/0126227608/?keywords=absolute%20value&v=search-inside "Absolute Values"], Academic Press (1997) ISBN 0-12-622760-8
* {{MathWorld | urlname=AbsoluteValue | title=Absolute Value}}
* {{PlanetMath | urlname=AbsoluteValue | title=absolute value | id=448}}
</div>

== See also ==
* [[Absolute value (algebra)]]
* [[Valuation (mathematics)]]

[[Category:Numeration]]
[[Category:Elementary special functions]]

[[ar:قيمة مطلقة]]
[[bs:Apsolutna vrijednost]]
[[bg:Абсолютна стойност]]
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[[de:Betragsfunktion]]
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[[fa:قدر مطلق (ریاضی)]]
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[[zh-classical:絕對值]]
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[[he:ערך מוחלט]]
[[hu:Abszolútérték-függvény]]
[[nl:Absolute waarde]]
[[ja:絶対値]]
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[[pl:Wartość bezwzględna]]
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[[ru:Абсолютная величина]]
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[[sl:Absolutna vrednost]]
[[sr:Апсолутна вредност]]
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[[fi:Itseisarvo]]
[[sv:Absolutbelopp]]
[[th:ค่าสัมบูรณ์]]
[[vi:Giá trị tuyệt đối]]
[[tr:Mutlak değer]]
[[uk:Абсолютна величина]]
[[zh:绝对值]]