Editing Absolute value (section)

==Generalizations==

===Ordered rings===
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:<ref>Mac Lane, [https://books.google.com/books?id=L6FENd8GHIUC&pg=PA264 p.&nbsp;264].</ref>

:<math>|a| = \left\{
   \begin{array}{rl}
     a, & \text{if }  a \geq 0 \\
     -a, & \text{if } a < 0.
   \end{array}\right.
 </math>

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.

===Fields===
{{Main|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&nbsp;{{mvar|v}} on a [[field (mathematics)|field]]&nbsp;{{mvar|F}} is called an ''absolute value'' (also a ''modulus'', ''magnitude'', ''value'', or ''valuation'')<ref>Shechter, [https://books.google.com/books?id=eqUv3Bcd56EC&pg=PA260 p.&nbsp;260]. This meaning of ''valuation'' is rare. Usually, a [[valuation (algebra)|valuation]] is the logarithm of the inverse of an absolute value</ref> 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>
|Multiplicativity
|-
|<math>v(a+b)  \le v(a) + v(b)  </math>
|Subadditivity or the triangle inequality
|}

Where '''0''' denotes the [[additive identity]] of&nbsp;{{mvar|F}}. It follows from positive-definiteness and multiplicativity that {{math|1=''v''('''1''') = 1}}, where '''1''' denotes the [[multiplicative identity]] of&nbsp;{{mvar|F}}. The real and complex absolute values defined above are examples of absolute values for an arbitrary field.

If {{mvar|v}} is an absolute value on&nbsp;{{mvar|F}}, then the function&nbsp;{{mvar|d}} on {{math|''F'' × ''F''}}, defined by {{math|1=''d''(''a'', ''b'') = ''v''(''a'' − ''b'')}}, is a metric and the following are equivalent:

* {{mvar|d}} satisfies the [[ultrametric]] inequality <math>d(x, y) \leq \max(d(x,z),d(y,z))</math> for all {{mvar|x}}, {{mvar|y}}, {{mvar|z}} in&nbsp;{{mvar|F}}.
* <math display="inline"> \left\{ v\left( \sum_{k=1}^n \mathbf{1}\right) : n \in \N \right\} </math> is [[bounded set|bounded]] in&nbsp;'''R'''.
* <math> v\left({\textstyle \sum_{k=1}^n } \mathbf{1}\right) \le 1\ </math> for every <math>n \in \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 \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>Shechter, [https://books.google.com/books?id=eqUv3Bcd56EC&pg=PA260 pp.&nbsp;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 generalise the notion to an arbitrary vector space.

A real-valued function on a [[vector space]]&nbsp;{{mvar|V}} over a field&nbsp;{{mvar|F}}, represented as {{math|{{norm}}}}, is called an '''absolute value''', but more usually a [[Norm (mathematics)|'''norm''']], if it satisfies the following axioms:

For all&nbsp;{{mvar|a}} in&nbsp;{{mvar|F}}, and {{math|'''v'''}}, {{math|'''u'''}} in&nbsp;{{mvar|V}},

:{| 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}\| = \left|a\right| \left\|\mathbf{v}\right\| </math>
|Absolute homogeneity or positive scalability
|-
|<math>\|\mathbf{v} + \mathbf{u}\| \le \|\mathbf{v}\| + \|\mathbf{u}\| </math>
|Subadditivity or the triangle inequality
|}

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

In the case of [[Euclidean space]] <math>\mathbb{R}^n</math>, the function defined by

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

is a norm called the Euclidean norm. When the real numbers <math>\mathbb{R}</math> are considered as the one-dimensional vector space <math>\mathbb{R}^1</math>, the absolute value is a [[Norm (mathematics)|norm]], and is the {{mvar|p}}-norm (see [[L^p space#Definition|L<sup>p</sup> space]]) for any&nbsp;{{mvar|p}}. In fact the absolute value is the "only" norm on <math>\mathbb{R}^1</math>, in the sense that, for every norm {{math|{{norm}}}} on <math>\mathbb{R}^1</math>, {{math|1={{norm|''x''}} = {{norm|1}} ⋅ {{abs|''x''}}}}.

The complex absolute value is a special case of the norm in an [[inner product space]], which is identical to the Euclidean norm when the complex plane is identified as the [[Euclidean plane]]&nbsp;<math>\mathbb{R}^2</math>.

===Composition algebras===
{{Main|Composition algebra}}
Every composition algebra ''A'' has an [[involution (mathematics)|involution]] ''x'' → ''x''* called its '''conjugation'''. The product in ''A'' of an element ''x'' and its conjugate ''x''* is written ''N''(''x'') = ''x x''* and called the '''norm of x'''.

The real numbers <math>\mathbb{R}</math>, complex numbers <math>\mathbb{C}</math>, and quaternions <math>\mathbb{H}</math> are all composition algebras with norms given by [[definite quadratic form]]s. The absolute value in these [[division algebra]]s is given by the square root of the composition algebra norm.

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'').