Editing Absolute value (section)

==Distance==
{{See also|Metric space}}
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, \dots , a_n) </math>

and

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

in [[Euclidean space|Euclidean {{mvar|n}}-space]] is defined as:
:<math>\sqrt{\textstyle\sum_{i=1}^n(a_i-b_i)^2}. </math>

This can be seen as a generalisation, since for <math>a_1</math> and <math>b_1</math> real, i.e. in a 1-space, according to the alternative definition of the absolute value,

:<math>|a_1 - b_1| = \sqrt{(a_1 - b_1)^2} = \sqrt{\textstyle\sum_{i=1}^1(a_i-b_i)^2},</math>

and for <math> a = a_1 + i a_2 </math> and <math> b = b_1 + i b_2 </math> complex numbers, i.e. in a 2-space,
 
:{|
|-
|<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} = \sqrt{\textstyle\sum_{i=1}^2(a_i-b_i)^2}.</math>
|}

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 a [[distance function]] as follows:

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:<ref>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'')}}.</ref>
:{|
|-
|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
|}