Editing Complex number (section)

===Other number systems===
{{main|Cayley–Dickson construction|Quaternion|Octonion}}

{| class="wikitable"
|+ Number systems
|-
!
! rational numbers <math>\Q</math>
! real numbers <math>\R</math>
! complex numbers <math>\C</math>
! quaternions <math>\mathbb H</math>
! octonions <math>\mathbb O</math>
! sedenions <math>\mathbb S</math>
|-
! [[complete metric space|complete]]
| {{no}} || {{yes}} || {{yes}} || {{yes}} || {{yes}} || {{yes}}
|-
! [[dimension (vector space)|dimension]] as an <math>\R</math>-vector space
| [does not apply] || 1 || 2 || 4 || 8 || 16 
|-
! [[ordered field|ordered]] 
| {{yes}} || {{yes}} || {{no}} || {{no}} || {{no}} || {{no}}
|-
! multiplication commutative {{nowrap|1=(<math>xy=yx</math>)}}
| {{yes}} || {{yes}} || {{yes}} || {{no}} || {{no}} || {{no}}
|-
! multiplication associative {{nowrap|1=(<math>(xy)z=x(yz)</math>)}}
| {{yes}} || {{yes}} || {{yes}} || {{yes}} || {{no}} || {{no}}
|-
! [[normed division algebra]] {{nowrap|1=(over <math>\R</math>)}}
| [does not apply] || {{yes}} || {{yes}} || {{yes}} || {{yes}} || {{no}}
|}

The process of extending the field <math>\mathbb R</math> of reals to <math>\mathbb C</math> is an instance of the ''Cayley–Dickson construction''. Applying this construction iteratively to <math>\C</math> then yields the [[quaternion]]s, the [[octonion]]s,<ref>{{cite book |first=Kevin |last=McCrimmon |authorlink=Kevin McCrimmon|year=2004 |title=A Taste of Jordan Algebras |page=64 |series=Universitext |publisher=Springer |isbn=0-387-95447-3}} {{mr|id=2014924}}</ref> the [[sedenion]]s, and the [[trigintaduonion]]s. This construction turns out to diminish the structural properties of the involved number systems. 

Unlike the reals, <math>\Complex</math> is not an [[ordered field]], that is to say, it is not possible to define a relation {{math|''z''<sub>1</sub> < ''z''<sub>2</sub>}} that is compatible with the addition and multiplication. In fact, in any ordered field, the square of any element is necessarily positive, so {{math|1=''i''<sup>2</sup> = −1}} precludes the existence of an [[total order|ordering]] on <math>\Complex.</math>{{sfn|Apostol|1981|p=25}} Passing from <math>\C</math> to the quaternions <math>\mathbb H</math> loses commutativity, while the octonions (additionally to not being commutative) fail to be associative. The reals, complex numbers, quaternions and octonions are all [[normed division algebra]]s over <math>\mathbb R</math>. By [[Hurwitz's theorem (normed division algebras)|Hurwitz's theorem]] they are the only ones; the [[sedenion]]s, the next step in the Cayley–Dickson construction, fail to have this structure.

The Cayley–Dickson construction is closely related to the [[regular representation]] of <math>\mathbb C,</math> thought of as an <math>\mathbb R</math>-[[Algebra (ring theory)|algebra]] (an <math>\mathbb{R}</math>-vector space with a multiplication), with respect to the basis {{math|(1, ''i'')}}. This means the following: the <math>\mathbb R</math>-linear map
<math display=block>\begin{align}
   \mathbb{C} &\rightarrow \mathbb{C} \\
   z &\mapsto wz
 \end{align}</math>
for some fixed complex number {{mvar|w}} can be represented by a {{math|2 × 2}} matrix (once a basis has been chosen). With respect to the basis {{math|(1, ''i'')}}, this matrix is
<math display=block>\begin{pmatrix}
  \operatorname{Re}(w) & -\operatorname{Im}(w) \\
  \operatorname{Im}(w) &  \operatorname{Re}(w)
 \end{pmatrix},</math>
that is, the one mentioned in the section on matrix representation of complex numbers above. While this is a [[linear representation]] of <math>\mathbb C</math> in the 2 × 2 real matrices, it is not the only one. Any matrix
<math display=block>J = \begin{pmatrix}p & q \\ r & -p \end{pmatrix}, \quad p^2 + qr + 1 = 0</math>
has the property that its square is the negative of the identity matrix: {{math|1=''J''<sup>2</sup> = −''I''}}. Then
<math display=block>\{ z = a I + b J : a,b \in \mathbb{R} \}</math>
is also isomorphic to the field <math>\mathbb C,</math> and gives an alternative complex structure on <math>\mathbb R^2.</math> This is generalized by the notion of a [[linear complex structure]].

[[Hypercomplex number]]s also generalize <math>\mathbb R,</math> <math>\mathbb C,</math> <math>\mathbb H,</math> and <math>\mathbb{O}.</math> For example, this notion contains the [[split-complex number]]s, which are elements of the ring <math>\mathbb R[x]/(x^2-1)</math> (as opposed to <math>\mathbb R[x]/(x^2+1)</math> for complex numbers). In this ring, the equation {{math|1=''a''<sup>2</sup> = 1}} has four solutions.

The field <math>\mathbb R</math> is the completion of <math>\mathbb Q,</math> the field of [[rational number]]s, with respect to the usual [[absolute value]] [[metric (mathematics)|metric]]. Other choices of [[metric (mathematics)|metrics]] on <math>\mathbb Q</math> lead to the fields <math>\mathbb Q_p</math> of [[p-adic number|{{mvar|p}}-adic numbers]] (for any [[prime number]] {{mvar|p}}), which are thereby analogous to <math>\mathbb{R}</math>. There are no other nontrivial ways of completing <math>\mathbb Q</math> than <math>\mathbb R</math> and <math>\mathbb Q_p,</math> by [[Ostrowski's theorem]]. The algebraic closures <math>\overline {\mathbb{Q}_p}</math> of <math>\mathbb Q_p</math> still carry a norm, but (unlike <math>\mathbb C</math>) are not complete with respect to it. The completion <math>\mathbb{C}_p</math> of <math>\overline {\mathbb{Q}_p}</math> turns out to be algebraically closed. By analogy, the field is called {{mvar|p}}-adic complex numbers.

The fields <math>\mathbb R,</math> <math>\mathbb Q_p,</math> and their finite field extensions, including <math>\mathbb C,</math> are called [[local field]]s.