Editing Hypercomplex number (section)

=== Cayley–Dickson construction ===
{{Further|Cayley–Dickson construction}}
[[File:Cayley_Q8_multiplication_graph.svg|thumb|link={{filepath:Cayley_Q8_multiplication_graph.svg}}|Cayley Q8 graph of quaternion multiplication showing cycles of multiplication of ''i'' (red), ''j'' (green) and ''k'' (blue). In [{{filepath:Cayley_Q8_quaternion_multiplication_graph.svg}} the SVG file,] hover over or click a path to highlight it.]]
All of the Clifford algebras Cl<sub>''p'',''q''</sub>(<math>\mathbb{R}</math>) apart from the real numbers, complex numbers and the quaternions contain non-real elements that square to +1; and so cannot be division algebras.  A different approach to extending the complex numbers is taken by the [[Cayley–Dickson construction]].  This generates number systems of dimension 2<sup>''n''</sup>, ''n'' = 2, 3, 4, ..., with bases <math>\left\{1, i_1, \dots, i_{2^n-1}\right\}</math>, where all the non-real basis elements anti-commute and satisfy <math>i_m^2 = -1</math>. In 8 or more dimensions ({{nowrap|''n'' ≥ 3}}) these algebras are non-associative. In 16 or more dimensions ({{nowrap|''n'' ≥ 4}}) these algebras also have [[zero-divisor]]s.

The first algebras in this sequence include the 4-dimensional [[quaternion]]s, 8-dimensional [[octonion]]s, and 16-dimensional [[sedenion]]s. An algebraic symmetry is lost with each increase in dimensionality: quaternion multiplication is not [[commutative]], octonion multiplication is non-[[associative]], and the [[norm (mathematics)|norm]] of [[sedenion]]s is not multiplicative. After the sedenions are the 32-dimensional [[trigintaduonion]]s (or 32-nions), the 64-dimensional sexagintaquatronions (or 64-nions), the 128-dimensional centumduodetrigintanions (or 128-nions), the 256-dimensional ducentiquinquagintasexions (or 256-nions), and ''[[ad infinitum]]'', as summarized in the table below.<ref>{{cite journal | last=Cariow | first=Aleksandr | title=An unified approach for developing rationalized algorithms for hypercomplex number multiplication | journal=Przegląd Elektrotechniczny | publisher=Wydawnictwo SIGMA-NOT | volume=1 | issue=2 | date=2015 | issn=0033-2097 | doi=10.15199/48.2015.02.09 | pages=38–41}}</ref>

{| class="wikitable"
|-
! Name !! No. of<br>[[dimension]]s !! Dimensions<br>([[Powers of two|2<sup>n</sup>]]) !! Symbol
|-
| [[real number]]s || 1 || 2<sup>0</sup> || <math>\mathbb R</math>
|-
| [[complex number]]s || 2 || 2<sup>1</sup> || <math>\mathbb C</math>
|-
| [[quaternion]]s || 4 || 2<sup>2</sup> || <math>\mathbb H</math>
|-
| [[octonion]]s || 8 || 2<sup>3</sup> || <math>\mathbb O</math>
|-
| [[sedenion]]s || 16 || 2<sup>4</sup> || <math>\mathbb S</math>
|-
| [[trigintaduonion]]s || 32 || 2<sup>5</sup> || <math>\mathbb T</math>
|-
| sexagintaquatronions || 64 || 2<sup>6</sup> || 
|-
| centumduodetrigintanions || 128 || 2<sup>7</sup> || 
|-
| ducentiquinquagintasexions || 256 || 2<sup>8</sup> || 
|}

The Cayley–Dickson construction can be modified by inserting an extra sign at some stages. It then generates the "split algebras" in the collection of [[composition algebra]]s instead of the division algebras:
: [[split-complex number]]s with basis <math>\{ 1,\, i_1 \}</math> satisfying <math>\ i_1^2 = +1</math>,
: [[split-quaternion]]s with basis <math>\{ 1,\, i_1,\, i_2,\, i_3 \}</math> satisfying <math>\ i_1^2 = -1,\, i_2^2 = i_3^2 = +1</math>, and
: [[split-octonion]]s with basis <math>\{ 1,\, i_1,\, \dots,\, i_7 \}</math> satisfying <math>\ i_1^2 = i_2^2 = i_3^2 = -1</math>, <math>\ i_4^2 = i_5^2 = i_6^2 = i_7^2 = +1 .</math>

Unlike the complex numbers, the split-complex numbers are not [[algebraically closed field|algebraically closed]], and further contain nontrivial [[zero divisor]]s and nontrivial [[idempotent]]s. As with the quaternions, split-quaternions are not commutative, but further contain [[nilpotent]]s; they are isomorphic to the [[square matrices]] of dimension two. Split-octonions are non-associative and contain nilpotents.