Editing Hypercomplex number (section)

== Higher-dimensional examples (more than one non-real axis) ==

=== Clifford algebras ===
A [[Clifford algebra]] is the unital associative algebra generated over an underlying vector space equipped with a [[quadratic form]]. Over the real numbers this is equivalent to being able to define a symmetric scalar product, {{nowrap|1=''u'' ⋅ ''v'' = {{sfrac|1|2}}(''uv'' + ''vu'')}} that can be used to [[orthogonalization|orthogonalise]] the quadratic form, to give a basis {{nowrap|{{mset|''e''<sub>1</sub>, ..., ''e''<sub>''k''</sub>}}}} such that:
<math display="block">\frac{1}{2} \left(e_i e_j + e_j e_i\right) = \begin{cases}
  -1, 0, +1 & i = j, \\
          0 & i \not = j.
\end{cases}</math>

Imposing closure under multiplication generates a multivector space spanned by a basis of 2<sup>''k''</sup> elements, {{mset|1, ''e''<sub>1</sub>, ''e''<sub>2</sub>, ''e''<sub>3</sub>, ..., ''e''<sub>1</sub>''e''<sub>2</sub>, ..., ''e''<sub>1</sub>''e''<sub>2</sub>''e''<sub>3</sub>, ...}}.  These can be interpreted as the basis of a hypercomplex number system. Unlike the basis {{mset|''e''<sub>1</sub>, ..., ''e''<sub>''k''</sub>}}, the remaining basis elements need not [[Anticommutative property|anti-commute]], depending on how many simple exchanges must be carried out to swap the two factors.  So {{nowrap|1=''e''<sub>1</sub>''e''<sub>2</sub> = −''e''<sub>2</sub>''e''<sub>1</sub>}}, but {{nowrap|1=''e''<sub>1</sub>(''e''<sub>2</sub>''e''<sub>3</sub>) = +(''e''<sub>2</sub>''e''<sub>3</sub>)''e''<sub>1</sub>}}.

Putting aside the bases which contain an element ''e''<sub>''i''</sub> such that {{nowrap|1=''e''<sub>''i''</sub><sup>2</sup> = 0}} (i.e. directions in the original space over which the quadratic form was [[degenerate form|degenerate]]), the remaining Clifford algebras can be identified by the label Cl<sub>''p'',''q''</sub>(<math>\mathbb{R}</math>), indicating that the algebra is constructed from ''p'' simple basis elements with {{nowrap|1=''e''<sub>''i''</sub><sup>2</sup> = +1}}, ''q'' with {{nowrap|1=''e''<sub>''i''</sub><sup>2</sup> = −1}}, and where <math>\mathbb{R}</math> indicates that this is to be a Clifford algebra over the reals—i.e. coefficients of elements of the algebra are to be real numbers.

These algebras, called [[geometric algebra]]s, form a systematic set, which turn out to be very useful in physics problems which involve [[rotation]]s, [[phase (waves)|phase]]s, or [[Spin (physics)|spin]]s, notably in [[classical mechanics|classical]] and [[quantum mechanics]], [[electromagnetic theory]] and [[theory of relativity|relativity]].

Examples include: the [[complex number]]s Cl<sub>0,1</sub>(<math>\mathbb{R}</math>), [[split-complex number]]s Cl<sub>1,0</sub>(<math>\mathbb{R}</math>), [[quaternion]]s Cl<sub>0,2</sub>(<math>\mathbb{R}</math>), [[split-biquaternion]]s Cl<sub>0,3</sub>(<math>\mathbb{R}</math>), [[split-quaternion]]s {{nowrap|Cl<sub>1,1</sub>(<math>\mathbb{R}</math>) ≈ Cl<sub>2,0</sub>(<math>\mathbb{R}</math>)}} (the natural algebra of two-dimensional space); Cl<sub>3,0</sub>(<math>\mathbb{R}</math>) (the natural algebra of three-dimensional space, and the algebra of the [[Pauli matrices]]); and the [[spacetime algebra]] Cl<sub>1,3</sub>(<math>\mathbb{R}</math>).

The elements of the algebra Cl<sub>''p'',''q''</sub>(<math>\mathbb{R}</math>) form an even subalgebra Cl{{su|lh=1em|p=[0]|b=''q''+1,''p''}}(<math>\mathbb{R}</math>) of the algebra Cl<sub>''q''+1,''p''</sub>(<math>\mathbb{R}</math>), which can be used to parametrise rotations in the larger algebra.  There is thus a close connection between complex numbers and rotations in two-dimensional space; between quaternions and rotations in three-dimensional space; between split-complex numbers and (hyperbolic) rotations ([[Lorentz transformations]]) in 1+1-dimensional space, and so on.

Whereas Cayley–Dickson and split-complex constructs with eight or more dimensions are not associative with respect to multiplication, Clifford algebras retain associativity at any number of dimensions.

In 1995 [[Ian R. Porteous]] wrote on "The recognition of subalgebras" in his book on Clifford algebras. His Proposition 11.4 summarizes the hypercomplex cases:<ref>{{citation |author-link=Ian R. Porteous |first=Ian R. |last=Porteous |title=Clifford Algebras and the Classical Groups |publisher=[[Cambridge University Press]] |year=1995 |isbn=0-521-55177-3 |pages=88–89 }}</ref>
: Let ''A'' be a real associative algebra with unit element 1. Then
:* 1 generates <math>\mathbb{R}</math> ([[real number|algebra of real numbers]]),
:* any two-dimensional subalgebra generated by an element ''e''<sub>0</sub> of ''A'' such that {{nowrap|1=''e''<sub>0</sub><sup>2</sup> = −1}} is isomorphic to <math>\mathbb{C}</math> ([[complex number|algebra of complex number]]s),
:* any two-dimensional subalgebra generated by an element ''e''<sub>0</sub> of ''A'' such that {{nowrap|1=''e''<sub>0</sub><sup>2</sup> = 1}} is isomorphic to <math>\mathbb{R}</math><sup>2</sup> (pairs of real numbers with component-wise product, isomorphic to the [[split-complex number|algebra of split-complex numbers]]),
:* any four-dimensional subalgebra generated by a set {{mset|''e''<sub>0</sub>, ''e''<sub>1</sub>}} of mutually anti-commuting elements of ''A'' such that <math>e_0 ^2 = e_1 ^2 = -1</math> is isomorphic to <math>\mathbb{H}</math> ([[quaternion|algebra of quaternions]]),
:* any four-dimensional subalgebra generated by a set {{mset|''e''<sub>0</sub>, ''e''<sub>1</sub>}} of mutually anti-commuting elements of ''A'' such that <math>e_0 ^2 = e_1 ^2 = 1</math> is isomorphic to M<sub>2</sub>(<math>\mathbb{R}</math>) (2 × 2 [[real matrices]], [[coquaternion]]s),
:* any eight-dimensional subalgebra generated by a set {{mset|''e''<sub>0</sub>, ''e''<sub>1</sub>, ''e''<sub>2</sub>}} of mutually anti-commuting elements of ''A'' such that <math>e_0 ^2 = e_1 ^2 = e_2 ^2 = -1</math> is isomorphic to <sup>2</sup><math>\mathbb{H}</math> ([[split-biquaternion]]s),
:* any eight-dimensional subalgebra generated by a set {{mset|''e''<sub>0</sub>, ''e''<sub>1</sub>, ''e''<sub>2</sub>}} of mutually anti-commuting elements of ''A'' such that <math>e_0 ^2 = e_1 ^2 = e_2 ^2 = 1</math> is isomorphic to M<sub>2</sub>(<math>\mathbb{C}</math>) ({{nowrap|2 × 2}} complex matrices, [[biquaternion]]s, [[Pauli algebra]]).

{{for|extension beyond the classical algebras|Classification of Clifford algebras}}

=== 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.

=== Tensor products ===
The [[tensor product]] of any two algebras is another algebra, which can be used to produce many more examples of hypercomplex number systems.

In particular taking tensor products with the complex numbers (considered as algebras over the reals) leads to four-dimensional [[bicomplex number]]s <math>\mathbb{C} \otimes_\mathbb{R} \mathbb{C}</math> (isomorphic to tessarines <math>\mathbb{C} \otimes_\mathbb{R} D</math>), eight-dimensional [[biquaternion]]s <math>\mathbb{C} \otimes_\mathbb{R} \mathbb{H}</math>, and 16-dimensional [[octonion|complex octonion]]s <math>\mathbb{C} \otimes_\mathbb{R} \mathbb{O}</math>.

=== Further examples ===
* [[bicomplex number]]s: a 4-dimensional vector space over the reals, 2-dimensional over the complex numbers, isomorphic to tessarines.
* [[multicomplex number]]s: 2<sup>''n''</sup>-dimensional vector spaces over the reals, 2<sup>''n''−1</sup>-dimensional over the complex numbers
* [[composition algebra]]: algebra with a [[quadratic form]] that composes with the product