View source for Hypercomplex number

{{Short description|Element of a unital algebra over the field of real numbers}}
{{Distinguish|surcomplex number}}
{{Redirect|Hypernumber|the extension of the real numbers used in [[non-standard analysis]]|Hyperreal number}}
In [[mathematics]], '''hypercomplex number''' is a traditional term for an [[element (mathematics)|element]] of a finite-dimensional [[Algebra over a field#Unital algebra|unital]] [[algebra over a field|algebra]] over the [[field (mathematics)|field]] of [[real number]]s. 
The study of hypercomplex numbers in the late 19th century forms the basis of modern [[group representation]] theory.

== History ==
[[File:Common_associative_algebras_over_the_reals.svg|thumb|right|300px|A [[lattice (order)|lattice]] of some hypercomplex number systems and the corresponding lattice of [[group (mathematics)|group]]s generated by the basis elements]]

In the nineteenth century, [[number system]]s called [[quaternion]]s, [[tessarine]]s, [[coquaternion]]s, [[biquaternion]]s, and [[octonion]]s became established concepts in mathematical literature, extending the real and [[complex number]]s. The concept of a hypercomplex number covered them all, and called for a discipline to explain and classify them.

The cataloguing project began in 1872 when [[Benjamin Peirce]] first published his ''Linear Associative Algebra'', and was carried forward by his son [[Charles Sanders Peirce]].<ref>{{citation |title=Linear Associative Algebra |journal=[[American Journal of Mathematics]] |volume=4 |issue=1 |pages=221–6 |year=1881 |jstor=2369153|last1= Peirce|first1= Benjamin|doi=10.2307/2369153 |url=http://archive.org/details/linearassocalgeb00pierrich }}</ref>  Most significantly, they identified the [[nilpotent]] and the [[idempotent element (ring theory)|idempotent element]]s as useful hypercomplex numbers for classifications. The [[Cayley–Dickson construction]] used [[involution (mathematics)|involution]]s to generate complex numbers, quaternions, and octonions out of the real number system. Hurwitz and Frobenius proved theorems that put limits on hypercomplexity: [[Hurwitz's theorem (normed division algebras)|Hurwitz's theorem]] says finite-dimensional real [[composition algebra]]s are the reals <math>\mathbb{R}</math>, the complexes <math>\mathbb{C}</math>, the quaternions <math>\mathbb{H}</math>, and the octonions <math>\mathbb{O}</math>, and the [[Frobenius theorem (real division algebras)|Frobenius theorem]] says the only real [[associative division algebra]]s are <math>\mathbb{R}</math>, <math>\mathbb{C}</math>, and <math>\mathbb{H}</math>. In 1958 [[Frank Adams|J. Frank Adams]] published a further generalization in terms of Hopf invariants on ''H''-spaces which still limits the dimension to 1, 2, 4, or 8.<ref name="Adams1958">{{citation | jstor=1970147 | title=On the Non-Existence of Elements of Hopf Invariant One | author=Adams, J. F. | journal=Annals of Mathematics |date=July 1960  | volume=72 | issue=1 | pages=20–104 | doi=10.2307/1970147| url=http://www.math.rochester.edu/people/faculty/doug/otherpapers/Adams-HI1.pdf | citeseerx=10.1.1.299.4490 }}</ref>

It was [[matrix (mathematics)|matrix algebra]] that harnessed the hypercomplex systems. For instance, 2 x 2 [[real matrix|real matrices]] were found [[Isomorphism|isomorphic]] to [[coquaternion]]s. Soon the matrix paradigm began to explain several others as they were represented by matrices and their operations.  In 1907 [[Joseph Wedderburn]] showed that associative hypercomplex systems could be represented by [[square matrices]], or [[direct product]]s of algebras of square matrices.<ref>{{citation |author=J.H.M. Wedderburn |author-link=Joseph Wedderburn | title=On Hypercomplex Numbers |journal=Proceedings of the London Mathematical Society |volume=6 | pages=77–118 |year=1908 | doi= 10.1112/plms/s2-6.1.77 |url=https://zenodo.org/record/1447798 }}</ref><ref>[[Emil Artin]] later generalized Wedderburn's result so it is known as the [[Artin–Wedderburn theorem]]</ref> From that date the preferred term for a ''hypercomplex system'' became ''[[associative algebra]]'', as seen in the title of Wedderburn's thesis at [[University of Edinburgh]]. Note, however, that non-associative systems like octonions and  [[hyperbolic quaternion]]s represent another type of hypercomplex number.

As [[Thomas W. Hawkins Jr.|Thomas Hawkins]]<ref>{{citation |first=Thomas |last=Hawkins |title=Hypercomplex numbers, Lie groups, and the creation of group representation theory |journal=[[Archive for History of Exact Sciences]] |volume=8 |pages=243–287 |year=1972 |issue=4 |doi=10.1007/BF00328434 |s2cid=120562272 }}</ref> explains, the hypercomplex numbers are stepping stones to learning about [[Lie group]]s and [[group representation]] theory. For instance, in 1929 [[Emmy Noether]] wrote on "hypercomplex quantities and representation theory".<ref>{{citation | last = Noether | first = Emmy | year = 1929 | title = Hyperkomplexe Größen und Darstellungstheorie | trans-title = Hypercomplex Quantities and the Theory of Representations | journal = Mathematische Annalen | volume = 30 | pages = 641–92 | doi = 10.1007/BF01187794 | s2cid = 120464373 | language = de | url = http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=GDZPPN002371448&L=1 | access-date = 2016-01-14 | archive-url = https://web.archive.org/web/20160329230805/http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=GDZPPN002371448&L=1 | archive-date = 2016-03-29 | url-status = dead }}</ref> In 1973 [[Isaiah Kantor|Kantor]] and Solodovnikov published a textbook on hypercomplex numbers which was translated in 1989.<ref name=KS78>Kantor, I.L., Solodownikow (1978), ''Hyperkomplexe Zahlen'', BSB B.G. Teubner Verlagsgesellschaft, Leipzig</ref><ref>{{Citation | last1=Kantor | first1=I. L. | last2=Solodovnikov | first2=A. S. | title=Hypercomplex numbers | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-0-387-96980-0 | mr=996029 | year=1989 | url-access=registration | url=https://archive.org/details/hypercomplexnumb0000kant }}</ref>

[[Karen Parshall]] has written a detailed exposition of the heyday of hypercomplex numbers,<ref>{{citation |author-link=Karen Parshall |first=Karen |last=Parshall |title=Joseph H. M. Wedderburn and the structure theory of algebras |journal=Archive for History of Exact Sciences |volume=32 |pages=223–349 |year=1985 |issue=3–4 |doi=10.1007/BF00348450 |s2cid=119888377 }}</ref> including the role of mathematicians including [[Theodor Molien]]<ref>{{citation |author-link=Theodor Molien |first=Theodor |last=Molien |title=Ueber Systeme höherer complexer Zahlen |journal=Mathematische Annalen |volume=41 |issue=1 |pages=83–156 |year=1893 |doi=10.1007/BF01443450 |s2cid=122333076 |url=https://zenodo.org/record/2029540}}</ref> and [[Eduard Study]].<ref>{{citation |author-link=Eduard Study |first=Eduard |last=Study |year=1898 |chapter=Theorie der gemeinen und höhern komplexen Grössen |title=[[Klein's encyclopedia|Encyclopädie der mathematischen Wissenschaften]] |volume=I A |issue=4 |pages=147–183}}</ref> For the transition to [[Abstract algebra|modern algebra]], [[Bartel van der Waerden]] devotes thirty pages to hypercomplex numbers in his ''History of Algebra''.<ref>{{citation |author-link=B.L. van der Waerden |first=B.L. |last=van der Waerden |year=1985 |title=A History of Algebra |chapter=10. The discovery of algebras, 11. Structure of algebras |publisher=Springer |isbn=3-540-13610X}}</ref>

== Definition ==
A definition of a '''hypercomplex number''' is given by {{harvtxt|Kantor|Solodovnikov|1989}} as an element of a [[unital algebra|unital]], but not necessarily [[associative property|associative]] or [[commutative property|commutative]], finite-dimensional algebra over the real numbers. Elements are generated with real number coefficients <math>(a_0, \dots, a_n)</math> for a basis <math>\{ 1, i_1, \dots, i_n \}</math>. Where possible, it is conventional to choose the basis so that <math>i_k^2 \in \{ -1, 0, +1 \}</math>. A technical approach to hypercomplex numbers directs attention first to those of [[dimension]] two.

== Two-dimensional real algebras ==
'''Theorem:'''<ref name=KS78/>{{rp|14,15}}<ref>{{citation |author-link=Isaak Yaglom |first=Isaak |last=Yaglom |year=1968 |title=Complex Numbers in Geometry |pages=10–14}}</ref><ref>{{citation |editor-first=John H. |editor-last=Ewing |year=1991 |title=Numbers |page=237 |publisher=Springer |isbn=3-540-97497-0}}</ref> Up to isomorphism, there are exactly three 2-dimensional unital algebras over the reals: the ordinary [[complex number]]s, the [[split-complex number]]s, and the [[dual number]]s. In particular, every 2-dimensional unital algebra over the reals is associative and commutative.

Proof: Since the algebra is 2-dimensional, we can pick a basis {{nowrap|{{mset|1, ''u''}}}}.  Since the algebra is [[closure (mathematics)|closed]] under squaring, the non-real basis element ''u'' squares to a linear combination of 1 and ''u'':
: <math>u^2 = a_0 + a_1 u</math>
for some real numbers ''a''<sub>0</sub> and ''a''<sub>1</sub>.

Using the common method of [[completing the square]] by subtracting ''a''<sub>1</sub>''u'' and adding the quadratic complement ''a''{{su|b=1|p=2}}{{nnbsp}}/{{nnbsp}}4 to both sides yields
: <math>u^2 - a_1 u + \frac{1}{4}a_1^2 = a_0 + \frac{1}{4}a_1^2.</math>

Rewriting the left half of the above equation as <math display="inline">\left(u - \frac{1}{2}a_1\right)^2</math> and then substituting in new element ''ũ'' where <math display="inline">\tilde{u} = u - \frac{1}{2}a_1</math>, one obtains
: <math>\tilde{u}^2 = a_0 + \frac{1}{4}a_1^2.</math>

The three cases depend on this real value:
* If {{nowrap|1=4''a<sub>0</sub>'' = −''a''<sub>1</sub><sup>2</sup>}}, the above formula yields {{nowrap|1=''ũ''<sup>2</sup> = 0}}. Hence, ''ũ'' can directly be identified with the [[nilpotent]] element <math>\varepsilon</math> of the basis <math>\{ 1, ~\varepsilon \}</math> of the dual numbers.
* If {{nowrap|4''a<sub>0</sub>'' > −''a''<sub>1</sub><sup>2</sup>}}, the above formula yields {{nowrap|''ũ''<sup>2</sup> > 0}}. This leads to the split-complex numbers which have normalized basis <math>\{ 1 , ~j \}</math> with <math>j^2 = +1</math>. To obtain ''j'' from ''ũ'', the latter must be divided by the positive real number <math display="inline">a \mathrel{:=} \sqrt{a_0 + \frac{1}{4}a_1^2}</math> which has the same square as ''ũ'' has.
* If {{nowrap|4''a<sub>0</sub>'' < −''a''<sub>1</sub><sup>2</sup>}}, the above formula yields {{nowrap|''ũ''<sup>2</sup> < 0}}. This leads to the complex numbers which have normalized basis <math>\{ 1 , ~i \}</math> with <math>i^2 = -1</math>. To yield ''i'' from ''ũ'', the latter has to be divided by the positive real number <math display="inline">a \mathrel{:=} \sqrt{-(a_0 + \frac{1}{4}a_1^2)}</math> which squares to the negative of ''ũ''<sup>2</sup>.

The complex numbers are the only [[quadratic algebra]] that is a [[Field (mathematics)|field]]. [[Split algebra]]s such as the split-complex numbers that include non-real roots of 1 also contain [[idempotent element|idempotent]]s <math display="inline">\frac{1}{2}(1 \pm j)</math> and [[zero divisor]]s <math>(1 + j)(1 - j) = 0</math>, so such algebras cannot be [[division algebra]]s. However, these properties can turn out to be very meaningful, for instance in representing a [[light cone]] with a [[null cone]].

In a 2004 edition of ''[[Mathematics Magazine]]'' the 2-dimensional real algebras have been styled the "generalized complex numbers".<ref>{{citation |first1=Anthony A. |last1=Harkin |first2=Joseph B. |last2=Harkin |title=Geometry of Generalized Complex Numbers |journal=[[Mathematics Magazine]] |volume=77 |issue=2 |pages=118–129 |year=2004 |doi=10.1080/0025570X.2004.11953236 |s2cid=7837108 |url=http://people.rit.edu/harkin/research/articles/generalized_complex_numbers.pdf}}</ref> The idea of [[cross-ratio]] of four complex numbers can be extended to the 2-dimensional real algebras.<ref>{{citation |first=Sky |last=Brewer |title=Projective Cross-ratio on Hypercomplex Numbers |journal=[[Advances in Applied Clifford Algebras]] |volume=23 |issue=1 |pages=1–14 |year=2013 |doi=10.1007/s00006-012-0335-7 |arxiv=1203.2554|s2cid=119623082 }}</ref>

== 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>.

Another example is the tensor product of two quaternion algebras <math>\mathbb{H}^{\otimes 2}=\mathbb{H}\otimes_\mathbb{R}\mathbb{H} </math> (isomorphic to the Clifford algebra <math>Cl_{3,1}\mathbb{(R)}</math> and to <math>4\times 4</math> real matrices <math>M(4,\mathbb{R})</math>) leading to applications in relativistic physics.<ref>{{cite book |last1=Girard |first1=P. R. |title= Quaternions, Clifford algebras and relativistic physics |date=2007 |publisher=Springer Science & Business Media |location= |language= |url=https://link.springer.com/book/10.1007/978-3-7643-7791-5}}</ref><ref>{{cite book |last1=G'''ü'''rlebeck |first1=K. |last2=Spr'''ö'''ssig |first2=W. |title=Quaternionic and Clifford calculus for physicists and engineers |date=1997 |publisher=John Wiley & Sons |location=New York |language= |url=https://www.wiley.com/Quaternionic+and+Clifford+Calculus+for+Physicists+and+Engineers-p-9780471962007 }}</ref>

More generally, one defines <math>\mathbb{H}^{\otimes m} </math> (and its subalgebras) as ”hyperquaternion algebras”. In particular, <math>\mathbb{H}^{\otimes 3}=M(4,\mathbb{H})</math> yields a [[quaternionic matrix]] and its even subalgebra <math>\mathbb{H}^{\otimes 2} \otimes _\mathbb{R}\mathbb{C} </math> ([[Dirac algebra]]).<ref>{{cite journal |last1=Girard |first1=P.R. |last2=Clarysse |first2=P. |last3=Pujol |first3=R. |last4=Delachartre |first4=P. |date=2025 |title=Hyperquaternionic unitary symplectic groups: A unifying tool for physics|url= |journal=Advances in Applied Clifford Algebras |volume=35 |issue=40 |publisher= Springer |pages= |jstor= |doi=10.1007/s00006-025-01402-w |access-date=}}</ref><ref>{{cite book |last1=Spr'''ö'''ssig |first1=W. |title=Advancements in Complex Analysis: From Theory to Practice |chapter=Some new aspects in hypercomplex analysis |date=2020 |pages=497–518 |publisher=Springer |location= |language= |url=https://doi.org/10.1007/978-3-030-40120-7}}</ref>

=== 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

== See also ==
* [[Thomas Kirkman]]
* [[Georg Scheffers]]
* [[Richard Brauer]]
* [[Hypercomplex analysis]]

== References ==
{{reflist}}

== Further reading ==
{{refbegin}}
* {{citation |first=Daniel |last=Alfsmann |chapter=On families of 2^N dimensional hypercomplex algebras suitable for digital signal processing |chapter-url=http://www.eurasip.org/proceedings/eusipco/eusipco2006/papers/1568981962.pdf |title=14th European Signal Processing Conference, Florence, Italy |year=2006 |pages=1–4 |url=https://ieeexplore.ieee.org/xpl/conhome/7065145/proceeding }}
* {{citation |author-link=Emil Artin |first=Emil |last=Artin |orig-year=1928 |chapter=Zur Theorie der hyperkomplexen Zahlen; Zur Arithmetik hyperkomplexer Zahlen  |title=The Collected Papers of Emil Artin |publisher=[[Addison-Wesley]] |year=1965 |pages=301–345 |editor-link=Serge Lang |editor-first=Serge |editor-last=Lang |editor2-link=John T. Tate |editor2-first=John T. |editor2-last=Tate}}
* {{Citation | last1=Baez | first1=John | author1-link=John Baez | title=The Octonions | url=http://math.ucr.edu/home/baez/octonions/octonions.html | year=2002 | journal=[[Bulletin of the American Mathematical Society]] | issn=0002-9904 | volume=39 | issue=2 | pages=145–205 | doi=10.1090/S0273-0979-01-00934-X | arxiv=math/0105155 | s2cid=586512 }}
* {{citation |author-link=Élie Cartan |first=Élie |last=Cartan |chapter=Les systèmes de nombres complex et les groupes de transformations |title=Encyclopédie des sciences mathématiques pures et appliquées |volume=I 1 |year=1908 }}. and ''Ouvres Completes'' T.2 pt. 1, pp 107–246.
* {{citation |author-link=Max Herzberger |first=Max |last=Herzberger |year=1923 |url=https://www.deutsche-digitale-bibliothek.de/item/4V6F5ID5PHYD7UJ4F4ZDWHD2SBEXGCGQ |title=Ueber Systeme hyperkomplexer Grössen |work=Doctoral Dissertation |publisher=[[Humboldt University of Berlin|Friedrich Wilhelm University]] |access-date=2015-09-20 |archive-date=2021-01-30 |archive-url=https://web.archive.org/web/20210130183638/https://www.deutsche-digitale-bibliothek.de/item/4V6F5ID5PHYD7UJ4F4ZDWHD2SBEXGCGQ |url-status=dead }}
* {{citation |first=Jeanne |last=La Duke |chapter=The study of linear associative algebras in the United States, 1870–1927 |pages=147–159 |editor-first=B. |editor-last=Srinivasan |editor2-first=J. |editor2-last=Sally |title=Emmy Noether in Bryn Mawr: Proceedings of a Symposium Sponsored by the Association for Women in Mathematics in Honor of Emmy Noether's 100th Birthday |url=https://books.google.com/books?id=b7NYvQEACAAJ |date=1983 |publisher=Springer |isbn=978-0-387-90838-0 }}
* {{citation |first=Silviu |last=Olariu |title=Complex Numbers in N Dimensions |publisher=[[Elsevier]] |year=2002 |isbn=0-444-51123-7 |series=North-Holland Mathematics Studies |volume=190}}
* {{citation |editor-first=Irene |editor-last=Sabadini|editor-link=Irene Sabadini |editor2-first=Michael |editor2-last=Shapiro |editor3-first=Frank |editor3-last=Sommen |title=Hypercomplex Analysis and Applications |publisher=Birkhauser |year=2009 |isbn=978-3-7643-9892-7 }}
* {{citation |author-link=Henry Taber |first=Henry |last=Taber |title=On Hypercomplex Number Systems |journal=[[Transactions of the American Mathematical Society]] |volume=5 |issue=4 |pages=509–548 |year=1904 |jstor=1986280|doi=10.2307/1986280}}
* {{citation |author-link=Joseph Wedderburn |first=J.H. |last=MacLagan Wedderburn |title=On Hypercomplex Numbers |journal=Proceedings of the London Mathematical Society |volume=s2-6 |issue=1 |pages=77–118 |year=1908 |doi=10.1112/plms/s2-6.1.77 |url=https://zenodo.org/record/1447798 }}
{{refend}}

== External links ==
{{wikibooks|Abstract Algebra|Hypercomplex numbers}}
* {{springer|title=Hypercomplex number|id=p/h048390}}
* {{mathworld|urlname=HypercomplexNumber|title=Hypercomplex number}}
* {{citation |url=http://neo-classical-physics.info/uploads/3/0/6/5/3065888/study_-_complex_numbers_and_transformation_groups.pdf |first=E. |last=Study |title=On systems of complex numbers and their application to the theory of transformation groups}} (English translation)
* {{citation |url=http://neo-classical-physics.info/uploads/3/0/6/5/3065888/frobenius_-_hypercomplex_i.pdf |first=G. |last=Frobenius |title=Theory of hypercomplex quantities}} (English translation)

{{Number systems}}
{{Dimension topics}}
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{{DEFAULTSORT:Hypercomplex Number}}
[[Category:Hypercomplex numbers| ]]
[[Category:History of mathematics]]
[[Category:Historical treatment of quaternions]]