Hypercomplex number#Definition

File:Common_associative_algebras_over_the_reals.svg of some hypercomplex number systems and the corresponding lattice of groups generated by the basis elements]]

In the nineteenth century, number systems called quaternions, tessarines, coquaternions, biquaternions, and octonions became established concepts in mathematical literature, extending the real and complex numbers. 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.{{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 }} Most significantly, they identified the nilpotent and the idempotent elements as useful hypercomplex numbers for classifications. The Cayley–Dickson construction used involutions 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 says finite-dimensional real composition algebras are the reals $\backslash mathbb\{R\}$, the complexes $\backslash mathbb\{C\}$, the quaternions $\backslash mathbb\{H\}$, and the octonions $\backslash mathbb\{O\}$, and the Frobenius theorem says the only real associative division algebras are $\backslash mathbb\{R\}$, $\backslash mathbb\{C\}$, and $\backslash mathbb\{H\}$. In 1958 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.{{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 }}

It was matrix algebra that harnessed the hypercomplex systems. For instance, 2 x 2 real matrices were found isomorphic to coquaternions. 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 products of algebras of square matrices.{{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 }}Emil Artin later generalized Wedderburn's result so it is known as the Artin–Wedderburn theorem 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 quaternions represent another type of hypercomplex number.

As Thomas Hawkins{{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 }} explains, the hypercomplex numbers are stepping stones to learning about Lie groups and group representation theory. For instance, in 1929 Emmy Noether wrote on "hypercomplex quantities and representation theory".{{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 }} In 1973 Kantor and Solodovnikov published a textbook on hypercomplex numbers which was translated in 1989.Kantor, I.L., Solodownikow (1978), Hyperkomplexe Zahlen, BSB B.G. Teubner Verlagsgesellschaft, Leipzig{{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 }}

Karen Parshall has written a detailed exposition of the heyday of hypercomplex numbers,{{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 }} including the role of mathematicians including Theodor Molien{{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}} and Eduard Study.{{citation |author-link=Eduard Study |first=Eduard |last=Study |year=1898 |chapter=Theorie der gemeinen und höhern komplexen Grössen |title=Encyclopädie der mathematischen Wissenschaften |volume=I A |issue=4 |pages=147–183}} For the transition to modern algebra, Bartel van der Waerden devotes thirty pages to hypercomplex numbers in his History of Algebra.{{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}}

A definition of a hypercomplex number is given by {{harvtxt|Kantor|Solodovnikov|1989}} as an element of a unital, but not necessarily associative or commutative, finite-dimensional algebra over the real numbers. Elements are generated with real number coefficients $(a\_0,\; \backslash dots,\; a\_n)$ for a basis $\backslash \{\; 1,\; i\_1,\; \backslash dots,\; i\_n\; \backslash \}$. Where possible, it is conventional to choose the basis so that $i\_k^2\; \backslash in\; \backslash \{\; -1,\; 0,\; +1\; \backslash \}$. A technical approach to hypercomplex numbers directs attention first to those of dimension two.

Theorem:{{rp|14,15}}{{citation |author-link=Isaak Yaglom |first=Isaak |last=Yaglom |year=1968 |title=Complex Numbers in Geometry |pages=10–14}}{{citation |editor-first=John H. |editor-last=Ewing |year=1991 |title=Numbers |page=237 |publisher=Springer |isbn=3-540-97497-0}} Up to isomorphism, there are exactly three 2-dimensional unital algebras over the reals: the ordinary complex numbers, the split-complex numbers, and the dual numbers. 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 closed under squaring, the non-real basis element u squares to a linear combination of 1 and u:

: $u^2\; =\; a\_0\; +\; a\_1\; u$

for some real numbers a₀ and a₁.

Using the common method of completing the square by subtracting a₁u and adding the quadratic complement a{{su|b=1|p=2}}{{nnbsp}}/{{nnbsp}}4 to both sides yields

: $u^2\; -\; a\_1\; u\; +\; \backslash frac\{1\}\{4\}a\_1^2\; =\; a\_0\; +\; \backslash frac\{1\}\{4\}a\_1^2.$

Thus $\backslash left(u\; -\; \backslash frac\{1\}\{2\}a\_1\backslash right)^2\; =\; \backslash tilde\{u\}^2$ where $\backslash tilde\{u\}^2~\; =\; a\_0\; +\; \backslash frac\{1\}\{4\}a\_1^2.$

The three cases depend on this real value:

• If {{nowrap|1=4a₀ = −a₁²}}, the above formula yields {{nowrap|1=ũ² = 0}}. Hence, ũ can directly be identified with the nilpotent element $\backslash varepsilon$ of the basis $\backslash \{\; 1,\; ~\backslash varepsilon\; \backslash \}$ of the dual numbers.
• If {{nowrap|4a₀ > −a₁²}}, the above formula yields {{nowrap|ũ² > 0}}. This leads to the split-complex numbers which have normalized basis $\backslash \{\; 1\; ,\; ~j\; \backslash \}$ with $j^2\; =\; +1$. To obtain j from ũ, the latter must be divided by the positive real number $a\; \backslash mathrel\{:=\}\; \backslash sqrt\{a\_0\; +\; \backslash frac\{1\}\{4\}a\_1^2\}$ which has the same square as ũ has.
• If {{nowrap|4a₀ < −a₁²}}, the above formula yields {{nowrap|ũ² < 0}}. This leads to the complex numbers which have normalized basis $\backslash \{\; 1\; ,\; ~i\; \backslash \}$ with $i^2\; =\; -1$. To yield i from ũ, the latter has to be divided by a positive real number $a\; \backslash mathrel\{:=\}\; \backslash sqrt\{\backslash frac\{1\}\{4\}a\_1^2\; -\; a\_0\}$ which squares to the negative of ũ².

The complex numbers are the only 2-dimensional hypercomplex algebra that is a field.

Split algebras such as the split-complex numbers that include non-real roots of 1 also contain idempotents $\backslash frac\{1\}\{2\}(1\; \backslash pm\; j)$ and zero divisors $(1\; +\; j)(1\; -\; j)\; =\; 0$, so such algebras cannot be division algebras. 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".{{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}} The idea of cross-ratio of four complex numbers can be extended to the 2-dimensional real algebras.{{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 }}

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 orthogonalise the quadratic form, to give a basis {{nowrap|{{mset|e₁, ..., e_k}}}} such that:

$$\backslash frac\{1\}\{2\}\; \backslash left(e\_i\; e\_j\; +\; e\_j\; e\_i\backslash right)\; =\; \backslash begin\{cases\}$$

-1, 0, +1 & i = j, \\

0 & i \not = j.

\end{cases}

Imposing closure under multiplication generates a multivector space spanned by a basis of 2^k elements, {{mset|1, e₁, e₂, e₃, ..., e₁e₂, ..., e₁e₂e₃, ...}}. These can be interpreted as the basis of a hypercomplex number system. Unlike the basis {{mset|e₁, ..., e_k}}, the remaining basis elements need not anti-commute, depending on how many simple exchanges must be carried out to swap the two factors. So {{nowrap|1=e₁e₂ = −e₂e₁}}, but {{nowrap|1=e₁(e₂e₃) = +(e₂e₃)e₁}}.

Putting aside the bases which contain an element e_i such that {{nowrap|1=e_i² = 0}} (i.e. directions in the original space over which the quadratic form was degenerate), the remaining Clifford algebras can be identified by the label Cl_p,q($\backslash mathbb\{R\}$), indicating that the algebra is constructed from p simple basis elements with {{nowrap|1=e_i² = +1}}, q with {{nowrap|1=e_i² = −1}}, and where $\backslash mathbb\{R\}$ 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 algebras, form a systematic set, which turn out to be very useful in physics problems which involve rotations, phases, or spins, notably in classical and quantum mechanics, electromagnetic theory and relativity.

Examples include: the complex numbers Cl_0,1($\backslash mathbb\{R\}$), split-complex numbers Cl_1,0($\backslash mathbb\{R\}$), quaternions Cl_0,2($\backslash mathbb\{R\}$), split-biquaternions Cl_0,3($\backslash mathbb\{R\}$), split-quaternions {{nowrap|Cl_1,1($\backslash mathbb\{R\}$) ≈ Cl_2,0($\backslash mathbb\{R\}$)}} (the natural algebra of two-dimensional space); Cl_3,0($\backslash mathbb\{R\}$) (the natural algebra of three-dimensional space, and the algebra of the Pauli matrices); and the spacetime algebra Cl_1,3($\backslash mathbb\{R\}$).

The elements of the algebra Cl_p,q($\backslash mathbb\{R\}$) form an even subalgebra Cl{{su|lh=1em|p=[0]|b=q+1,p}}($\backslash mathbb\{R\}$) of the algebra Cl_q+1,p($\backslash mathbb\{R\}$), 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:{{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 }}

: Let A be a real associative algebra with unit element 1. Then

:* 1 generates $\backslash mathbb\{R\}$ (algebra of real numbers),

:* any two-dimensional subalgebra generated by an element e₀ of A such that {{nowrap|1=e₀² = −1}} is isomorphic to $\backslash mathbb\{C\}$ (algebra of complex numbers),

:* any two-dimensional subalgebra generated by an element e₀ of A such that {{nowrap|1=e₀² = 1}} is isomorphic to $\backslash mathbb\{R\}$² (pairs of real numbers with component-wise product, isomorphic to the algebra of split-complex numbers),

:* any four-dimensional subalgebra generated by a set {{mset|e₀, e₁}} of mutually anti-commuting elements of A such that $e\_0\; ^2\; =\; e\_1\; ^2\; =\; -1$ is isomorphic to $\backslash mathbb\{H\}$ (algebra of quaternions),

:* any four-dimensional subalgebra generated by a set {{mset|e₀, e₁}} of mutually anti-commuting elements of A such that $e\_0\; ^2\; =\; e\_1\; ^2\; =\; 1$ is isomorphic to M₂($\backslash mathbb\{R\}$) (2 × 2 real matrices, coquaternions),

:* any eight-dimensional subalgebra generated by a set {{mset|e₀, e₁, e₂}} of mutually anti-commuting elements of A such that $e\_0\; ^2\; =\; e\_1\; ^2\; =\; e\_2\; ^2\; =\; -1$ is isomorphic to ²$\backslash mathbb\{H\}$ (split-biquaternions),

:* any eight-dimensional subalgebra generated by a set {{mset|e₀, e₁, e₂}} of mutually anti-commuting elements of A such that $e\_0\; ^2\; =\; e\_1\; ^2\; =\; e\_2\; ^2\; =\; 1$ is isomorphic to M₂($\backslash mathbb\{C\}$) ({{nowrap|2 × 2}} complex matrices, biquaternions, Pauli algebra).

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

{{Further|Cayley–Dickson construction}}

File:Cayley_Q8_multiplication_graph.svg

All of the Clifford algebras Cl_p,q($\backslash mathbb\{R\}$) 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 2ⁿ, n = 2, 3, 4, ..., with bases $\backslash left\backslash \{1,\; i\_1,\; \backslash dots,\; i\_\{2^n-1\}\backslash right\backslash \}$, where all the non-real basis elements anti-commute and satisfy $i\_m^2\; =\; -1$. 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-divisors.

The first algebras in this sequence include the 4-dimensional quaternions, 8-dimensional octonions, and 16-dimensional sedenions. An algebraic symmetry is lost with each increase in dimensionality: quaternion multiplication is not commutative, octonion multiplication is non-associative, and the norm of sedenions is not multiplicative. After the sedenions are the 32-dimensional trigintaduonions (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.{{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}}

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 algebras instead of the division algebras:

: split-complex numbers with basis $\backslash \{\; 1,\backslash ,\; i\_1\; \backslash \}$ satisfying $\backslash \; i\_1^2\; =\; +1$,

: split-quaternions with basis $\backslash \{\; 1,\backslash ,\; i\_1,\backslash ,\; i\_2,\backslash ,\; i\_3\; \backslash \}$ satisfying $\backslash \; i\_1^2\; =\; -1,\backslash ,\; i\_2^2\; =\; i\_3^2\; =\; +1$, and

: split-octonions with basis $\backslash \{\; 1,\backslash ,\; i\_1,\backslash ,\; \backslash dots,\backslash ,\; i\_7\; \backslash \}$ satisfying $\backslash \; i\_1^2\; =\; i\_2^2\; =\; i\_3^2\; =\; -1$, $\backslash \; i\_4^2\; =\; i\_5^2\; =\; i\_6^2\; =\; i\_7^2\; =\; +1\; .$

Unlike the complex numbers, the split-complex numbers are not algebraically closed, and further contain nontrivial zero divisors and nontrivial idempotents. As with the quaternions, split-quaternions are not commutative, but further contain nilpotents; they are isomorphic to the square matrices of dimension two. Split-octonions are non-associative and contain nilpotents.

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 numbers $\backslash mathbb\{C\}\; \backslash otimes\_\backslash mathbb\{R\}\; \backslash mathbb\{C\}$ (isomorphic to tessarines $\backslash mathbb\{C\}\; \backslash otimes\_\backslash mathbb\{R\}\; D$), eight-dimensional biquaternions $\backslash mathbb\{C\}\; \backslash otimes\_\backslash mathbb\{R\}\; \backslash mathbb\{H\}$, and 16-dimensional complex octonions $\backslash mathbb\{C\}\; \backslash otimes\_\backslash mathbb\{R\}\; \backslash mathbb\{O\}$.

• bicomplex numbers: a 4-dimensional vector space over the reals, 2-dimensional over the complex numbers, isomorphic to tessarines.
• multicomplex numbers: 2ⁿ-dimensional vector spaces over the reals, 2ⁿ⁻¹-dimensional over the complex numbers
• composition algebra: algebra with a quadratic form that composes with the product

• Thomas Kirkman
• Georg Scheffers
• Richard Brauer
• Hypercomplex analysis

{{reflist}}

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{{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)

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Category:History of mathematics

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