Hypercomplex number

The term hypercomplex number has been used in mathematics for the elements of algebras that extend or go beyond complex number arithmetic. Hypercomplex numbers have had a long lineage of devotees including Hermann Hankel, Georg Frobenius, Eduard Study, and Elie Cartan. Study of particular hypercomplex systems leads to their representation with linear algebra. This article gives an overview of the key systems, including some not originally considered by the pioneers before modern insight from linear algebra. For details, references, and sources, please follow the particular number type link.

Arguably the most common use of the term hypercomplex number refers to algebraic systems with dimensionality (axes), as contained in the following list. For others (like transfinite number, superreal number, hyperreal number, surreal number) see also under number.

Despite their different algebraic properties, it is noted that none of these extensions form a field, because the field of complex numbers is algebraically closed — see fundamental theorem of algebra.

A comprehensive modern definition of hypercomplex number is given by Kantor and Solodovnikov ^[1] as unitary, distributive number systems that contain at least one non-real axis and are closed under addition and multiplication. Axes are generated through real number coefficients ${\displaystyle (a_{0},~...,a_{n})}$ to bases ${\displaystyle \{1,~i_{1},...,i_{n}\}}$ (${\displaystyle n\in \{1,2,3...\}}$). The coefficients distribute, associate, and commute with the real (1) and non-real(${\displaystyle ~i_{n}}$) bases. Three types of${\displaystyle ~i_{n}}$ are possible: ${\displaystyle i_{n}^{2}\in \{-1,0,+1\}}$.

From a geometric viewpoint, these numbers form a finite-dimensional algebras over the real numbers.

The following classifications fall under this category. At times, the term 'hypernumber' is used synonymously to 'hypercomplex number' as defined by Kantor and Solodovnikov (but see below for Musean hypernumbers, some of which are not distributive or don't include a real number axis).

Cayley-Dickson construction provides for the extension of complex numbers into number systems with dimensionality${\displaystyle ~2^{n}}$ (${\displaystyle n\in \{2,3,4,...\}}$). These include the four-dimensional quaternions, eight-dimensional octonions, and 16-dimensional sedenions. Increasing dimensionality introduces algebraic complications: Quaternion multiplication is not commutative anymore, octonion multiplication additionally is non-associative, and sedenions do not form a normed space with multiplicative norm.

In the definition of Kantor and Solodovnikov, these numbers correspond to anti-commutative bases of type ${\displaystyle i_{m}^{2}=~-1}$ (with ${\displaystyle m\in \{1,...,~2^{n}-1\}}$).

Because quaternions and octonions offer a (multiplicative) norm similar to lengths in four and eight dimensional Euclidean vector space respectively, these numbers can be referred to as points in some higher-dimensional Euclidean space. Beyond octonions, however, this analogy fails since these constructs are not normed anymore.

Dual numbers are to bases ${\displaystyle \{1,i\}}$ with nilpotent ${\displaystyle i^{2}=0}$.

Split-complex numbers refer to bases ${\displaystyle \{1,i\}}$ with ${\displaystyle i^{2}=+1}$ a non-real root of 1. They contain idempotents ${\displaystyle {\frac {1}{2}}(1~\pm i)}$ and zero divisors ${\displaystyle (1+i)(1-i)=~0}$.

A modified Cayley-Dickson construction leads to coquaternions (split-quaternions; e.g. to bases ${\displaystyle \{1,~i_{1},i_{2},i_{3}\}}$ with ${\displaystyle i_{1}^{2}=i_{2}^{2}=+1}$, ${\displaystyle i_{3}^{2}=-1}$) and split-octonions (e.g. to bases ${\displaystyle \{1,~i_{1},...,i_{7}\}}$ with ${\displaystyle i_{1}^{2}=i_{2}^{2}=i_{3}^{2}=-1}$, ${\displaystyle i_{4}^{2}=...=i_{7}^{2}=+1}$). Coquaternions contain nilpotents, have a non-commutative multiplication, and are isomorphic to real matrices (2 x 2). Split-octonions are non-associative.

All non-real bases of split-complex algebra are anti-commutative.

Clifford algebra is the unitary associative algebra over real, complex, and quaternionic vector spaces equipped with a quadratic form. Whereas Cayley-Dickson and split-complex constructs with eight or more dimensions are not associative anymore with respect to multiplication, Clifford algebras retain associativity at any dimensionality.

While for Cayley-Dickson constructs, split-complex algebra, and Clifford algebra all non-real bases are anti-commutative, use of a commutative imaginary base leads to four-dimensional tessarines, eight-dimensional biquaternions or Clifford biquaternions, and 16-dimensional conic sedenions.

Tessarines offer a commutative and associative multiplication, biquaternions are associative but not commutative, and conic sedenions are not associative and not commutative. They all contain idempotents and zero-divisors, are not normed, but offer a multiplicative modulus. Biquaternions contain nilpotents, conic sedenions are also not power associative.

With the exception of their idempotents, zero-divisors, and nilpotents, the arithmetic of these numbers is closed with respect to multiplication, division, exponentiation, and logarithms (see e.g. conic quaternions, which are isomorphic to tessarines).

The hyperbolic quaternions (after Alexander MacFarlane) have a non-associative and non-commutative multiplication. Nevertheless, they offer a ring structure somewhat richer than the Minkowski space of special relativity. All bases are roots of 1, i.e. ${\displaystyle i_{n}^{2}=+1}$ for ${\displaystyle n\in \{1,2,3\}}$.This structure is of historical and educational interest since it was a spectacle of the 1890s that presaged the spacetime revolution of the following decade.

While Kantor and Solodovnikov generalize multiplication for numbers of more than one dimension through distributive rectangular (Cartesian coordinate) products, hypernumbers after Charles A. Musès use an approach to generalization by means of absolutes and angles. Musean hypernumbers are organized in 'levels' which correspond to different algebraic properties. While arithmetics built on the first three levels (to real, imaginary ${\displaystyle i={\sqrt {-1}}}$, and counterimaginary ${\displaystyle \varepsilon ={\sqrt {+1}}\neq \pm 1}$ bases) are contained in the definition by Kantor and Solodovnikov (see hypernumbers for isomorphisms to numbers mentioned above), the remaining levels offer additional arithmetical properties. For example, they are not necessarily distributive, and not all have a real axis.

Multicomplex numbers are a commutative n dimensional algebra generated by one element e that satisfies ${\displaystyle ~e^{n}=-1}$. A special case are the bicomplex numbers which are isomorphic to tessarines, conic quaternions (from Musès' hypernumbers), and are also contained in the 'hypercomplex number' definition by Kantor and Solodovnikov.

• Weisstein, Eric W. "Hypercomplex number". MathWorld.
• Jeanne La Duke "The study of linear associative algebras in the United States, 1870 - 1927", see pp. 147-159 of Emmy Noether in Bryn Mawr Bhama Srinivasan & Judith Sally editors, Springer Verlag 1983.

• History of the Hypercomplexes on hyperjeff.com
• Hypercomplex.ru
• Clyde Davenport's Commutative Hypercomplex Math Page
• Tom Jewitt's "The Generalized Number System", "Hypercomplex Signal Processing", and "The Hypercomplex Kalman Filter"