Definitions

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 Élie 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.

Numbers with dimensionality

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.

Distributive numbers with one real and n non-real axes

A comprehensive modern definition of hypercomplex number is given by Kantor and Solodovnikov as unital, 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 $\left(a_0 ,~... , a_n\right)$ to bases $\left\{ 1,~i_1, dots, i_n \right\}$ ($n in \left\{ 1, 2, 3,dots \right\}$). The coefficients distribute, associate, and commute with the real (1) and non-real($~i_n$) bases. Three types of$~i_n$ are possible: $i_n^2 in \left\{ -1, 0, +1 \right\}$.

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).

One non-real axis

Split-complex numbers
Split-complex numbers are constructed from the bases $\left\{ 1 , ~j \right\}$ with $j^2 = +1$ a non-real root of 1.

Algebras that include such non-real roots of 1 contain idempotents $tfrac\left\{1\right\}\left\{2\right\} \left(1 pm j\right)$ and zero divisors $\left(1 + j\right)\left(1 - j\right) = 0$, so such algebras cannot be division algebras. However, these properties can turn out to be very meaningful, for instance in describing the Lorentz transformations of special relativity.

Dual numbers
Dual numbers have bases $\left\{ 1, epsilon \right\}$ with nilpotent $epsilon^2 = 0$.

More than one non-real axis

Clifford algebras

Clifford algebra is the unital associative algebra generated over an underlying vector space equipped with a quadratic form. This is equivalent to being able to define a symmetric scalar product, u.v = ½(uv + vu) that can be used to orthogonalise the quadratic form, to give a set of bases {e1...ek} such that:

$tfrac\left\{1\right\}\left\{2\right\} \left(e_i e_j + e_j e_i\right) = Bigg\left\{ begin\left\{matrix\right\} -1, 0, +1 & i=j,$
0 & i not = j end{matrix} Imposing closure under multiplication now generates a multivector space spanned by 2k bases, {1, e1, e2, e3, ... , e1e2, ... , e1e2e3, ...}. These can be interpreted as the bases of a hypercomplex number system. Unlike the bases {e1...ek}, the remaining bases may or may not anti-commute, depending on how many simple exchanges must be carried out to swap the two factors. So e1e2 = - e2e1; but e1(e2e3) = + (e2e3)e1.

Putting aside the bases for which ei2 = 0 (ie directions in the original space over which the quadratic form was degenerate), the remaining Clifford algebras can be identified by the label Cp,q(R) indicating that the algebra is constructed from p simple bases with ei2 = +1, q with ei2 = -1, and where R indicates that this is to be a Clifford algebra over the reals - ie 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 C0,1(R); split-complex numbers C1,0(R); quaternions C0,2(R); split-biquaternions C0,3(R); coquaternions C1,1(R) ≈ C2,0(R) (the natural algebra of 2d space); C3,0(R) (the natural algebra of 3d space, and the algebra of the Pauli matrices); and C1,3(R) the space-time algebra.

The elements of the algebra Cp,q(R) form an even subalgebra C0q+1,p(R) of the algebra Cq+1,p(R), which can be used to parametrise rotations in the larger algebra. There is thus a close connection between complex numbers and rotations in 2D space; between quaternions and rotations in 3D space; between split-complex numbers and (hyperbolic) rotations (Lorentz transformations) in 1+1 D space, and so on.

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.

Quaternion, octonion, and beyond: Cayley-Dickson construction

All of the Clifford algebras Cp,q(R) apart from the complex numbers and the quaternions contain non-real elements j 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 in {2, 3, 4, ...}, with bases $\left\{1, i_1, ..., i_\left\{2^n-1\right\}\right\}$, where all the non-real bases anti-commute and satisfy $i_m^2 = -1$.

The first algebras in this sequence are the four-dimensional quaternions, eight-dimensional octonions, and 16-dimensional sedenions. However, satisfying these requirements comes at a price: Each increase in dimensionality introduces new 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.

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.

=Modified Cayley-Dickson construction=

The Cayley-Dickson construction can be modified by starting with the split-complex numbers rather than the complex numbers. This leads to coquaternions (split-quaternions; e.g. to bases $\left\{ 1,~i_1, i_2, i_3 \right\}$ with $i_1^2 = -1, i_2^2 = i_3^2 = +1$, ) and split-octonions (e.g. to bases $\left\{ 1,~i_1, dots , i_7 \right\}$ with $i_1^2 = i_2^2 = i_3^2 = -1$, $i_4^2 = cdots = i_7^2 = +1$). The 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 Cayley-Dickinson algebras are anti-commutative.

=Complexified algebras: Tessarine, biquaternion, and conic sedenion=

While for the Cayley-Dickson constructs and the split Cayley-Dickson constructs all non-real bases are anti-commutative, use of a commutative imaginary base leads to four-dimensional tessarines $mathbb Cotimesmathbb C$, eight-dimensional biquaternions $mathbb Cotimesmathbb H$, and 16-dimensional conic sedenions $mathbb Cotimesmathbb O$.

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).

Alexander MacFarlane's hyperbolic quaternion

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. $i_n^2 = +1$ for $n in \left\{ 1, 2, 3 \right\}$.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.

Musean hypernumber

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 $i = sqrt\left\{-1\right\}$, and counterimaginary $varepsilon = sqrt\left\{+1\right\} ne pm 1$ 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 number

Multicomplex numbers are a commutative n-dimensional algebra generated by one element e that satisfies $~e^n = -1$. 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.

References

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• 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.