Hypercomplex number

Overview of some common sets of hypercomplex numbers with their respective dimensions and their subset relations.

Hypercomplex numbers are generalizations of complex numbers . In this article, hypercomplex numbers are considered as an algebraic structure . Sometimes the quaternions are also referred to as the hyper-complex numbers.

definition

A hypercomplex number is an element of an algebra of hypercomplex numbers. An algebra over the real numbers is called the algebra of hypercomplex numbers or hypercomplex system of rank , if ${\displaystyle A}$ ${\displaystyle n}$

• as vector space it has finite dimensions and if ${\displaystyle n}$
• it has a unity, that is, if one exists, such that the equation holds for all . ${\displaystyle e\in A}$ ${\displaystyle a\in A}$ ${\displaystyle e\cdot a=a\cdot e=a}$

Some authors require in addition that the algebra with respect to multiplication associative is. In particular, the real numbers are themselves an algebra of hyper-complex numbers. ${\displaystyle A}$

properties

• The commutative law and the associative law apply to the addition .
• The addition can be inverted .
• The left-hand and right-hand distributive law applies.

• The multiplication in a hypercomplex algebra is bilinear over the real numbers, i.e. i.e., it applies ${\displaystyle A}$ ${\displaystyle (\alpha x)(\beta y)=\alpha \beta (xy)~~\forall \alpha ,\beta \in \mathbb {R} ;~x,y\in A}$

The following properties are not required:

• The commutative law does not have to apply to the multiplication of hyper-complex numbers .
• Elements need not necessarily be invertible with regard to multiplication .
• The multiplication does not have to be zero divisors .

conjugation

Hypercomplex numbers can be represented as a sum as follows:

${\displaystyle a=a_{0}1+a_{1}\mathrm {i} _{1}+\dotsb +a_{n}\mathrm {i} _{n}}$.

The quantities for are called imaginary units . The number to be conjugated is created by replacing all imaginary units with their negative ( ). The complex number to be conjugated is represented by or . Your total representation is ${\displaystyle \mathrm {i} _{k}}$ ${\displaystyle k>0}$ ${\displaystyle a}$ ${\displaystyle \mathrm {i} _{k}\mapsto -\mathrm {i} _{k}}$ ${\displaystyle a}$ ${\displaystyle {\bar {a}}}$ ${\displaystyle a^{*}}$

${\displaystyle {\bar {a}}=a_{0}1-a_{1}\mathrm {i} _{1}-\dotsb -a_{n}\mathrm {i} _{n}}$.

The conjugation is an involution on the hypercomplex numbers, that is, that

${\displaystyle {\bar {\bar {a}}}=a}$.

Examples

Complex numbers

→ Main article : Complex number

The complex numbers are a hypercomplex number system that is characterized by ${\displaystyle \mathbb {C} }$

${\displaystyle z=a+b\mathrm {i} }$ With ${\displaystyle \mathrm {i} ^{2}=-1}$

is defined.

Abnormally complex numbers

→ Main article : Abnormal complex number

The abnormally complex numbers are defined by

${\displaystyle z=a+bj}$with . ${\displaystyle j^{2}=1}$

Dual numbers

→ Main article : Dual number

The dual numbers are defined by

${\displaystyle z=a+b\varepsilon }$with . ${\displaystyle \varepsilon ^{2}=0}$

Note that they have nothing to do with binary numbers .

Quaternions

→ Main article : Quaternion

The quaternions (symbol often after their discoverer WR Hamilton ) form a four-dimensional algebra with division and associative (but not commutative) multiplication. The quaternions are therefore an oblique body . ${\displaystyle \mathbb {H} }$ ${\displaystyle \mathbb {R} }$

Biquaternions

→ Main article : Biquaternion

The biquaternions are defined as quaternions with complex coefficients; That is, they form a four-dimensional vector space over just as the quaternions form a four-dimensional vector space over . ${\displaystyle \mathbb {C} }$ ${\displaystyle \mathbb {R} }$

Octonions

→ Main article : Octave (mathematics)

The octonions (symbol , also called octaves ) are eight-dimensional hyper-complex numbers with division and alternative multiplication. ${\displaystyle \mathbb {O} }$

Sedenions

→ Main article : Sedenion

The Sedenions (symbol ) are sixteen-dimensional hyper-complex numbers. Their multiplication is neither commutative, associative nor alternative. Nor do they have a division; instead they have zero divisors . ${\displaystyle \mathbb {S} }$

Square matrices

→ Main article : Matrix (mathematics)

Be a natural number. That is then an algebra with the - identity matrix as a unitary element - thus also a hyper-complex algebra. More precisely, it is an associative hyper-complex algebra and thus also a ring and as such also unitary . The real-number multiples of the identity matrix form a sub-algebra that is too isomorphic. ${\displaystyle n}$ ${\displaystyle \mathbb {R} ^{n\times n}}$ ${\displaystyle n\times n}$ ${\displaystyle \mathbb {R} }$

In the case there are subalgebras that are isomorphic to the three two-dimensional algebras mentioned above; They are characterized by the fact that the main diagonal elements always match (which corresponds to the real part) and rules apply to the elements of the secondary diagonal that determine the algebra shown: ${\displaystyle n=2}$

• A secondary diagonal element is 0 → The algebra is isomorphic to the dual numbers
• Both secondary diagonal elements match → The algebra is isomorphic to the binary numbers
• Each secondary diagonal element is the negative of the other → The algebra is isomorphic to the complex numbers

Note: Every matrix of the third type, divided by the determinant, is a rotation matrix of two-dimensional space; every matrix of the second type by its determinant (if this is different from 0) corresponds to a Lorentz transformation in a 1 + 1-dimensional Minkowski space .

Remarks

• The doubling process (also known as the Cayley-Dickson process) can be used to generate new hyper-complex number systems whose dimensions are twice as large as those of the original number system.
• Every Clifford algebra is an associative hypercomplex number system.

See also

• Coquaternion
• Bicomplex number

literature

• Ulf von Rauchhaupt : From natural to hyper-complex Frankfurter Allgemeine Sonntagszeitung, June 11, 2017
• Heinz-Dieter Ebbinghaus et al .: Numbers (basic knowledge of mathematics). 3rd edition Springer, Berlin 1992, ISBN 3-54055-654-0 .
• Isaj L. Kantor, Alexander S. Solodownikow: Hypercomplex numbers ( mathematical student library ; Vol. 95). BG Teubner, Leipzig 1978.
• NN Vil'yams: Hypercomplex number . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).