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...]], where the argument is a [[quaternion]] (in this case, the sub-field of hypercomplex analysis is called [[quaternionic analysis]]). A second instance involves In [[mathematical physics]], there are hypercomplex systems called [[Clifford algebra]]s. The study of functions with arguments ...

...pers.org/rec/EDMNCO Nine-vectors, complex octonion/quaternion hypercomplex numbers, Lie groups and the ‘real’ world], [[Foundations of Physics]] 8(3-4): 303–1 * D.G. Kabe (1984) "Hypercomplex Multivariate Normal Distribution", [[Metrika]] 31(2):63−76 {{mr|id=744966}} ...

==Hypercomplex numbers== {{main|hypercomplex number}} ...

...ral numbers. Natural numbers including 0 are also sometimes called ''whole numbers''.<ref> ...ath>\mathbb{Z}</math>): Positive and [[negative number|negative]] counting numbers, as well as zero: {..., −3, −2, −1, 0, 1, 2, 3, ...}. ...

...the imaginary unit <math>i</math> satisfies <math>i^{2}=1</math>, for dual numbers the imaginary unit <math>\epsilon ...umbers in place of complex numbers. As an advantage, the method using dual numbers to compute sensitivities is step size independent; therefore, a typical ste ...

{{Short description|Hypercomplex number system}} | type = [[Hypercomplex number|Hypercomplex]] [[algebra over a field|algebra]] ...

{{Short description|Four-dimensional algebra over the real numbers}} ...{Citation | last1=Matsuda|first1=Genki|title=Anti-commutative Dual Complex Numbers and 2D Rigid Transformation|date=2014| work=Mathematical Progress in Expres ...

...6106|doi-access=free}}</ref> When studying these transformations, the dual numbers are often interpreted as representing [[oriented line]]s on the plane.<ref dual number projective line]], which adjoins to the dual numbers a set of points at infinity. Topologically, this projective line is equival ...

...sements to twentieth century algebraists, and they soon retired the term ''hypercomplex'' by displaying the structure of algebras. ...

...at if the quadratic form defines a [[homomorphism]] into the positive real numbers on the non-zero part of the algebra, then the algebra must be [[isomorphic] ...gebraic topology]] to problems on [[vector fields on spheres#Radon–Hurwitz numbers|vector fields on spheres]] and the [[homotopy group]]s of the [[classical g ...

...ter operator. Further some aspects of Clifford analysis are referred to as hypercomplex analysis. ...ns in '''R'''^''n'' expressed through 2 × 2 matrices of Clifford numbers | journal=Complex Variables |volume=5 | pages=215–224 | year=1986 | issue=2 ...

...ry algebra]], the use of [[variable (mathematics)|variables]] to represent numbers in computation and reasoning. The abstract perspective on algebra has becom ...pter=An exposition of some properties of the class number of ideal complex numbers}}</ref> similar to the modern laws for a finite [[abelian group]].{{sfn|Kle ...

...ex numbers and manually simulates complex-number operations using two real numbers, for those who do not have a [[complex data type]]. The program may be simp ...ive-url=https://web.archive.org/web/20210323153331/https://www.evl.uic.edu/hypercomplex/html/book/book.pdf |url-status=live }}</ref> ...

{{Short description|Array of numbers}} The numbers, symbols, or expressions in the matrix are called its ''entries'' or its '' ...