Definition & Meaning | English word HYPERCOMPLEX

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Definitions of HYPERCOMPLEX

1. (mathematics) Describing any of several types of higher-dimensional numbers having some characteristics of complex numbers.
2. (math) A hypercomplex number.

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Examples of Using HYPERCOMPLEX in a Sentence

• As -algebras, they generalize the real numbers, complex numbers, quaternions and several other hypercomplex number systems.
• In mathematics, the octonions are a normed division algebra over the real numbers, a kind of hypercomplex number system.
• In mathematics, hypercomplex number is a traditional term for an element of a finite-dimensional unital algebra over the field of real numbers.
• Thus, for example, the studies of "hypercomplex numbers", such as considered by the Quaternion Society, were put onto an axiomatic footing as branches of ring theory (in this case, with the specific meaning of associative algebras over the field of complex numbers).
• Following William Kingdon Clifford who had extended quaternions to dual quaternions, McAulay made a special study of this hypercomplex number system.
• He developed a multi-dimensional extension of Pompeiu's areolar derivative, and studied monogenic functions of one hypercomplex variable with applications to mechanics.
• Much earlier (in 1955) Morio Obata studied affine connection associated with almost hypercomplex structures (under the former terminology of Charles Ehresmann of almost quaternionic structures).
• The Hopf surface is the only compact hypercomplex manifold of quaternionic dimension 1 which is not hyperkähler.
• Quaternions continued to be a well-studied mathematical structure in the twentieth century, as the third term in the Cayley–Dickson construction of hypercomplex number systems over the reals, followed by the octonions, the sedenions, the trigintaduonions; they are also a useful tool in number theory, particularly in the study of the representation of numbers as sums of squares.
• Fractals have been generated on computers using the following methods: Menger sponge, Hypercomplex manifold, Brownian tree, Brownian motion, Decomposition, L-systems, Lyapunov fractals, Newton fractals, Pickover stalks and Strange attractors.
• Giovanni Battista Rizza (7 February 1924 – 15 October 2018), officially known as Giambattista Rizza, was an Italian mathematician, working in the fields of complex analysis of several variables and in differential geometry: he is known for his contribution to hypercomplex analysis, notably for extending Cauchy's integral theorem and Cauchy's integral formula to complex functions of a hypercomplex variable, the theory of pluriharmonic functions and for the introduction of the now called Rizza manifolds.
• For volume 41 of Mathematische Annalen Scheffers contributed a further short note, this time including reference to 1867 work by Edmond Laguerre on linear systems, a rich source of hypercomplex numbers.
• Explorative articles on hypercomplex numbers, mentioned by Bottazzini and Gray, written by Eduard Study (1898) and Elie Cartan (1908), served as advertisements to twentieth century algebraists, and they soon retired the term hypercomplex by displaying the structure of algebras.

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