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Element of a unital algebra over the field of real numbers
In mathematics, the hypercomplex number is a traditional term for an element of a finite-dimensional unital algebra over the field of real numbers. The
Hypercomplex_number
Topics referred to by the same term
Hypercomplex may refer to: Hypercomplex cell Hypercomplex analysis Hypercomplex manifold Hypercomplex number This disambiguation page lists articles associated
Hypercomplex
Branch of mathematical analysis
In mathematics, hypercomplex analysis is the extension of complex analysis to the hypercomplex numbers. The first instance is functions of a quaternion
Hypercomplex_analysis
Manifold equipped with a quaternionic structure
In differential geometry, a hypercomplex manifold is a manifold with the tangent bundle equipped with an action by the algebra of quaternions in such a
Hypercomplex_manifold
Used to count, measure, and label
are explicitly referred to as numbers (such as the p-adic numbers and hypercomplex numbers) while others are not, but this is more a matter of convention
Number
German mathematician (1882–1935)
epoch (1927–1935), she published works on noncommutative algebras and hypercomplex numbers and united the representation theory of groups with the theory
Emmy_Noether
Branch of mathematics
Noncommutative ring theory began with extensions of the complex numbers to hypercomplex numbers, specifically William Rowan Hamilton's quaternions in 1843. Many
Abstract_algebra
Even-dimensional Hopf manifolds admit hypercomplex structure. The Hopf surface is the only compact hypercomplex manifold of quaternionic dimension 1 which
Hopf_manifold
Four-dimensional number system
Quaternion Association, devoted to the study of quaternions and other hypercomplex number systems. From the mid-1880s, quaternions began to be displaced
Quaternion
Mathematical table
examples, see group. Hypercomplex number multiplication tables show the non-commutative results of multiplying two hypercomplex imaginary units. The simplest
Multiplication_table
Neuron in the cerebral cortex used for visual processing
A hypercomplex cell (currently called an end-stopped cell) is a type of visual processing neuron in the mammalian cerebral cortex. Initially discovered
Hypercomplex_cell
Hypercomplex number system
octonions are a normed division algebra over the real numbers, a kind of hypercomplex number system. The octonions are usually represented by the capital letter
Octonion
Algebra based on a vector space with a quadratic form
generalize the real numbers, complex numbers, quaternions and several other hypercomplex number systems. The theory of Clifford algebras is intimately connected
Clifford_algebra
German-American mathematician
the National Medal of Science. Eduard Study had written an article on hypercomplex numbers for Klein's encyclopedia in 1898. This article was expanded for
Richard_Brauer
German mathematician (1866–1945)
In §14 (p 386) Scheffers reviews both German and English authors on hypercomplex numbers. In particular, he cites Eduard Study’s work of 1889. For volume
Georg_Scheffers
Notation for expressing numbers
as the system of real numbers, the system of complex numbers, various hypercomplex number systems, the system of p-adic numbers, etc. Such systems are,
Numeral_system
Geometric space with five dimensions
Demihypercube Hypersphere Cross-polytope Simplex Hyperpyramid Number systems Hypercomplex numbers Cayley–Dickson construction Dimensions by number Zero One Two
Five-dimensional_space
Hypercomplex number system
triginta 'thirty' + duo 'two' + the suffix -nion, which is used for hypercomplex number systems. Other names include 32-ion, 32-nion, 25-ion, and 25-nion
Trigintaduonion
theorem: 0.107648 < d < 0.49094093, Romanov conjectured that it is 0.434 Hypercomplex number is a term for an element of a unital algebra over the field of
List_of_numbers
Delimited medium where some stimuli can evoke neuronal responses
of cells in the visual cortex into simple cells, complex cells, and hypercomplex cells. Simple cell receptive fields are elongated, for example with an
Receptive_field
Natural number
{\displaystyle {\tfrac {1}{2}}.} The trigintaduonions form a 32-dimensional hypercomplex number system. An international calling code for Belgium. 32 is the ninth
32_(number)
Branch of mathematics
quaternion difference p – q also produces a segment equipollent to pq. Other hypercomplex number systems also used the idea of a linear space with a basis. Arthur
Linear_algebra
Generalized sphere of dimension n (mathematics)
Demihypercube Hypersphere Cross-polytope Simplex Hyperpyramid Number systems Hypercomplex numbers Cayley–Dickson construction Dimensions by number Zero One Two
N-sphere
Area of geometry, about angles and lengths
Ring theory Universal Analysis Calculus Real analysis Complex analysis Hypercomplex analysis Differential equations Functional analysis Harmonic analysis
Trigonometry
Natural number
first stellation is the cube-octahedron compound. The octonions are a hypercomplex normed division algebra that are an extension of the complex numbers
8
Involutive change of basis in linear algebra
symmetric, involutive, linear operation on 2m real numbers (or complex, or hypercomplex numbers, although the Hadamard matrices themselves are purely real).
Hadamard_transform
Topics referred to by the same term
where one road bears three numbers Triplex (mathematics), a type of Hypercomplex number Triplex, a cinema multiplex with three screens Triplex (software)
Triplex
Number of vectors in any basis of the vector space
Demihypercube Hypersphere Cross-polytope Simplex Hyperpyramid Number systems Hypercomplex numbers Cayley–Dickson construction Dimensions by number Zero One Two
Dimension_(vector_space)
Fractal named after mathematician Benoit Mandelbrot
been shown that the generalized Mandelbrot set in higher-dimensional hypercomplex number spaces (i.e. when the power α {\displaystyle \alpha } of the iterated
Mandelbrot_set
Branch of algebra
theory began with attempts to extend the complex numbers to various hypercomplex number systems. The genesis of the theories of commutative and noncommutative
Ring_theory
Method for producing composition algebras
(2015). "An unified approach for developing rationalized algorithms for hypercomplex number multiplication". Przegląd Elektrotechniczny. 1 (2). Wydawnictwo
Cayley–Dickson_construction
Natural number
× 4. {\displaystyle 4\times 4.} The sedenions form a 16-dimensional hypercomplex number system. Sixteen is the base of the hexadecimal number system,
16_(number)
imaginary numbers, and sums and differences of real and imaginary numbers. Hypercomplex numbers include various number-system extensions: quaternions ( H {\displaystyle
List_of_types_of_numbers
Invariant measure of fractal dimension
Demihypercube Hypersphere Cross-polytope Simplex Hyperpyramid Number systems Hypercomplex numbers Cayley–Dickson construction Dimensions by number Zero One Two
Hausdorff_dimension
Type of functional equation (mathematics)
Ring theory Universal Analysis Calculus Real analysis Complex analysis Hypercomplex analysis Differential equations Functional analysis Harmonic analysis
Differential_equation
In mathematics, dimension of a ring
Demihypercube Hypersphere Cross-polytope Simplex Hyperpyramid Number systems Hypercomplex numbers Cayley–Dickson construction Dimensions by number Zero One Two
Krull_dimension
Italian mathematician
Sabadini is an Italian mathematician specializing in complex analysis, hypercomplex analysis and the analysis of superoscillations. She is a professor of
Irene_Sabadini
Hungarian and American mathematician and physicist (1903–1957)
"the cold, wet, rain-wet streets of Göttingen" after class discussing hypercomplex number systems and their representations. Von Neumann's habilitation
John_von_Neumann
Fundamental space of geometry
Demihypercube Hypersphere Cross-polytope Simplex Hyperpyramid Number systems Hypercomplex numbers Cayley–Dickson construction Dimensions by number Zero One Two
Euclidean_space
Mathematical space with two coordinates
Demihypercube Hypersphere Cross-polytope Simplex Hyperpyramid Number systems Hypercomplex numbers Cayley–Dickson construction Dimensions by number Zero One Two
Two-dimensional_space
Branch of mathematics
Ring theory Universal Analysis Calculus Real analysis Complex analysis Hypercomplex analysis Differential equations Functional analysis Harmonic analysis
Geometry
function: a function whose domain is quaternionic. Hypercomplex function: a function whose domain is hypercomplex (e.g. quaternions, octonions, sedenions, trigintaduonions
List_of_types_of_functions
and led to a subsequent analytical theory; they also began the use of hypercomplex numbers. Karl Weierstrass and others carried out the arithmetization
19th_century_in_science
Book on the history of mathematics by Michael J. Crowe
the book in a competition for "a study on the history of complex and hypercomplex numbers" twenty-five years after his book was first published. The book
A_History_of_Vector_Analysis
Number with a real and an imaginary part
^{2}.} This is generalized by the notion of a linear complex structure. Hypercomplex numbers also generalize R , {\displaystyle \mathbb {R} ,} C , {\displaystyle
Complex_number
Convex polytope, the n-dimensional analogue of a square and a cube
Demihypercube Hypersphere Cross-polytope Simplex Hyperpyramid Number systems Hypercomplex numbers Cayley–Dickson construction Dimensions by number Zero One Two
Hypercube
Hyperkomplexe Größen und Darstellungstheorie, in arithmetischer Auffassung Hypercomplex Quantities and the Theory of Representations, from an Arithmetic Perspective§
Emmy_Noether_bibliography
In mathematics, a module that has a basis
Demihypercube Hypersphere Cross-polytope Simplex Hyperpyramid Number systems Hypercomplex numbers Cayley–Dickson construction Dimensions by number Zero One Two
Free_module
Commutative, associative algebra of two complex dimensions
hypercomplex numbers. In 1848 James Cockle introduced the tessarines in a series of articles in Philosophical Magazine. A tessarine is a hypercomplex
Bicomplex_number
Property of a space in which the local dimensionality is the same everywhere
Demihypercube Hypersphere Cross-polytope Simplex Hyperpyramid Number systems Hypercomplex numbers Cayley–Dickson construction Dimensions by number Zero One Two
Equidimensionality
Faster-than-light travel in science fiction
Demihypercube Hypersphere Cross-polytope Simplex Hyperpyramid Number systems Hypercomplex numbers Cayley–Dickson construction Dimensions by number Zero One Two
Hyperspace
and led to a subsequent analytical theory; they also began the use of hypercomplex numbers. Karl Weierstrass and others carried out the arithmetization
History_of_science
Benzodiazepine drug
gaps and hypercomplex automatisms after a single oral dose of benzodiazepines: clinical and medico-legal aspects]" [Memory gaps and hypercomplex automatisms
Bromazepam
German mathematician (1862 – 1930)
trigonometry. He is also known for contributions to space geometry, hypercomplex numbers, and criticism of early physical chemistry. Study was born in
Eduard_Study
Multi-dimensional generalization of triangle
Demihypercube Hypersphere Cross-polytope Simplex Hyperpyramid Number systems Hypercomplex numbers Cayley–Dickson construction Dimensions by number Zero One Two
Simplex
Geometric space with six dimensions
Demihypercube Hypersphere Cross-polytope Simplex Hyperpyramid Number systems Hypercomplex numbers Cayley–Dickson construction Dimensions by number Zero One Two
Six-dimensional_space
Study of Lie groups, Lie algebras and differential equations
Ring theory Universal Analysis Calculus Real analysis Complex analysis Hypercomplex analysis Differential equations Functional analysis Harmonic analysis
Lie_theory
Type of Riemannian manifold
1 {\displaystyle I^{2}=J^{2}=K^{2}=IJK=-1} . In particular, it is a hypercomplex manifold. All hyperkähler manifolds are Ricci-flat and are thus Calabi–Yau
Hyperkähler_manifold
Invariant of topological spaces
Demihypercube Hypersphere Cross-polytope Simplex Hyperpyramid Number systems Hypercomplex numbers Cayley–Dickson construction Dimensions by number Zero One Two
Inductive_dimension
Branch of mathematics studying functions of a complex variable
complex spaces is in quantum mechanics as wave functions. Complex geometry Hypercomplex analysis List of complex analysis topics Monodromy theorem Riemann–Roch
Complex_analysis
Manifold or algebraic variety of dimension n in a space of dimension n+1
Demihypercube Hypersphere Cross-polytope Simplex Hyperpyramid Number systems Hypercomplex numbers Cayley–Dickson construction Dimensions by number Zero One Two
Hypersurface
Completion of the usual space with "points at infinity"
Demihypercube Hypersphere Cross-polytope Simplex Hyperpyramid Number systems Hypercomplex numbers Cayley–Dickson construction Dimensions by number Zero One Two
Projective_space
Number of independent parameters of a system
Demihypercube Hypersphere Cross-polytope Simplex Hyperpyramid Number systems Hypercomplex numbers Cayley–Dickson construction Dimensions by number Zero One Two
Degrees_of_freedom
Matrices important in quantum mechanics and the study of spin
In mathematical physics and mathematics, the Pauli matrices are a set of three 2 × 2 {\displaystyle 2\times 2} complex matrices that are traceless, Hermitian
Pauli_matrices
Concept in geometry
M} together with a quaternionic structure on M {\displaystyle M} . A hypercomplex manifold is a quaternionic manifold with a torsion-free GL ( n , H
Quaternionic_manifold
Array of numbers
linear algebra, partially due to their use in the classification of the hypercomplex number systems of the previous century. The inception of matrix mechanics
Matrix_(mathematics)
Classification of semi-simple rings and algebras
{\displaystyle k} . Maschke's theorem Brauer group Jacobson density theorem Hypercomplex number Emil Artin Joseph Wedderburn By the definition used here, semisimple
Wedderburn–Artin_theorem
Property of a mathematical space
Demihypercube Hypersphere Cross-polytope Simplex Hyperpyramid Number systems Hypercomplex numbers Cayley–Dickson construction Dimensions by number Zero One Two
Dimension
Geometric space with four dimensions
Demihypercube Hypersphere Cross-polytope Simplex Hyperpyramid Number systems Hypercomplex numbers Cayley–Dickson construction Dimensions by number Zero One Two
Four-dimensional_space
since some novelty in the subject lingered there. Research turned to hypercomplex numbers more generally. For instance, Thomas Kirkman and Arthur Cayley
History_of_quaternions
Branch of mathematics
Ring theory Universal Analysis Calculus Real analysis Complex analysis Hypercomplex analysis Differential equations Functional analysis Harmonic analysis
Algebraic_geometry
surpassed in the 19th century through considerations of parameter space and hypercomplex numbers. Abel and Galois's investigations into the solutions of various
History_of_mathematics
Fundamental object of geometry
Demihypercube Hypersphere Cross-polytope Simplex Hyperpyramid Number systems Hypercomplex numbers Cayley–Dickson construction Dimensions by number Zero One Two
Point_(geometry)
Academic subfield of computer science
Ring theory Universal Analysis Calculus Real analysis Complex analysis Hypercomplex analysis Differential equations Functional analysis Harmonic analysis
Theory_of_computation
One hundred years, from 1801 to 1900
and led to a subsequent analytical theory; they also began the use of hypercomplex numbers. Karl Weierstrass and others carried out the arithmetization
19th_century
Mathematical model combining space and time
Demihypercube Hypersphere Cross-polytope Simplex Hyperpyramid Number systems Hypercomplex numbers Cayley–Dickson construction Dimensions by number Zero One Two
Spacetime
Area of mathematics
Ring theory Universal Analysis Calculus Real analysis Complex analysis Hypercomplex analysis Differential equations Functional analysis Harmonic analysis
Dynamical_systems_theory
German mathematician (1849–1917)
Frobenius", MacTutor History of Mathematics Archive, University of St Andrews G. Frobenius, "Theory of hypercomplex quantities" (English translation)
Ferdinand_Georg_Frobenius
Geometric model of the physical space
came with William Rowan Hamilton's development of the quaternions, a hypercomplex number system. For this purpose, Hamilton coined the terms scalar and
Three-dimensional_space
needed. A new transform, Hypercomplex Wavelet transform was developed in order to address this issue. The dual tree hypercomplex wavelet transform (HWT)
Wavelet for multidimensional signals analysis
Wavelet_for_multidimensional_signals_analysis
Algebraic variety that is a moduli space for principally polarized abelian varieties
Ring theory Universal Analysis Calculus Real analysis Complex analysis Hypercomplex analysis Differential equations Functional analysis Harmonic analysis
Siegel_modular_variety
split-complex numbers and dual numbers form alternative topological rings. See hypercomplex numbers for other low-dimensional examples. In commutative algebra, the
Topological_ring
Correspondence between quaternions and 3D rotations
Patrick J. Ryan, Cambridge University Press, Cambridge, 1987. I.L. Kantor. Hypercomplex numbers, Springer-Verlag, New York, 1989. Andrew J. Hanson. Visualizing
Quaternions and spatial rotation
Quaternions_and_spatial_rotation
N-dimensional generalisation of a pyramid
Demihypercube Hypersphere Cross-polytope Simplex Hyperpyramid Number systems Hypercomplex numbers Cayley–Dickson construction Dimensions by number Zero One Two
Hyperpyramid
Branch of mathematics
Arithmetization of analysis Constructive analysis History of calculus Hypercomplex analysis Multiple rule-based problems Multivariable calculus Paraconsistent
Mathematical_analysis
Branch of elementary mathematics
Ring theory Universal Analysis Calculus Real analysis Complex analysis Hypercomplex analysis Differential equations Functional analysis Harmonic analysis
Arithmetic
Branch of algebraic geometry
Ring theory Universal Analysis Calculus Real analysis Complex analysis Hypercomplex analysis Differential equations Functional analysis Harmonic analysis
Arithmetic_geometry
trigonometry. Hypercomplex analysis the extension of real analysis and complex analysis to the study of functions where the argument is a hypercomplex number
Glossary of areas of mathematics
Glossary_of_areas_of_mathematics
1994 book by Michio Kaku
Demihypercube Hypersphere Cross-polytope Simplex Hyperpyramid Number systems Hypercomplex numbers Cayley–Dickson construction Dimensions by number Zero One Two
Hyperspace_(book)
Arithmetical operation
commutative for matrices and quaternions. Hurwitz's theorem shows that for the hypercomplex numbers of dimension 8 or greater, including the octonions, sedenions
Multiplication
Branch of mathematics
Ring theory Universal Analysis Calculus Real analysis Complex analysis Hypercomplex analysis Differential equations Functional analysis Harmonic analysis
Order_theory
Hypercomplex number system
) ( e 6 − e 15 ) {\displaystyle (e_{3}+e_{10})(e_{6}-e_{15})} . All hypercomplex number systems after sedenions that are based on the Cayley–Dickson construction
Sedenion
Geometric object with flat sides
Demihypercube Hypersphere Cross-polytope Simplex Hyperpyramid Number systems Hypercomplex numbers Cayley–Dickson construction Dimensions by number Zero One Two
Polytope
Topics referred to by the same term
Octave, an IT risk management method Octonion, originally octave, in hypercomplex algebra Octave (given name) including a list of people with the name
Octave_(disambiguation)
a locally compact group Heyting algebra Hopf algebra Hurwitz algebra Hypercomplex algebra Incidence algebra Iwahori–Hecke algebra Jordan algebra Kac–Moody
List_of_algebras
Quaternions with complex number coefficients
biquaternions with non-zero square modulus. Biquaternion algebra Hypercomplex number Hypercomplex analysis Joachim Lambek MacFarlane's use Quotient ring Quaternion
Biquaternion
Ideal ring structure
Matematicheskii Sbornik (in Russian). 33: 13–26. Wedderburn, J.H.M. (1908). "On Hypercomplex Numbers". Proceedings of the London Mathematical Society. 6 (1): 77–118
Radical_of_a_ring
American-Israeli mathematician
field theory, general relativity, representations of quantum theory on hypercomplex Hilbert modules, group theory and functional analysis and stochastic
Lawrence_Paul_Horwitz
b c {\displaystyle ad-bc} is not a zero divisor. A dual number is a hypercomplex number of the form x + y ε {\displaystyle x+y\varepsilon } where ε 2
Laguerre_transformations
Word that etymologically derives from at least two languages
Latin!".) Hyperactive – from Greek ὑπέρ (hyper) 'over' and Latin activus Hypercomplex – from Greek ὑπέρ (hyper) 'over' and Latin complexus 'an embrace' Hypercorrection
Hybrid_word
HYPERCOMPLEX
HYPERCOMPLEX
HYPERCOMPLEX
HYPERCOMPLEX
Boy/Male
Afghan, Australian, Gaelic, Irish
Red-haired; Red
Girl/Female
Russian French
Masculine.
Surname or Lastname
English
English : probably a habitational name from a place so called in Gloucestershire.
Boy/Male
Indian, Punjabi, Sanskrit, Sikh
The Image or Symbol of God
Boy/Male
Hindu
Who is Happy always
Male
Celtic
, hereditary king.
Boy/Male
Bengali, Gujarati, Hindu, Indian, Kannada, Kashmiri, Malayalam, Marathi, Sanskrit, Tamil, Traditional
Lord of the Truth
Boy/Male
Arabic, Muslim, Sindhi
Righteous; Pious
Male
Greek
Variant form of Greek Lapidot, LAPIDOS means "torches."Â
Girl/Female
Hindu
Name of a Raga
HYPERCOMPLEX
HYPERCOMPLEX
HYPERCOMPLEX
HYPERCOMPLEX
HYPERCOMPLEX