About: Biquaternion

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In abstract algebra, the biquaternions are the numbers w + x i + y j + z k, where w, x, y, and z are complex numbers, or variants thereof, and the elements of {1, i, j, k} multiply as in the quaternion group and commute with their coefficients. There are three types of biquaternions corresponding to complex numbers and the variations thereof: * Biquaternions when the coefficients are complex numbers. * Split-biquaternions when the coefficients are split-complex numbers. * Dual quaternions when the coefficients are dual numbers.

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rdfs:label	Biquaternion (de) Biquaternion (en) Bikuaternion (in) Biquaternion (fr) Bikwaterniony (pl) Бикватернион (ru) 複四元數 (zh) Бікватерніони (uk)
rdfs:comment	Die Biquaternionen sind ein hyperkomplexes Zahlensystem, das von William Kingdon Clifford in der zweiten Hälfte des 19. Jahrhunderts beschrieben wurde. Vor Clifford hatte Arthur Cayley bereits die Quaternionen mit komplexen Koeffizienten (also die Menge ) als Biquaternionen bezeichnet. (de) Бикватернионы — комплексификация (расширение) обычных (вещественных) кватернионов. (ru) 抽象代數中，複四元數（英語：Biquaternion）為一數值w + x i + y j + z k，其中w、x、y、z為複數，而{1, i, j, k}等元素的乘積方式同四元群。其與相對論中的勞侖茲群有關。 (zh) Бікватерніони — комплексифікація (розширення) звичайних (дійсних) кватерніонів. (uk) In abstract algebra, the biquaternions are the numbers w + x i + y j + z k, where w, x, y, and z are complex numbers, or variants thereof, and the elements of {1, i, j, k} multiply as in the quaternion group and commute with their coefficients. There are three types of biquaternions corresponding to complex numbers and the variations thereof: * Biquaternions when the coefficients are complex numbers. * Split-biquaternions when the coefficients are split-complex numbers. * Dual quaternions when the coefficients are dual numbers. (en) Bikuaternion (atau kuaternion ganda) adalah bilangan hiperkompleks dan merupakan suatu kuaternion di mana variabelnya w, x, y, dan z adalah bilangan kompleks. Perkalian elemen dasar bikuaternion {i, i, j, k} sama dengan perkalian elemen dasar kuaternion berbilangan cacah (bilangan riil). Karena ada terdapat 3 jenis bilangan kompleks, maka begitu pula dengan bikuaternion, yaitu: * Bikuaternion dengan bilangan kompleks biasa * Bikuaternion hiperbolik dengan bilangan kompleks hiperbolik (split-complex biquaternion) * Bikuaternion rangkap dengan bilangan rangkap (dual quaternion). (in) En mathématiques, un biquaternion (ou quaternion complexe) est un élément de l'algèbre des quaternions sur les nombres complexes. Le concept d'un biquaternion fut mentionné la première fois par William Rowan Hamilton au XIXe siècle.William Kingdon Clifford utilisa le même nom à propos d'une algèbre différente. Article détaillé : biquaternion de Clifford. (fr) Bikwaterniony – liczby postaci gdzie współczynniki wszystkie należą do jednej z opisanych niżej „struktur quasi-zespolonych”, zaś elementy tworzą grupę kwaternionów ze względu na mnożenie, a zarazem są przemienne ze współczynnikami (dokonawszy odpowiednich utożsamień element zwykle pomija się w zapisie). Ze względu na rodzaj liczb pełniących rolę współczynników wyróżnia się: * bikwaterniony (klasyczne, zwykłe), w przypadku liczb zespolonych, * , w przypadku liczb podwójnych, * , w przypadku liczb dualnych. (pl)
dcterms:subject	Articles containing proofs Ring theory Quaternions Special relativity Composition algebras William Rowan Hamilton
Wikipage page ID	1207070 (xsd:integer)
Wikipage revision ID	1124902126 (xsd:integer)
Link from a Wikipage to another Wikipage	American Journal of Mathematics Proceedings of the Royal Irish Academy Projective representation Quantum mechanics Quasi-sphere Quaternion Quaternions and spatial rotation One-parameter group Euclidean metric Europhysics Letters Basis (linear algebra) Algebra over a field Hyperbolic angle Hyperboloid Pauli matrices Representation theory of the Lorentz group Ring isomorphism Charles Jasper Joly Unit (ring theory) University of Dublin Velocity Velocity of light Versor Adrian Albert Lie group Lie theory Quotient ring Quaternions Articles containing proofs Ring theory Commutative Commutativity Commutator Complex number Complexification Coquaternion Cornelius Lanczos SO(3) Normal subgroup Quotient group Clifford algebra Minkowski space Lie algebra Linear algebra Lorentz boost Lorentz group Lorentz transformation Magma (algebra) Majorana spinor Six-dimensional space Standard model Closure (mathematics) Composition algebra Zero divisor Élie Cartan Identity matrix Mathematical structure Subgroup Mathematical physics Matrix ring

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