In abstract algebra, the biquaternions are the numbers w + x i + y j + z k, where w, x, y, and z are complex numbers and the elements of {1, i, j, k} multiply as in the quaternion group. As there are three types of complex number, so there are three types of biquaternion: * (Ordinary) biquaternions when the coefficients are (ordinary) complex numbers * Split-biquaternions when w, x, y, and z are split-complex numbers * Dual quaternions when w, x, y, and z are dual numbers.

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  • 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)
  • In abstract algebra, the biquaternions are the numbers w + x i + y j + z k, where w, x, y, and z are complex numbers and the elements of {1, i, j, k} multiply as in the quaternion group. As there are three types of complex number, so there are three types of biquaternion: * (Ordinary) biquaternions when the coefficients are (ordinary) complex numbers * Split-biquaternions when w, x, y, and z are split-complex numbers * Dual quaternions when w, x, y, and z are dual numbers. This article is about the ordinary biquaternions named by William Rowan Hamilton in 1844 (see Proceedings of Royal Irish Academy 1844 & 1850 page 388). Some of the more prominent proponents of these biquaternions include Alexander Macfarlane, Arthur W. Conway, Ludwik Silberstein, and Cornelius Lanczos. As developed below, the unit quasi-sphere of the biquaternions provides a presentation of the Lorentz group, which is the foundation of special relativity. The algebra of biquaternions can be considered as a tensor product C ⊗ H (taken over the reals) where C is the field of complex numbers and H is the algebra of (real) quaternions. In other words, the biquaternions are just the complexification of the (real) quaternions. Viewed as a complex algebra, the biquaternions are isomorphic to the algebra of 2×2 complex matrices M2(C). They can be classified as the Clifford algebra Cℓ2(C) = Cℓ03(C). This is also isomorphic to the Pauli algebra Cℓ3,0(R), and the even part of the spacetime algebra Cℓ01,3(R). (en)
  • 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. Il y a aussi une autre notion de biquaternions, distincte : une algèbre de biquaternions sur un corps commutatif K est une algèbre qui est isomorphe au produit tensoriel de deux algèbres de quaternions sur K (sa dimension est 16 sur K, et non pas 8 sur R). (fr)
  • 這是與数学相關的小作品。你可以通过编辑或修订扩充其内容。 (zh)
  • Бикватернионы — комплексификация (расширение) обычных (вещественных) кватернионов. (ru)
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http://purl.org/linguistics/gold/hypernym
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  • 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)
  • 這是與数学相關的小作品。你可以通过编辑或修订扩充其内容。 (zh)
  • Бикватернионы — комплексификация (расширение) обычных (вещественных) кватернионов. (ru)
  • In abstract algebra, the biquaternions are the numbers w + x i + y j + z k, where w, x, y, and z are complex numbers and the elements of {1, i, j, k} multiply as in the quaternion group. As there are three types of complex number, so there are three types of biquaternion: * (Ordinary) biquaternions when the coefficients are (ordinary) complex numbers * Split-biquaternions when w, x, y, and z are split-complex numbers * Dual quaternions when w, x, y, and z are dual numbers. (en)
  • 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)
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  • Biquaternion (en)
  • Biquaternion (de)
  • Biquaternion (fr)
  • Бикватернион (ru)
  • 複四元數 (zh)
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