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biqua
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There are 5 letters in BIQUA ( A1B3I1Q10U1 )
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BIQUA - 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 e...
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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.This article is about the ordinary biquaternions named by William Rowan Hamilton in 1844 (see Proceedings of the 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 representation 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 division algebra of (real) quaternions. In other words, the biquaternions are just the complexification of the quaternions. Viewed as a complex algebra, the biquaternions are isomorphic to the algebra of 2 × 2 complex matrices M2(ℂ). They are also isomorphic to several Clifford algebras including ℍ(ℂ) = Cℓ03(ℂ) = Cℓ2(ℂ) = Cℓ1,2(ℝ), the Pauli algebra Cℓ3,0(ℝ), and the even part Cℓ01,3(ℝ) = Cℓ03,1(ℝ) of the spacetime algebra. |