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Hypercomplex Numbers: Quaternion, Octonion, Sedenian, and beyond





Quaternions (4-ions)
Octonions (8-ions)
Sedenions (16-ions)
Tricenibinions / trigintabinions (32-ions)
Sexageniquaternions / sexagintaquaternions (64-ions)
Centeniduodetricenions / centumduodetricenions (128-ions)
Duceniquinquagenisenions / ducentiquinquagintasenions (256-ions)

Quingeniduodenions / quingentiduodenions (512-ions)
Miliaviceniquaternions / millevigintiquaternions (1024-ions)
Binamiliaduodequinquagenions / duomiliaduodequinquagenions (2048-ions)
Quaternamilianonagenisenions / quattuormilianonagintasenions (4096-ions)


Cayley-Dickson Construction First, we define the reals R where a ∈ R implies that a* = a. Given an algebra A of diminsion n, we create, using the Cayley-Dickson construction, an algebra of dimension 2n by taking pairs (a, b) ∈ A × A and thus define the standard operations:

1 = (1, 0)
−(a, b) = (−a, −b)
(a, b)* = (a*, −b)
(a, b) + (c, d) = (a + c, b + d)
(a, b)(c, d) = (ac − d*b, da + bc*)

In order to maintain a standard behaviour, a variation of the Cayley-Dickson construction shown in wikipedia is used for calculating products in higher dimensions. The justification is that it would be nice if the original quaternion identity ijk = -1 holds — the variation on wikipedia defines ijk = 1.



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