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## Hypercomplex Numbers: Quaternion, Octonion, Sedenian, and beyond

http://math.ucr.edu/home/baez/octonions/node5.html

 ```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 â 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) â A Ã A and thus define the standard operations: ```1 = (1, 0) â(a, b) = (âa, âb) (a, b)* = (a*, âb) (a, b) + (c, d) = (a + c, b + d) (a, b)(c, d) = (ac â 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 â the variation on wikipedia defines ijk = 1.