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Why is 8 so special?

First of all let me state that I am not a mathematician but I work in computer science and engineering.

I was reading about hypercomplex numbers, and in particular about quaternions and octonions.

My question is is there anything special with the 4 dimensions of the quaternions and with the 8 of the octonions? Why can't we also do hypercomplex numbers with any other arbitrary sum of imaginary parts? For example in 5, 6 or 3 dimensions?

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    See http://math.stackexchange.com/questions/39522/why-is-8-so-special and the links within. – Samuel Apr 25 '12 at 23:41
  • If you're still interested in the answer, "complexification" doubles the dimension each time, making C, H, O, etc. (The existing space remains the same, then a copy includes the next generator, e.g. quaternions introduce j to a copy of C.) There are 2^N combinations of N generators, per the powerset from combinatorics. Any less than 2^N dimensions means some combinations are missing, e.g. quaternions could be nonreal in 4-1, octonions in 8-1; for what it's worth, those could use the 3D/7D cross products. – John P Apr 10 '22 at 18:37

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