Now I understand how natural numbers work and the basics of complex numbers like i and i\^4 and stuff. But what is a hypercomplex number? I’ve tried searching google but I always had no idea what it was talking about. There are even things like quaternions and octonions, like what is that?
In: Mathematics
“Hypercomplex numbers” are a dated terminology for things “such as” the complex numbers, but potentially more… complex.
**Real** numbers are 1D. They form a “number lines” along which they are arranged.
**Complex** numbers are 2D: a real and an imaginary axis. 1 and i. The basic rule is i² = -1 and all else follows from that. Geometrically they correspond to scaling and rotating things in 2D.
**Quaternions** are 4D: a real and three imaginary axes. 1 and i, j, k. The rules are nnow i² = j² = k² = -1 and ij = k = -ji, jk = i = -kj, ki = j = -ik. They can be used in computer graphics to do 3D rotations.
**Octonions** are 8D: a real and seven imaginary axes (names omitted for brevity and my sanity). They are more of an interesting thing that “just exists”.
Others exist such as the 2D **dual numbers**: a real and an infinitesimal axis. 1 and ε, where ε² = 0. There is also an n-D version where we only require ε^^n = 0 yet ε^^n-1 is non-zero.
Or _silly numbers_ (my name): a real and another real axis. 1 and u, where u² = 1.
You can calculate in all those somewhat: +, -, · and sometimes even /. However, those larger and larger things lose more and more rules you might be used to:
– in complex numbers you cannot tell which is larger or smaller.
– quaternions violate the rule a·b = b·a.
– octonions even violate a·(b·c) = (a·b)·c as well.
– any other systems don’t allow division (this is a very deep result).
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