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).
Also, irrelevant side question about this subreddit. Do you need certain things to post here like account needs to be a certain age or something?
i*i is defined as -1.
Quaternions etc just define j*j as also -1, and k*k as -1 but then they also define i*j=k, and k*j=-i. In doing so they make a number system with useful properties, especially for rotations.
But if you are looking for a “real” thing, they are just defined to be the way they are to be useful.
For “beyond”, read up on geometric algebra, (also Clifford algebra(s) and Grassmann algebra(s)). Too much to go into here, but these are umbrellas under which complex, hypercomplex, dual, and all the others lie.
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