I think I have a good grasp of complex numbers and their applications. Quick Google searches show quarternions and octonions. I figured beyond complex numbers would be needed for higher dimensions, but why are the next more complex numbers seemingly 4 and 8 dimensional rather than 3, 4, 5, etc. TIA
In: Mathematics
We use complex numbers when we want to describe rotation in 1 dimension. We write these numbers as a+bi, where a is just some scalar and the bi term is a vector. But this only describes rotation in 1 dimension, what if we want to describe it in 3? Well you could use Hamiltonian mechanics, which applies the concept of quaternions to classical mechanics(as well as some other stuff). These are written as a + bi + cj + dk where a is again a scalar, a,b,c,d are all real numbers and i, j and k are unit-vectors pointing along the three spatial axes. This allows us to solve very complex problems in a much more elegant way.
Octonions are used in abstract algebra to classification and construction other mathematical entities. I don’t have any good examples of this that I understand enough to ELI5.
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