Just a tiny rant: In my view complex numbers are really about the concept of "orthogonality". The complex 'dimension' is orthogonal to the 'real' dimension, but anything in reality that's a continuum of values can be seen as a dimension, and therefore each one must have an orthogonal. That is, whenever you have a direction in a higher dimensional space (regardless of dimensionality) any vector will have a normal direction (perpendicular direction).
What basic complex numbers represent is a way of doing rotations where something moves from one direction towards it's orthogonal. That's what Euler's Formula is about also, which shows the relationship of 'e' and 'i' in this of course.
Now what Quaternions represents is the realization that if complex numbers have two components (real, imaginary) then we can treat each of those as a base vector and find a sort of 'next level up' orthogonality to each one individually.
I'm not good enough at math/geometry to know if this kind of 'next level up' bifurcation of dimensionality extends up past Quaternions or not (like something called Octernions, 16ions, 32ions, 64ions, etc), but it seems like is would?
What basic complex numbers represent is a way of doing rotations where something moves from one direction towards it's orthogonal. That's what Euler's Formula is about also, which shows the relationship of 'e' and 'i' in this of course.
Now what Quaternions represents is the realization that if complex numbers have two components (real, imaginary) then we can treat each of those as a base vector and find a sort of 'next level up' orthogonality to each one individually.
I'm not good enough at math/geometry to know if this kind of 'next level up' bifurcation of dimensionality extends up past Quaternions or not (like something called Octernions, 16ions, 32ions, 64ions, etc), but it seems like is would?