Why is it that with quaternions $ij \neq ji$? I've been using rotations in 3d space lately for simulations. Today I came across the quaternion, which from what I understand will be a much better alternative to my cross/dot product methods.
Now I was messing with the algebra, and I can't seem to wrap my head around how $ij$ does not equal $ji$. In elementary school, I learned to treat $i^2=-1$, and to treat it like any other variable with that caveat. So does someone have an alebraic/serious proof of how we can fairly say that $ij$ does not equal $ji$? I understand rotations aren't associative, but that doesn't explain the algebra.

The thing is...not everything is commutative, in particular rotations in 3d do not commute. 
So to answer your question it is not "provably true" that $ij\neq ji$ in the quaternion world, more the quaternion world is defined to be a world in which $ij \neq ji$ (so as to model 3d rotations well).
