i is basically "rotate by 90" in the same way -1 is "head left from 0 instead of right"
[1]: https://math.stackexchange.com/questions/444475/whats-the-di...
It happens to be true that this field called the complex numbers is a two dimensional real vector space. It took humans quite a long time to come up with 0 and negative numbers. There’s no way the jump to the complex numbers was going to be as easy and it isn’t as simple as, “numbers in 2d”. Why isn’t 3D a finite dimensional associative division algebra?
I'm not sure that's true. It's a very good way to visualize complex numbers at first, but the set of points in a 2D plane does not define complex numbers.
That's a very interesting question actually, what is the difference between R^2 != C ? Here's someone's attempt at showing this (top answer): https://math.stackexchange.com/questions/444475/whats-the-di...
To me, trying to visualize complex numbers is an easy way to forget that R^2 != C.... They look the same but don't behave the same way (how do you multiply to members of R^2 with each other ? you need to define it, whereas multiplication in C is already defined). C doesn't behave like 2D vectors either. Since multiplying 2 complex is not the same as multiplying 2 vectors (whatever that means) although their components could be the same.
structure structure structure.
You can use the geometric product, and C falls out as a result.
There is no reason to limit to the plane either[0]. Or four dimensions for that matter[1]. Complex numbers have ways in which they are "more fundamental" than quaternions in the same ways that reals are "more fundamental" than complex numbers.