> If the Reals are fundamentally the set of all points that make up a line, the "imaginary numbers" are the set of all 2d points in a plane I'm not sure that'

> If the Reals are fundamentally the set of all points that make up a line, the "imaginary numbers" are the set of all 2d points in a plane 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-difference-between-mathbbr2-and-the-complex-plane 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.