>>11039712They have a multiplication rule that R^2 as a vector space does not. The special part of C is their algebraic structure (in particular, being a division ring, which R^2 is not).
Now if you want to map C to a vector space over the reals, you need a subspace of R^(2x2). You need the multiplication operation to preserve the structure of C. But it is unwieldy to constantly multiply 2x2 matrices, when the polar coordinate picture is so much simpler to do computations with, algebraically and geometrically.
So there is an explicit isomorphism between the fields C and that subset of R^(2x2), and as any good mathematician knows, once you have a structure-preserving isomorphism, it literally doesn't matter which representation you want to work with. "A rose by any other name", as they say. In other words, no one ever gives a fuck, have fun doing your matrix multiplication, I'll just add phases and multiply scalars, and we'll both get the same result at the end of the day.
>>11040370What the fuck
