Comment by ActorNightly

I still think you misunderstanding what im saying. i don't have a problem with the math working out, its that i have a problem with the math being able to be applied in the first place.

Let me try a different way.

Suppose you start with just the i number line with the rules that exist. You have zero, integer i's, rational is, and even irrational i's. All seems good. Then you start to define operations. i*i is undefined (because it goes to -1, and the reals are outside of your domain that you are working with currently). And this means you can't effectively do any sort of futher work in defining exponentiation, or limits, because you can have purely complex polynomials with just undefined terms.

So like you said, complex is R+, not something like RxR. The definition of complex numbers is intrinsic to real numbers - its an enhancement on real numbers. And by extension, all the math works out when you do taylor series with e^i and such.

But this pretty much means its a rigid definition, i.e you are defining something in a certain construction to supplement reals.

And as for geometric claim, my argument with that is that just like when you have x, and then you add <x,y> in some form and way, you are defining geometry. So in defining a+bi, you are defining geometry.*