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I think C={z: z=a+ib , i^2=-1} it has no such defn. It is just similarity. It is isomorph to R^2.
I'm not so sure if the two spaces are isomorphic. They "look alike".. but there are theorems that are true in C that just aren't true in R^2.. e.g. a function that is differentiable is infinitely differentiable.
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Both spaces are isomorphic but the product in both cases is very different and that changes a lot of things
In R^2 (a,b) * (c,d) = (ac,bd)
In C if a,c are the real part and b,d the imaginary part:
(a,b)*(c,d) = (ac-bd,ac+bd) -
Topological properties will be very similar in both spaces as it is indeed the same space. But once you start multiplying everything is very different as in R2 both components (a,b) are independent whilst in C they are related as i^2 = -1
End of conversation
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Exactly!
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