The purpose of proving uniqueness of representation?

Question

So, suppose we define the complex numbers as (a, b).
We can prove this representation is equivalent to a + bi.
And that makes sense, we are showing two representations are equivalent.

But what is the purpose in provign something like:
a + bi = a' + bi' forces a = a' and b = b'?
It just seems odd for this not to be the case.
Is this just to prove that two complex numbers are only equvilant if their real and imaginary parts are the same?
If so, regardless, what is the purpose in proving things like this?

Also, in Ahlfors:
They consider F  which is a super set of R and has a solution for x^2 + 1 = 0.
They let C be the subset of all elements of the form a + bi. [first, I don't see how they can let that be so, because how do we know elements in F are even of that form?]

They then state C is a subfield of F, and that makes sense via some mechanical proofs for a field.
They then prove the structure of C is independent of F.
They consider F' which is another superset of R and a root i' being the solution to x^2 + 1 =0.
The corresponding C' is formed by all elements a + b i'  .
They show there is a bijection and consequent isomorphism, but what does it mean for the structure to be independent of F?

Anonymous · Accepted Answer

I used to have a copy of Ahlfors but it somehow disappeared from my possession sometime in the distant past. It was a good text, so far as I can recollect. 
There are some entities which can be defined as ordered pairs of integers, (m, n) with (m, n) = (m', n') and m ≠ m' and n ≠ n'. In fact in approaches in which everything is reduced to set theory, a fraction is defined as the equivalence class of all similar fractions. (m, n) and (m', n') are similar or equal if mn' = m'n. Depending on how complex numbers are specified by Ahlfors, there may be some point in giving a formal proof that (x, y) = (x', y') ⇔ x = x', y = y'. And when the number system is extended from the natural numbers → signed integers → rationals → reals → complex numbers, the rationals (for instance) are not an actual subset of the reals but are isomorphic to one that is an actual subset of it. Abstractly, all isomorphic sets are considered the same, inasmuch as they have the same abstract algebraic structure, even though the nature of their elements may differ. I can't remember how Ahlfors characterised the complex numbers, but he must have done so in an approach which necessitated a formal proof that x + iy = x' + iy' ⇔ x = x', y = y'. This reminds me of x + √y = x' + √y', where y, y' are not perfect squares. Of course you would expect x = x', y = y', and indeed this is so. But though the proof isn't difficult, you have to do some work to get there. Thus the conclusion is not on the surface but is buried somewhat beneath it.

I don't expect my rambling comments will have shed any light on your puzzlement. Maybe the next poster will.

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The purpose of proving uniqueness of representation?

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