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When I first learned about representation of a complex number by a point in a $2D$ plane, I wondered: what if it's redundant? What if a line is sufficient?

Apparently, it's not, but I still wonder: what is the proof of this? When was the fact first realized? Or was it totally obvious from the start of development of the theory of complex numbers?

Ruslan
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    With a bijection from $\Bbb R\to \Bbb R^2$, one could argue that a line is still sufficient, but that doesn't mean it would be practical to work with... – abiessu Dec 13 '14 at 06:38
  • Complex Numbers founded by algebraists who were working with algebraic expressions and encountered negative numbers under radicals. They just defined a number whose squares are negative and there were no need to make a total order on them. But as @abiessu stated you can equip $\mathbb{C}$ with a total ordering relation. In addition if you accept axiom of choice, you can equip complex numbers a well-ordering relation :) – Fardad Pouran Dec 13 '14 at 06:51
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    @abiessu: If I am not mistaken, there bijections from ℝ to ℂ do exist (e.g., take every second digit for the real part and every other for the imaginary part), however they do not preserve any useful algebraic structure and are thus not useful. – Wrzlprmft Dec 13 '14 at 10:36
  • @Wrzlprmft: That doesn't give a bijection.. For example 0.100000... and 0.0090909... map to the same complex number under your map! – user21820 Dec 13 '14 at 12:09
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    @user21820: You are right, it’s more complicated. Still gives a rough idea, what these bijections look like. This question addresses how to explicitly construct such bijections. – Wrzlprmft Dec 13 '14 at 12:16

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History
Complex numbers were introduced by Cardan (Girolamo Cardano) in his Ars Magna in 1545.
They were however described only purely algebraically as a means for solving polynomial equations of degrees 3 or 4.
The first geometric description of $\mathbb C$ as points of a plane was given in 1799 by Wessel, a Danish cartographer, and independently by Argand, a Swiss-French bookshop owner, in 1806.
Argand was also the first to prove the "fundamental theorem of algebra" according to which a real non constant polynomial has at least one complex root and he was the first to notice that the theorem also applies to polynomials with complex coefficients.

Facts
The field $\mathbb C$ has dimension 2 (i.e. is a plane) as a real vector space : this is an easy result.
It also has dimension 2 as a differential manifold, which is a bit harder to show.
It has dimension 2 as a topological space: this is really hard since already the very notion of topological dimension is quite sophisticated.

Conclusion
To sum up and answer one of your questions:
No it was definitely not "totally obvious from the start of development of the theory of complex numbers" that they have dimension two.

Ittay Weiss
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Georges Elencwajg
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  • The FTA was stated and partially proved by D'Alembert; the first rigorous proof was given by Gauss in his graduation thesis, in 1799. In this thesis Gauss represented complex numbers with points in the plane; Wessel and Argand did some more work on the topic. – egreg Dec 13 '14 at 14:38
  • @egreg: I know that Gauss is usually credited for the first rigorous proof but Steven Smale, the well-known topologist and Fields medalist, explains here on page 4 that Gauss's proof contains an "immense gap", only filled by Ostrowski more than 120 years later. As explained by Smale, the subtle point not proved by Gauss is that an algebraic curve cannot enter a disk without leaving it. – Georges Elencwajg Dec 13 '14 at 17:53