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I just started studying Complex analysis and have in fact just switched field to mathematics recently and so please forgive me if this is question seems trivial for a mathematics student to ask.

Question: Why do we use Complex numbers instead of another algebraic field or number system? i.e. It is "natural" to have the hierarchy $\mathbb{N}$, $\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{R}$, $\mathbb{C}$ and not others.

Attempt: I have often been told that $\mathbb{C}$ is algebraically closed (every polynomial with complex coefficients have a root in $\mathbb{C}$) and that it contains $\mathbb{R}$ as a isomorphic subfield.

But should there be other fields or number systems with the same properties as well? Did I miss anything "extra" about $\mathbb{C}$ that makes it special?

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If an algebraically closed field contains (a copy of) the real numbers, then it contains (a copy of) the complex numbers, so $\mathbb{C}$ is the minimum such field. – egreg Jan 4 '14 at 18:24
@egreg - That was the key insight that I was missing! Thanks! +1 – Jean Valjean Jan 4 '14 at 18:25
Because these real numbers are of two kinds: rational and irrational; and the latter are also of two kinds: algebraic and transcendental. But these two are self-coherent and self-contained within $\mathbb{C}$, not within $\mathbb{R}$, since $\mathbb{C}=\mathbb{A}\cup\mathbb{T}$. – Lucian Jan 5 '14 at 0:30
up vote 5 down vote accepted

If an algebraically closed field $F$ contains (a copy of) the real numbers, then $F$ also contains (a copy of) the complex numbers.

Indeed, if we have $\mathbb{R}\subset F$ and $F$ is algebraically closed, then $F$ contains a root $j$ of the polynomial $X^2+1\in \mathbb{R}[X]$. Therefore it contains $\mathbb{R}[j]$, but this field is isomorphic to $\mathbb{C}$: just define $a+bi\in\mathbb{C}\to a+bj\in \mathbb{R}[j]$ and check it's an isomorphism.

The key point is that $\mathbb{C}$ is algebraically closed and algebraic over $\mathbb{R}$.

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Thanks for writing this up! Accepted and +1. – Jean Valjean Jan 4 '14 at 18:30

Reading up further, thanks to prompting in the comments, I discovered this:

Using Zorn's lemma, it can be shown that every field has an algebraic closure,[1] and that the algebraic closure of a field K is unique up to an isomorphism that fixes every member of K. Because of this essential uniqueness, we often speak of the algebraic closure of K, rather than an algebraic closure of K.

$\mathbb{C}$ is the unique algebraic closure of $\mathbb{R}$.

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"Algebraically closed" is a heavy loaded notion. Think of the extension ${\mathbb R}\rightsquigarrow{\mathbb C}$ this way instead: We already have the real numbers ${\mathbb R}$ and are extremally successful with them in analysis, geometry, and physics. Unfortunately the equation $x^2+1=0$, and similar ones which turn up all the time don't have a solution in ${\mathbb R}$. Therefore we need a field $K\supset{\mathbb R}$ containing a number $i$ with $i^2=-1$ as everyday computing environment. It turns out that there are such fields. The simplest one of these is ${\mathbb C}$, consisting of no more than the obviously necessary numbers $x+ y\,i$, where $x$, $y\in{\mathbb R}$.

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And $x + yi$ is obviously necessary because the field has to be closed under addition and multiplication? That is a very nice way to put it. Thanks! +1 – Jean Valjean Jan 4 '14 at 18:54

I feel it is worth mentioning that there are other number systems that are of interest to mathematicians, although for different reasons, such as the $p$-adics and the quaternions. The complex numbers are most widely used though because of the algebraic properties you have already discussed, along with the fact that they play nicely with the Euclidean norm.

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