# Why are complex numbers so magical? [closed]

Both the real numbers and the complex numbers have a whole bunch of really nice properties: For reals, we have ordering, the intermediate value theorem, etc. Complex numbers are algebraically closed, and we have nice calculus results like the Cauchy–Goursat theorem, holomorphicity implies analyticity, etc. These results are to be contrasted with the case of e.g. higher dimensional spaces like $\mathbb{R}^n$ or other structures like the quaternions.

My feeling is that all the nice properties of the reals can be traced to completeness and ordering. However, I don't have a feeling for why the complex numbers have such miraculous analytical properties. Since $\mathbb{C}$ is defined essentially as the algebraic closure of $\mathbb{R}$, I might naively suspect that closedness is the crucial property, but I don't see how that manifests itself in proving theorems like Cauchy-Goursat. Perhaps Cauchy-Goursat is itself the essential property?

So my question is: Are there one or two fundamental properties of the complex numbers which beget all the other miracles of complex analysis? In other words, what makes complex analysis so different from real analysis or quaternionic analysis?

## closed as too broad by Clayton, John Gowers, Adam Hughes, Shailesh, NamasteDec 18 '16 at 0:59

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• because they have an imaginary part to them? – Jorge Fernández Hidalgo Dec 17 '16 at 20:04
• See here for a related question. – Dietrich Burde Dec 17 '16 at 20:10
• It's worth noting that the algebraic closure of $\mathbb C$ is often proved using Liouville's theorem (though it can certainly be proved with much less) - so the 'nice' analytic properties of $\mathbb C$ often come before the algebraic properties. – John Gowers Dec 17 '16 at 20:16
• Multiplication adds a constraint that forces locally differentiable functions to be locally analytic. That is pretty amazing to me. – copper.hat Dec 17 '16 at 20:33
• So your question is essentially why complex analysis (theory of holomorphic functions $\mathbb{C} \to \mathbb{C}$) is so "miraculous" compared to analysis in other spaces ? Maybe a proof that "differentiable $\implies$ analytic" is true only in (dense subspaces of) $\mathbb{C}$ is what you want ? The essential property is that if $f(z)$ is differentiable on $U$ and $\gamma \simeq \gamma'$ homotopically on $U$, then $\int_\gamma f(z)dz =\int_{\gamma'} f(z)dz$, plus the homotopy groups of open sets $U \subset \mathbb{C}$ – reuns Dec 17 '16 at 21:13