TITLE: Why should we expect the connection between complex arithmetic and geometry?
QUESTION [20 upvotes]: I realized that I take it for granted that properties of complex numbers have clear geometric interpretations. Visualizing complex numbers with the help of the complex plane really helps to understand complex arithmetic better and those mysterious properties of holomorphic functions (conformality, Maximum-Modulus Theorem, Argument Principle to name a few) make perfect sense once one knows that complex multiplication is simply a rotation and scaling. But lately I have been asking myself why there should be a connection between complex arithmetic and geometry at all? Of course there is nothing stopping us from interpreting these numbers as points in the plane (after all they are pairs of real numbers) but I am still bewildered by the fact that once we think of them this way, everything else related to complex numbers seems to find a perfect geometrical explanation! For example without a geometrical picture, the only way to understand complex multiplication is the distributive law. But the geometrical interpretation of complex multiplication turns out to be much more elegant and it is almost like it was always meant to be thought of that way. I am really curious to hear your thoughts about this.

REPLY [11 votes]: You can arrive at complex arithmetic from geometric intuition if you start with transformations of the plane.
It is well known that matrices which preserve angles (i.e. map shapes to similar shapes) and orientation are of the form $cR(\theta)$, where $c$ is a positive number and $R(\theta)$ is a rotation matrix. That is,
$$cR(\theta) = c\pmatrix{\cos \theta & -\sin \theta\\ \sin \theta & \cos \theta}. $$
Since $c$ and $\theta$ are arbitrary, these are all matrices of the form $$\pmatrix{a  & -b \\ b & a}$$ for $a, b \in \mathbb R$ (except for the null matrix).
Now, after adjoining the null matrix, this set becomes a vector space of dimension two, closed under matrix multiplication, and where all non-null elements have a multiplicative inverse ($c^{-1} R(-\theta)$).
The interesting part is that we can choose a basis like this:
$$ \pmatrix{a  & -b \\ b & a} = a \pmatrix{1  & 0 \\ 0 & 1} + b \pmatrix{0  & -1 \\ 1 & 0} = a I + b J,$$ where $I$ is the identity matrix and $J=\pmatrix{0  & -1 \\ 1 & 0}$ is a matrix which, under matrix multiplication, has the property $J^2=-I$. That is, it is in some sense the "square root" of $-I$. It also represents rotation in 90º (like the complex $i$ does). Indeed: $J = R(\pi/2)$ and, as expected, $J e_1 = e_2$ and $J e_2 = -e_1$.
Moreover, if you work out the product rule, it is exactly the one that arises in complex numbers:
$$ \pmatrix{a  & -b \\ b & a} \cdot \pmatrix{c  & -d \\ d & c} \\ = \pmatrix{ac-bd  & -(ad+bc) \\ ad+bc & ac-bd} \\ = (ac-bd)I + (ad+bc)J.$$
Furthermore, we can define subtraction, division, and all arithmetic operations for them in a way parallel to how they are defined for complex numbers.
Finally, add to this that the subspace generated by $I$ is an algebraic copy of $\mathbb R$, so you can view the full space as an extension of $\mathbb R$.
To sum up

Angle and orientation preserving linear transformations carry great geometric meaning (similarity).
They form a two-dimensional space, which you can think of as an algebraically compatible extension of $\mathbb R$.
They have two components, one in the direction of the identity/unit and one in the direction of a $\pi/2$ rotation.
This space can be constructed from $\mathbb R$ just by adjoining an outside element $J$ such that $J^2$ is minus the identity (and extending the usual algebraic rules).
This is essentially the same recipe as the one for constructing the complex numbers.