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.
[Math] we expect the connection between complex arithmetic and geometry
complex numbers
Best Answer
We can start way earlier to get a geometric interpretation, at the real numbers. Multiplication by a real number is a combination of scaling and mirroring. Multiplying by a a positive number is scaling the real line, multiplying by $-1$ is mirroring it at the origin. On an abstract level, a core feature of mirroring is that doing it twice returns the original picture. This gives rise to the interpretation that multiplication by $-1$ is a mirroring, since $(-1)^2=1$, so multiplying by $-1$ twice is the identity.
The complex numbers give rise to a similar interpretation. We can still view multiplication by $-1$ as mirroring the plane at the origin, but in a 2d context, we can also see it as a $180^\circ$ rotation. They are really the same. But we also get a new element, $\mathrm i$. Its basic feature is that $\mathrm i^2=-1$, that is, multiplying by $\mathrm i$ twice is rotation by $180^\circ$. But that's also a core feature of rotation by $90^\circ$: rotating by that amount twice is the same as rotating by $180^\circ$ once. So that's a good hint that complex multiplication can have something to do with rotations. We just need to find a fitting topology (a scalar product to describe angles, most importantly) which makes multiplication by $\mathrm i$ an actual $90^\circ$ rotation. And it turns out that the scalar product wrt which $1$ and $\mathrm i$ form an orthonormal basis does just that. So it's a good idea to choose those as a basis of $\mathbb C$ as a real vector space, making them span the coordinate axes. In this picture, multiplication by $\mathrm i$ will be guaranteed to be a $90^\circ$ rotation. And using some algebra, all other complex multiplications can then be shown to also be rotations and scalings.