The geometry of complex numbers

Complex numbers are usually introduced as a way to find the square root of negative numbers, using the imaginary unit $i = \sqrt{-1}$. But there's a much nicer geometric meaning behind them. In this article, I'll explain this geometric interpretation of complex numbers, and also connect it with Euler's formula.

Complex numbers as vectors

Consider a vector in the plane, represented by its Cartesian coordinates $(x, y)$. We can define addition of vectors $\textbf{a} = (a_x, a_y)$ and $\textbf{b} = (b_x, b_y)$ as:
$$ \textbf{a} + \textbf{b} \equiv (a_x + b_x, a_y + b_y). $$
In other words, to add two vectors, add their corresponding $x$ and $y$ coordinates[1].
Vectors can also be rotated and scaled. Scaling is multiplying both coordinates by a constant, and only changes the length, making the vector longer or shorter. Rotation only changes the direction. We'll define the positive direction of rotation as counterclockwise.

FIG. 1. Rotation of a vector by an angle $\phi$.
The key insight behind complex numbers is that a vector can represent both a point in space and a "rotation + scaling" operation.
Let's see how this is done. First, let's rewrite the vector in polar coordinates $(r, \theta)$. $r$ is the length of the vector, and $\theta$ is the angle measured counterclockwise from the positive x axis (we allow $\theta$ to be negative).

FIG. 2. Relationship between polar and Cartesian coordinates. (Source: Wikipedia).
From the figure, we see that $x$ and $y$ are related to $r$ and $\theta$ by:
$$ \begin{aligned} x &= r \cos \theta \\ y &= r \sin \theta \end{aligned}. $$
We can define multiplication of two vectors $\textbf{a} = (a_r, a_\theta)$ and $\textbf{b} = (b_r, b_\theta)$ as the following, calling the product $\textbf{c} = (c_r, c_\theta)$:
$$ \begin{aligned} c_r &= a_r b_r \\ c_\theta &= a_\theta + b_\theta. \end{aligned} $$
In this view, $a_r$ is a scaling factor and $a_\theta$ is an angle to rotate by. In other words, $\textbf{a}$ acts as a "rotation + scaling" on $\textbf{b}$. Note that $\textbf{a} \textbf{b} = \textbf{b} \textbf{a}$. In other words, it doesn't matter which order we multiply — multiplication is commutative.
Now, to convert $\textbf{c}$ back to $(x, y)$ coordinates, use equation $(2)$:
$$ \begin{aligned} c_x &= c_r \cos(c_\theta) = a_r b_r \cos(a_\theta + b_\theta) \\ c_y &= c_r \sin(c_\theta) = a_r b_r \sin(a_\theta + b_\theta) \end{aligned} $$
But this uses the polar coordinates of $\textbf{a}$ and $\textbf{b}$; we'd like to use Cartesian coordinates for those too.
Use the trig identities:
$$ \begin{aligned} \sin(u + v) &= \sin u \cos v + \cos u \sin v \\ \cos(u + v) &= \cos u \cos v - \sin u \sin v \end{aligned} $$
to express $c_x$ and $c_y$ in terms of $a_x$, $a_y$, $b_x$, and $b_y$.
Solution: $$ \begin{aligned} c_x &= a_x b_x - a_y b_y \\ c_y &= a_y b_x + a_x b_y \end{aligned} $$
Show/hide full solution
$$ \begin{aligned} c_x &= a_r b_r \cos(a_\theta + b_\theta) \\ &= a_r b_r (\cos a_\theta \cos b_\theta - \sin a_\theta \sin b_\theta) \\ &= (a_r \cos a_\theta) (b_r \cos b_\theta) - (a_r \sin a_\theta) (b_r \sin b_\theta) \\ &= a_x b_x - a_y b_y \\ \end{aligned} $$
$$ \begin{aligned} c_y &= a_r b_r \sin(a_\theta + b_\theta) \\ &= a_r b_r (\sin a_\theta \cos b_\theta + \cos a_\theta \sin b_\theta) \\ &= (a_r \sin a_\theta) (b_r \cos b_\theta) + (a_r \cos a_\theta) (b_r \sin b_\theta) \\ &= a_y b_x + a_x b_y \end{aligned} $$
Therefore, multiplication is defined as:
$$ \textbf{a} \textbf{b} \equiv (a_x b_x - a_y b_y, a_y b_x + a_x b_y) $$
in Cartesian coordinates.
Now, instead of writing $\textbf{a}$ as $(a_x, a_y)$, write it in the more familiar notation $a_x + a_y i$, and write $\textbf{b}$ as $b_x + b_y i$. This is just a different way to write the same vector. Then note that the "usual" rule for complex multiplication is exactly equation $(9)$[2]. So this geometric idea of multiplication checks out.
The essential point here is that complex numbers are two-dimensional vectors, with a multiplication operation defined geometrically as rotation + scaling of one vector by the other[3].
Some important results that follow:

The complex conjugate

The complex conjugate of a complex number $\textbf{a} = a + b i$ is defined as:
$$ \textbf{a}^* \equiv a - b i $$
It simply negates the imaginary component, i.e. reflects the vector across the x-axis[4]. Note that this is equivalent to negating its angle.
Show that the length of a vector, $\left|\textbf{a}\right|$, equals $\sqrt{\textbf{a}^*\textbf{a}}$.
Show/hide solution
$\sqrt{\textbf{a}^*\textbf{a}} = \sqrt{(a - b i) (a + b i)} = \sqrt{a^2 + b^2} = \left|\textbf{a}\right|$.
Visually, $\textbf{a}^*\textbf{a}$ rotates $\textbf{a}$ by its angle negated, moving it onto the positive x-axis, and also scales it by its length, making its total length $\left|\textbf{a}\right|^2$. Taking the square root yields $\left|\textbf{a}\right|$.

Euler's formula

The famous physicist (and all around genius) Richard Feynman called Euler's formula "the most remarkable formula in mathematics". It relates the function $e^x$ to the $\sin$ and $\cos$ functions, an unexpected twist, since these two kinds of operations seem pretty unrelated. There are some purely algebraic derivations of it that are short and slick, but for this article a more easily visualized one seems appropriate, and provides more insight. It comes from V.I. Arnold's Ordinary Differential Equations[5].
The number $e$ is defined as:
$$ e \equiv \lim_{k \rightarrow \infty} \left(1 + \frac{1}{k}\right)^k $$
and so
$$ e^x = \lim_{k \rightarrow \infty} \left(1 + \frac{1}{k}\right)^{kx}. $$
Substituting $n = kx$, we get:
$$ e^x = \lim_{n \rightarrow \infty} \left(1 + \frac{x}{n}\right)^n. $$
We'll take this to be the definition of the function $e^x$ for both real and complex numbers.
So what is $e^{i\phi}$ (for some real number $\phi$)? Plug it into the definition:
$$ e^{i\phi} = \lim_{n \rightarrow \infty} \left(1 + \frac{i\phi}{n}\right)^n $$
Think about what this equation is doing — repeatedly multiplying $\left(1 + \frac{i\phi}{n}\right)$ by itself $n$ times, which corresponds to repeated rotation and scaling by the same factor. So, perhaps we can simplify it by calculating the magnitude and angle separately.
Start with the magnitude. Each multiplication scales the result by $\left| \left(1 + \frac{i\phi}{n}\right) \right|$, so the final magnitude is:
$$ \begin{aligned} \left| e^{i\phi} \right| &= \lim_{n \rightarrow \infty} \left| \left(1 + \frac{i\phi}{n}\right) \right|^n \\ &= \lim_{n \rightarrow \infty} \left( \sqrt{1 + \frac{\phi^2}{n^2}} \right)^n \\ &= \lim_{n \rightarrow \infty} \left( 1 + \frac{\phi^2}{n^2} \right)^{n / 2} \\ &= \lim_{m \rightarrow \infty} \left( 1 + \frac{1}{m} \right)^{\phi \sqrt{m} / 2} \; \; &\text{(defining } \frac{\phi^2}{n^2} = \frac{1}{m}, \text{so } n = \phi \sqrt{m}) \\ &= \lim_{m \rightarrow \infty} e^{\frac{\phi}{2 \sqrt{m}}} \; \; &\text{(using eq. (12))} \\ &= 1. \end{aligned} $$
So the magnitude is $1$ regardless of $\phi$[6].
Now for the angle. Each multiplication rotates the result by $\angle(1 + \frac{i\phi}{n})$[7] and so adds this amount to the final angle:
$$ \begin{aligned} \angle e^{i\phi} &= \lim_{n \rightarrow \infty} n \angle\left(1 + \frac{i\phi}{n}\right) \\ &= \lim_{n \rightarrow \infty} n \tan^{-1}{\frac{\phi}{n}} \end{aligned} $$
Note that $\frac{\phi}{n} \rightarrow 0$ as $n \rightarrow \infty$. If you look at the graph of $\tan^{-1}(x)$, you'll see that $\tan^{-1}(x) \approx x$ when $x$ is near 0, which comes from the Taylor series of $\tan^{-1}(x)$ around $x = 0$. So:
$$ \begin{aligned} \angle e^{i\phi} &= \lim_{n \rightarrow \infty} n \frac{\phi}{n} \\ &= \lim_{n \rightarrow \infty} \phi \\ &= \phi. \end{aligned} $$
But the magnitude and angle are just the $r$ and $\theta$ of polar coordinates. So
$$ \begin{aligned} (e^{i\phi})_r &= 1 \\ (e^{i\phi})_\theta &= \phi \end{aligned} $$
and from $(2)$,
$$ \begin{aligned} (e^{i\phi})_x &= \cos\phi \\ (e^{i\phi})_y &= \sin\phi \end{aligned} $$
$$ e^{i\phi} = \cos\phi + i \sin\phi. $$
This is Euler's formula. An animation is helpful at this point, showing the convergence of the limit $(14)$:

FIG. 3. Convergence of $e^{i \pi / 3} = \cos{\pi / 3} + i \sin{\pi / 3}$ using definition $(14)$. Each dot corresponds to an additional rotation/scaling of $\left(1 + \frac{i\theta}{n}\right)$ (Source: Wikipedia).
In practice, Euler's formula basically provides a nicer way to write complex numbers in polar coordinates: $\textbf{a} = (a_r, a_\theta) = a_r e^{i a_\theta}$. The nice thing is that you can multiply them just like ordinary exponentials, since the lengths multiply and the angles add through the normal product law $e^a e^b = e^{a + b}$.
A few tips for complex number arithmetic:


1 Some notation:
2 And, of course, the usual rule for complex addition is $(1)$.
3 In more advanced mathematical jargon, the set of 2D vectors is called $\mathbb R^2$ and, with vector addition and scaling defined, constitutes a vector space. We have equipped this space with a multiplication operation satisfying the field axioms (one of which is commutativity), which also makes it a [field]( (usually called $\mathbb C$).
4 Open-ended question: why don't we need an operator that reflects it across the y-axis?
5 V.I. Arnold was a mathematician who once gave an interesting talk/rant against high abstraction in mathematics, worth checking out.
6 Replacing $n$ by $m$ above is not valid if $\phi = 0$, but in that case the result is obviously $1$ from the definition.
7 $\angle x$ means the angle of $x$, measured counterclockwise from the positive x-axis, as always.