Linear Algebra 0: Complex Numbers

July 3, 2026

Complex numbers are usually taught as taking the form $a + bi$, where $a$ is the real part and $b$ is the imaginary part. However, I want to start somewhere else: circles.

Circles and complex numbers fit naturally together, but to see why, we’ll need to play around with circles a bit more. Here’s a circle below. Drag around the red point and see how it relates to the variables $R$ and $\theta$.

As we can see, the position of $z$ depends on $R$ and $\theta$ in the relation $z = Re^{i \theta}$. You may have noticed that our graph is not drawn with $x$ and $y$ axes but with $\text{Re}$ and $\text{Im}$ axes. What do $e$ and $i$ have to do with circles, and how do we recover the conventional $a + bi$ formulation of a complex number from this circle business? That’s what we’ll learn here. We’ll learn about Euler’s formula along the way. After all of this, we’ll have enough experience with complex numbers to start discussing vector spaces with complex numbers. The goal is to understand what complex numbers are actually about, not just how they’re defined.

Defining $e$

We have this way to define a point on a circle:

$ \Large z = R e^{i \theta} $

What is $e$ and why does it fit in here?