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.

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 $b_y$.

$$
\begin{aligned}
c_x &= a_r b_r \cos(a_\theta + b_\theta) \\
&= a_r b_r (\cos a_\theta \cos b_\theta - \sin a_\theta \sin b_\theta) \\
&= (a_r \cos a_\theta) (b_r \cos b_\theta) - (a_r \sin a_\theta) (b_r \sin b_\theta) \\
&= a_x b_x - a_y b_y \\
\end{aligned}
$$

$$
\begin{aligned}
c_y &= a_r b_r \sin(a_\theta + b_\theta) \\
&= a_r b_r (\sin a_\theta \cos b_\theta + \cos a_\theta \sin b_\theta) \\
&= (a_r \sin a_\theta) (b_r \cos b_\theta) + (a_r \cos a_\theta) (b_r \sin b_\theta) \\
&= a_y b_x + a_x b_y
\end{aligned}
$$

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 imaginary unit $i$ is the vector $(x = 0, y = 1)$, which represents a scaling by $1$ (i.e. no scaling), since it has length $1$, and a rotation counter-clockwise by $90^\circ$ ($\pi / 2$ radians), since it's $90^\circ$ from the positive x-axis.
- The real numbers are represented by vectors on the x-axis, since $(1)$ and $(9)$ essentially reduce to real addition and multiplication when $y = 0$ (you can verify this geometrically as well).

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 $\left|\textbf{a}\right|$.

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 $(2)$,

$$
\begin{aligned}
(e^{i\phi})_x &= \cos\phi \\
(e^{i\phi})_y &= \sin\phi
\end{aligned}
$$

or,

$$
e^{i\phi} = \cos\phi + i \sin\phi.
$$

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:

- When adding complex numbers, use rectangular form ($x + y i$).
- When multiplying, dividing, or taking powers, use polar form ($r e^{i \theta}$).
- When converting between the two forms, try to visualize the vector before plugging in numbers.

- Complex numbers are 2D vectors.
- Complex multiplication is defined geometrically as rotation + scaling of one vector by the other, calculated with $(9)$ in Cartesian coordinates.
- The complex conjugate flips the vector across the x-axis, which is the same as negating its angle.
- Euler's formula provides a better way to write a complex number in polar coordinates that obeys the natural law of multiplying exponentials.

- The symbol "$\equiv$" means "is defined as".
- I'll refer to coordinates of a vector using subscripts — $a_x$ means the $x$ coordinate of $\textbf{a}$.