This is not a standard simplifience article. It provides minimal intuition for $i$ necessary to understand Euler’s identity. Both $i$ and Euler’s identity are contrives, which are normally avoided.

$i$ is called the “imaginary number”, as if all numbers aren’t imaginary. Let’s correct that misconception now: $i$ is no more nor less imaginary than any other number.

Consider the number line. We’ve been filling it in for millennia.

First there were only whole numbers. You can’t really even call it a number line: it’s more of a collection of number dots.

There was nothing in between the numbers, which were whole and pure. Fractions came as a shock to everybody and really started filling things in:

But it didn’t stop there. A man was allegedly murdered for discovering irrationals. Zero took an embarrassingly long time to discover. Negative numbers were extremely controversial and were ignored for quite some time.

It’s important to remember that discovering new types of Number doesn’t make the old types of Number stop working: fractions are useful, but if you if you cut a living friend in half you won’t have half a living friend.

The new features of the number line are only useful when they apply. Negative numbers are great for counting money, but I bet you’ve never seen a negative cow. Before you use numbers you have to figure out how much Number applies to your specific situation.

We’ve been jamming new features into the term “Number” for millennia. It should come as no surprise that there are yet more features we can shove into the overburdened number line.

The so called “imaginary numbers” are a fairly modern extension. They take the boring old number line and extend it into a number plane.

There are many ways to turn the number line into a number plane. You’ve already encountered number planes for things like graphs and charts where you write points in the form $(1, 2)$ or $(x, y)$. Using imaginary numbers is just like using $(x, y)$ pairs. The confusing part of imaginary numbers is that we refer to a point on the number plane as a single two-dimensional number.

The “imaginary numbers” you’ve heard so much about are just the vertical axis of the number plane. Together with the original (“real”) number line this creates the complex plane. Points on this plane are two-dimensional numbers.

The entity $2 + 3i$ is written in two parts. Don’t let the notation confuse you: though disjoint, it is describing a single unified 2D number (marked by the red dot). Why do we write one number in two parts? Don’t ask me. It’s a silly convention.

There are a few different ways that you can construct two-dimensional numbers. The complex plane is a type of 2D number that is really good at talking about rotation. Rotation, as it happens, is prevalent in reality.

In the interest of clarity we’ll introduce a new syntax for these 2D numbers that makes it easy to see their rotation. We’ll write them like $3_{↺\frac{1}{8}}$, where $3$ is the size of the number and $\frac{1}{8}$ is how much it’s rotated.

The complex plane acts how you’d expect a number plane to act, with one stipulation: multiplication of “complex numbers” works by multiplying the lengths and adding the rotations.

For example, try multiplying $3_{↺0}$ (three horizontal 3) by $2_{↺\frac{1}{4}}$ (the vertical 2).

You get a $6_{↺\frac{1}{4}}$ which is the vertical six, also known as $6i$.

The complex number plane is a number plane where numbers also have angles. Multiplication scales the numbers and adds the angles.

The cool thing about a number plane is that the number $1$ has gotten a lot more promiscuous. It used to be that we had just one $1$. Now we have a whole bunch of ones at a all the different angles.

By convention, we still refer to the original one $(1_{↺0})$ as “one” and the half-turned one $(1_{↺\frac{1}{2}})$ as “negative one”. We have a special name for the vertical one $(1_{↺\frac{1}{4}})$. That special name is $i$.

There’s nothing special about imaginaries or complex numbers. Whenever we use imaginary numbers, we’re using a number plane instead of a number line. On this particular number plane, multiplication is scaling rotation.

There are some more subtleties to the complex number plane, but that’s all you need to know about $i$ in order to understand Euler’s identity.