is considered by many to be the most beautiful equation in mathematics—rightly, in my opinion. However, despite what Gauss might say, it's not the most obvious thing in the world, so let's perhaps try to sneak up on it, rather than land right on it with a bang.
It's possible to think of complex numbers simply as combinations of real values and imaginary values (that is, square roots of negative numbers). However, plotting them on the complex plane provides a kind of geometric intuition that can be valuable.
On the complex plane, a complex number
Multiplication is where things get a little unusual. Multiplication by real values is just as you'd expect, generalizing from the one-dimensional real number line to the two-dimensional complex plane: Just as
But multiplication by imaginary values is different. When you multiply something by
OK, let's step away from the complex plane for a moment, and proceed to the exponential function. We're going to start with the ordinary ol' real-valued exponential function,
If you graph
Another way to put that is that the derivative of
To be sure, all exponential functions do that basic thing. However, the very unusual thing about
It's very rare that an expression has that property. The function
The only functions that have that property have the form
There's another way to think of the derivative that is not the slope, although it's related. It has to do with the effect that incremental changes in
That means that if you make a small change in
Now, let's return to the complex plane, and put the whole thing together. Let's start with
Suppose we then consider making a small change to
But what happens if we add not
Symbolically, we would say
Now, suppose we added another
One thing to observe about the small steps that we've taken is that each one is at right angles to where we are from the origin. When we were directly east of the origin, our small step was directly northward. When we were just a tiny bit north of east from the origin, our small step was mostly northward, but a tiny bit westward, too.
What curve could we put around the origin, such that if we traced its path, the direction we're moving would always be at right angles to our direction from the origin? That curve is, as you might have guessed already, a circle. And since we start off
If we follow this line of reasoning, then the value of
The crucial observation is in how fast we make our way around the circle. When we made our first step, from
In order to get to
or, in its more common form,
The foregoing is not, by any means, a rigorous demonstration. It's an attempt to give some kind of intuition behind the mysterious-looking formula.