*What follows is a bit I did over at Math StackExchange. Posting it over here was an experiment in whether the mathematical typesetting would transfer correctly in a copy-and-paste. For the most part, as long as I leave it alone, it seems to have done so (modulo the line breaks being lost in the shuffle).*

Euler's equation

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

*rotate*it counter-clockwise, 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

*exactly itself*. Other exponential functions have derivatives that are itself multiplied by some constant. But only

*the*exponential function, with

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

*not*mean that

Suppose we then consider making a small change to

But what happens if we add not

*north*of

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

*unit circle*.

If we follow this line of reasoning, then the value of

*where*on the unit circle

*is*

The crucial observation is in how fast we make our way around the circle. When we made our first step, from

*magnitude*, of

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.

## No comments:

## Post a Comment