How to explain Euler's identity using trianglesand spirals

From: http://xkcd.com/179/

Is this you? Want a better explanation? Read on...

When we multiply two complex numbers in polar coordinates, we multiply the lengths and add the angles.

*If not, try this article: http://betterexplained.com/articles/a-visual-intuitive-guide-to-imaginary-numbers/

Hopefully you remember learning in school about multiplying complex numbers.* 

We can draw this as two similar triangles in the complex plane.

The square of a complex number is even easier to draw. Just make two similar triangles that share an edge.

We can raise a complex number to a higher power by stacking similar triangles to build a logarithmic spiral. 

The length of the spokes grows exponentially, but the angle between each two spokes is always the same.

As a special case, if we stack isosceles triangles then we go around a circle instead of a spiral.

0 1

A

A5

A×ASince the first spoke must have length 1 and all the other spokes are the same length, the vertices all lie on the unit circle.

As a special case of the special case, here's how to draw the definition of i by stacking two right icosceles triangles.

0 1

i

i2 = -1

Each right triangle sits on the hypotenuse of the previous triangle, so the triangles have to get bigger.

1

A

A×A

This spiral has a nice formula:

A6

If we stack right triangles in a different way, we again get a spiral. 

A×A×A

The parameters of the formula control which way the spiral will go.

A = 1 + xi

A×A = (1 + xi)2

A6

1

A×A×A = (1 + xi)3

The formula for stacking triangles is similar to the one for compound interest, except that each "interest payment" is rotated 90 degrees from the "principal".

1

A

A×A

A6

Suppose we "compound" more frequently? That increases the number of triangles and makes them skinnier.

1

A

A16

I'm setting x to π/n so that things work out later.

As we increase the number of triangles and make them proportionally skinnier, the length of the hypothenuse of each right triangle becomes very close to the length of the previous spoke. So the triangles don't grow as much, and spiral becomes closer to circular.

1

AA

24

Taking the limit, we get an infinite number of spokes that are all the same size, taking us around a circle.

1

And going π radians around the unit circle takes us to the opposite side.

~-1

The limit that we just took corresponds closely to the definition of the exponential function.

1~-1

It would be nice if we could extend the domain of the exponential function so that x could be a complex number. Then we could plug in x = π i to get Euler's identity.

1~-1

For the limit itself, there's no mystery. We don't need to raise a number to an imaginary exponent (whatever that means). We're just repeatedly multiplying complex numbers that are very close to 1.

1~-1

It's repeated multiplication because n is a positive integer.This is just an ordinary

complex number, a + bi.As n gets large, it gets closer to 1.

But there seems to be some sleight of hand: we're defining imaginary exponents to mean something new, circular movement. It fits together seamlessly, but raising a number to an imaginary power seems weird. Why would anyone want that?

1~-1

Let's take a step back and look at the powers of some simpler numbers. If we start with 1 and repeatedly double it, we can plot the path of y=2x.

1

2

4

8

1, 2, 4, 8...

Here we are defining the powers of 2 using repeated multiplication. There's a big jump from each point to the next. But we can fix that.

If we want to use the same equation (y=2x) but allow any real number for x, we cannot use repeated multiplication anymore. (What does it mean to multiply a number π times?) 

1.0

2.0

4.0

8.0

We need another way of defining exponentiation to generate a curve that connects the dots.

1.0

2.0

4.0

8.0

The real exponential function uses continuous compounding to create a curve that extends repeated multiplication. With a little adjustment, it can connect the dots laid by repeated multiplication of any positive real number. We can do this because all exponential growth curves are essentially the same.

Now let's take a complex number and repeatedly multiply it. We get a spiral trail of points on the complex plane. (We're stacking triangles, remember?)1

1 + i

2i-2 + 2i

-4

-4 - 4i

We want the exponential function to connect the dots laid by repeated multiplication. But e is a real number and raising it to any real power won't rotate.

To construct a spiral, we need a circular component to control the angle and a linear component to control the radius. Defining the imaginary powers of e to follow a circle gives us everything we need to construct any logarithmic spiral on the complex plane by raising e to a complex exponent.

The complex exponential function generates curves that connect the dots from repeated complex multiplication.

Just like the real exponential function generates curves that connect the dots from repeated real multiplication.

-1 1

-1 1

i

i2 = -1

So we can think of Euler's identity as showing how we can use continuous rotation on the unit circle to connect the dots plotted by repeated multiplication of i.

Rotation was there all along in the definition of i and complex multiplication. The exponential function fills in the curve.

Hope that helped!

Brian Slesinskybrian-web@slesinsky.orghttp://slesinsky.org/brian/https://plus.google.com/114156500057804356924/posts