Beautiful Math

Take a moment to gaze at Euler’s Identity:

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It has been called “exquisite” and likened to a “Shakespearean sonnet.” It has earned the titles “the most famous” and “the most beautiful” formula in all of mathematics, and, in a mere seven symbols, symbolizes much of its foundation.

Today we’re going to graze on it!

By the way, this post could be subtitled Inevitable Geometry as a follow-up to the Inevitable Math post. Parts of this will get kinda math-y, but I’ll leave most of that for the end.^[1] The main goal here is to see why Euler’s Identity is so beautiful.

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One key to seeing the deeper beauty of Euler’s Identity is in the unit circle — so named because its radius is set to one.

The first thing to notice is that there are five value symbols (from left to right: e, i, pi, 1, and 0).

These five values are all important mathematical constants. We’ll come back to them in a moment.

Next, notice that there is one addition, one multiplication (i × pi), one exponentiation, and one equality.

The first three represent three fundamental arithmetic operations. The last, the equality operator, is a cornerstone of mathematics.^[2]

Now let’s consider each of those five constants.

Zero is the additive identity. Any number added to zero is that number. It is also the first state — the state of nothing.

One is the multiplicative identity. Any number multiplied by one is that number. It is also the second state — the state of something.

So Euler’s Identity represents addition and multiplication, not only in the operations themselves, but by the identity values of those operations.

What’s more, zero and one (plus addition) gives us the natural numbers (and the third state: many). Given the obvious operations (specifically, division), we get the rational numbers. The right side of the Identity implies the countably infinite side of mathematics!

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Clik here to view.Further, information theory sees everything in terms of bits — the world of ones and zeros — so the realm of information processing is also implied.

Metaphorically, one and zero also represent a Yin-Yang pair — the ultimate cup pair — of something and nothing. All of discrete mathematics is ultimately based on these two simple constants.

The left side of Euler’s Identity is kind of freaky.^[3]

The most well-known of the three constants is the good old number pi (π) — the ratio between a circle’s radius and circumference. Even the ancient Greeks knew something was up with π. Turns out it’s a transcendental irrational number!

So is the number e (also known as Euler’s number). Not popular outside math circles (heh, heh), it’s the base of the natural logarithm (and logarithms are not so well-known, let alone the natural ones).

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The complex unit circle extends the XY version for the complex number plane.

Finally there is the infuriating number i, the “imaginary” number. It’s the square root of -1, an impossible number!

And all by itself, it brings in the realm of the complex numbers.

So the left side of the Identity, in contrast to the countably infinite world on the right side, represents the real number realm — the world of the uncountably infinite — including the somewhat magical transcendental and complex numbers.^[4]

This is an awful lot of ground for such a simple formula — just seven symbols in all — to cover. Stanford University mathematics professor Keith Devlin has said, “like a Shakespearean sonnet that captures the very essence of love, or a painting that brings out the beauty of the human form that is far more than just skin deep, Euler’s equation reaches down into the very depths of existence.”

The reason for that last phrase about reaching down in the very depths of existence is that Euler’s Formula shows how connected fundamental mathematics is. And I did just say Formula rather than Identity.

The Identity is a specific case of a more general formula by Euler. It looks like this:

e^ia = cos a + i sin a

That looks a little scary, but we can unpack it. And we’re not going to actually use it — just look at a some a few cases (so don’t bail just yet).

We’ve met e and i already, just think of them as magic; a is the angle around a circle (in radians). The sin and cos are the sine and cosine trigonometric functions you hated in high school.^[5]

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Setting the radius to 1 allows some simple formulas to describe the unit circle. In particular, we can always get both x and y from a.

Now when we want to calculate the x-y points on a circle, we use these two formulas:

x = cos a
y = sin a

For any angle, a, along the circle, we can get the x and y coordinates for the point at that angle.^[6]

We can replace the trig functions in Euler’s Formula to get this:

e^ia = x + i y

The right side now looks like a complex number.^[7] Which is exactly what it is. The right side plots the complex unit circle, the unit circle in the complex plane.

What’s astonishing is the left side — with a logarithm base and an imaginary number and the angle as an exponent — plots the same thing. As it turns out, π isn’t the only place the simple circle gets intensely complex!

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Another way to calculate points on a circle is to solve the equation: 1 = x² + y² (which is the Pythagorean theorem applied to the unit circle). The set of all x,y pairs generated describes a circle.

The angle, a, is in radians here, which are usually expressed in terms of π. (Dividing a circle into 360 degrees is arbitrary — a human invention. But radians are a natural unit based on π.)

A full (360°) circle is 2π radians. Half a circle is just π radians (that is, 3.1415… radians).

Now consider what happens if we use Euler’s Formula to calculate a point on that half circle:

e^iπ = cos π + i sin π
e^iπ = -1 + i 0
e^iπ = -1

And from there it’s a short jump to:

e^iπ + 1 = 0

Which is the Identity.

The cos π and sin π may seem alien if you’re not used to them, but remember that π is just a number. It works like this:

x  =  cos π (radians)  =  cos 180°  =  -1
y  =  sin π (radians)  =  sin 180°  =  0

We’re calculating the point halfway around the circle, the one on the far left side where x=-1 and y=0.

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The red point allows us to create Euler’s Identity. (We also consider the blue and yellow points.)

We can check out a couple of other easy-to-calculate points. For instance, if the angle, a, is zero (that is, the point on the far right side of the circle where x=+1 and y=0):

eⁱ⁰ = cos 0 + i sin 0
e⁰ = +1 + i (0)
1 = 1

At the 90 degrees around the circle (the point on top of the circle where x=0 and y=+1), we’d expect:

x  =  cos π/2  =  cos 90°  =  0
y  =  sin π/2  =  sin 90°  =  +1

Plugging that into Euler’s Formula gives us:

e^i(π/2) = cos π/2 + i sin π/2
e^i(π/2) = 0 + i (+1)
e^i(π/2) = i

And in the complex plane, that (0, +1) coordinate is the complex number {0 + 1i} or {0 + i} or just i.

So both these cases confirm the Formula works (and it has, in fact, been proven to work for all real angles, a).^[8]

If you find Euler’s Identity (or Formula) interesting, this eleven-minute video takes you through its derivation (which is based on the Taylor series for e and for cos and sin):

If that’s a bit much for you, here’s a much shorter pretty video with nice music that more or less covers the same territory:

^[1] And I’ll do the math anyway, so you can just follow along.

^[2] As a side note, it represents the Yin-Yang aspect of the universe. The equals operator splits the (formula’s) world into two parts — a true opposing pair — that must be in balance.

^[3] In case you were wondering, “Euler” is properly pronounced “oil-er” in German.

^[4] Which is a third way Yin-Yang appears in the Identity — the continuous versus discrete duality.

^[5] Did you realize that trigonometry is filled with sin! (But the fun kind that isn’t wrong.)

^[6] For now, all you need to know about trig is this:

^[7] A common form is {a + bi}, but in reality a and b are actually x and y in the complex number plane.

^[8] Euler’s Formula and Identity are just part of what Leonhard Euler contributed to mathematics. He is one of the seminal names in the field. For instance, if you read about the RSA encryption method, you encounter his Totient function.

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