Take a moment to gaze at Euler’s Identity:
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.
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!
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).
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]}
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!
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.
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):
e^{i0} = cos 0 + i sin 0
e^{0} = +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:
degrees | radians | equals | degrees | radians | equals | |
---|---|---|---|---|---|---|
cos 0° | cos 0 | +1 | sin 0° | sin 0 | 0 | |
cos 90° | cos 1/2 π | 0 | sin 90° | sin 1/2 π | +1 | |
cos 180° | cos π | -1 | sin 180° | sin π | 0 | |
cos 270° | cos 3/2 π | 0 | sin 270° | sin 3/2 π | -1 |
^{[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.
October 17th, 2015 at 5:14 pm
A good test of your inner math sense is whether, upon seeing Euler’s Identity for the first time, some part of you goes, “Hey,… wait a minute…”
It starts when you see that what the equation is saying is:
And the idea that a really weird number to the power of an imaginary number times another really weird number associated with circles… gives the result -1 is preposterous!
But it is, in fact, the essence of circles — π, cosine, & sine — that makes it work.
October 23rd, 2015 at 4:50 pm
Can’t say I followed that one. I never took trig. 🙂
October 23rd, 2015 at 5:42 pm
The trig does open the door to a big part of why this is so stunning, but hopefully you followed the part about the five key math constants and the three basic operators (add, multiply, exponent).
(And maybe the Yin-Yang parts?)
Or you can just take my word for it. Beautiful! Amazing! Stunning! Sexy! Surprising!
October 24th, 2015 at 12:00 pm
I got those parts, but later I got scared. 🙂
October 24th, 2015 at 5:10 pm
🙂 I’ll make it my life’s mission to help you get past your fear of math!
I did try to write the post so people could just get off the bus once the going got really math-y.
October 24th, 2015 at 6:05 pm
Oh no! Your life’s mission…that’s going to be a disappointment for you. 🙂
October 25th, 2015 at 12:26 pm
Well, you’re kind of fascinating on the topic of math. I’ve long believed any intelligent person can “do math” — I’ve long believed mathphobia is in someone’s head. But in all our conversations I don’t get that sense with you, so I’m forced to re-consider my long-held belief. Maybe our brain’s innate hardwiring is that powerful. For me, that’s a very interesting change in POV.
OTOH, if I can help you break through, then I was right all along, and I like it when that happens. XD
October 26th, 2015 at 9:12 am
My husband says he was never “good at math” (I think he probably was, he just made bad grades because he was that kid). I, however, always made good grades, usually the highest in the class in math. Yes, in math. BUT I never learned anything and it never came easy for me. I remember one guy in particular who went through the school system with me and ended up in a lot of my classes. He was a nice guy and always helped me when I asked questions. Really, he was the math tutor for the whole table. He was the kind of person who just “got it” instantly. I could just see it in him. It was like people who can paint or draw. Pure talent.
Totally irrelevant tangent…I had a little crush on him and couldn’t figure out why the ladies weren’t talking about him. He wasn’t geeky or super quiet or awkward in conversation. Just a very normal guy. Decades later, before I decided to get on Facebook, I was at a friend’s house and perusing my friend’s HS people. I saw his name and said, “Oh wait, I’ve got to check him out.” Turns out he’s an engineer. And still cute. 🙂
Now back on topic. My husband says if I had gone to St. John’s I would have learned that I can do math. They read all of Euclid, go through all the proofs and demonstrate each one. I read some Euclid…don’t think I would’ve made it through alive. But who knows. I was really motivated back in college. These days I’m really not terribly motivated, so that could pose a problem for you. 🙂
October 26th, 2015 at 11:36 am
And for that matter, I don’t know how much teaching motivation I have left in me these days (and my blog posts need what little reserve I have left in that area), so maybe I’ll just leave you to it. 🙂
As I think I’ve mentioned before, I think early presentation does does matter. Perhaps if you’d been exposed to math in a different way you’d feel differently about it. Who knows!
April 7th, 2022 at 7:07 am
[…] complex numbers are important in quantum math. [See Circular Math and its many links (especially Beautiful Math) for […]
August 12th, 2022 at 7:12 am
[…] been called a “sonnet” and it really is pretty awesome. [See Beautiful Math for details.] Part of what makes it so cool is the contrast between the transcendental numbers e […]
August 12th, 2022 at 8:44 am
Not only is trigonometry
sin
not wrong, it involves being (a) right (triangle)!