Euler’s Formula

Leonhard Euler

Leonhard Euler

Besides being an obvious lady killer, Swiss mathematician Leonhard Euler gifted the world with some pretty important mathematical concepts, notational conventions, and formulas. I almost feel bad about the fact that I couldn’t even spell his name correctly until I was well into adulthood.

You are probably thinking, “Sure, he had a bitchin’ robe, and for an old-timey dude he was pretty good looking, but was he really all that?”

I can’t find any quotes from the ladies, but his peers, both contemporary and modern have had a few things to say about him and his work.

Pierre-Simon Laplace, hardly a schlub himself in the field of mathematics said,

Read Euler, read Euler, he is the master of us all.

François Arago, chimed in,

He (Euler) calculated without any apparent effort, just as men breathe, as eagles sustain themselves in the air.

More recently,  Richard Feynman said that Euler’s Formula was

the most remarkable formula in mathematics.

Wikipedia summarizes the man and his achievements best,

Euler’s work touched upon so many fields that he is often the earliest written reference on a given matter. It has been said that, in an effort to avoid naming everything after Euler, discoveries and theorems are named after the first person after Euler to have discovered it. [link]

So, yeah, the ideas of this dude shouldn’t never be taken lightly or for granted. And without further ado, let’s get on with the show.

Euler’s Formula

    \[e^{ix} = \cos x + i\sin x\]

That’s it. Not much to look at, is it? It’s brevity makes it pretty simple to memorize, but the implications wrapped up in this deceptively short equation are as mind-bending as they are powerful. I would like to illustrate how Euler synthesized this equation, but we have to take a detour through complex numbers first.

Complex Numbers and their Geometry

I am not going to go into great detail about complex numbers. All of what you’ll need to know as it relates to Euler’s Formula you likely learned in grade school.  My intent is to hit the salient and relevant properties. If, however, it is news to you that i = \sqrt{-1}, then you should probably read over the Complex Number Wikipedia page before continuing here.

Complex numbers arose from the fact that there is no solution for x in the equation x^2 = -1 in \mathbb{R}, the set of real numbers. Early mathematicians being the devil-may-care mavericks that they were, were all like, “Screw it. Let’s just invent a new number. We’ll just call this number i and say that the solution is x = i.” Or, in other words, this new number i they imagined up (see what I did there) is equal to \sqrt{-1}.

I don’t know this part for a fact, but I am guessing the guardians of \mathbb{R} laughed these heretics out of the room when they vouched for i to join the club. You can’t blame them. \mathbb{R} does stand for “real” after all.  Being the rebels they’d already proven themselves to be, the cult of i started their own club, or field as the math geeks like to call them.  To show those \mathbb{R} chumps who was boss they decided to include the real numbers as well as imaginary numbers in this new field, which they dubbed the field of complex numbers, denoted \mathbb{C}.

Now, that i was defined, mathematicians could solve other troublesome problems:

x^2 = -16

x = \sqrt{-1 \cdot 16} = 4\sqrt{-1} = 4i

It turns out that any number containing i, like 4i, 10i, etc, is strictly speaking an imaginary number. But as I said,  \mathbb{C} subsumes \mathbb{R}, hence the “Complex” part of the name. What gives? Where are the real numbers?


(x - 1)^2 = -16

The solution is found like so:

x - 1 = \sqrt{-1 \cdot 16} = 4\sqrt{-1} = 4i

x = 1 + 4i

So the solution to the problem (x - 1)^2 = -16 is x = 1 + 4i. Notably,  x = 1 + 4i has both a real part, 1, and imaginary part, -4i. So there you go. A complex number  is a number with both a real part and an imaginary part.

complex number = real part + imaginary part

As it turns out the real part can be zero, which explains why all of the imaginary numbers are also complex numbers, or the imaginary part can be zero, meaning all of the real numbers are also complex numbers.

Leave a Reply

This site uses Akismet to reduce spam. Learn how your comment data is processed.