equation tooth

[repost from 2008]

I got a new tooth this summer. It’s porcelain.

I had been contemplating a tattoo of this equation, but when I found out I needed a crown, I thought it would be better in my mouth. I was inspired by a friend who has a nice “Led Zep” on one of her molars.

The equation is more typically written:

What does it mean?

It’s an identity with confounding implications. I was completely blown away when my calculus teacher wrote it on the board so long ago. In this simple statement, huge branches of mathematics came crashing together—natural logarithms, trigonometry, geometry, complex numbers, infinite series, and the concepts of one and zero. How the hell could an incredibly weird transcendental number be the same thing as negative one? No previous teacher had even hinted that the varied subjects in math were part of a larger whole. I was never the same after that—math became more than just random facts and puzzles. There was something very deep going on, and I ended up studying math seriously and going to grad school because of it. (Well that, and Gödel’s Incompleteness Theorem.)

What are the elements of the expression?

“e” is Euler’s number, roughly equivalent to 2.71828183… It is an irrational and transcendental number, never repeating. It is fundamental to natural logarithms, and is related to growth of populations and radioactive decay, among other natural processes.

“i” is the imaginary unit number, otherwise known as the square root of negative one. It is the basis of complex analysis, fundamental to physics, number theory, fluid dynamics, and fractals.

“pi” is the ratio of a circle’s circumference to its diameter, roughly equivalent to 3.14159265… It is also an irrational and transcendental number, fundamental to geometry, music & sound, trigonometry, and waves of all kinds.

There are several ways to prove that e^(i*pi) = -1, but none of them are really satisfying to me. One involves infinite series, and another involves graphing on the complex plane. I can understand and accept the proofs, but the proofs don’t reveal everything that is going on, at least for me. (Maybe that’s part of why I didn’t finish my PhD…)

Regardless, Euler’s equation is a beautiful expression that binds together fundamental aspects of our world, and it will always be close to my heart, er gums.