24 December 2006

A complex story, part 2

(See part 1.)

Roger Cotes died of a sudden, violent fever in the summer of 1716. He was 33 years old. Isaac Newton is said to have remarked on Cotes's passing, “If he had lived we would have known something.”

Cotes left to the world a handful of unpublished math papers. Among them, this formula:

ln (cos x + i sin x) = ix

No such connection between trigonometry and logarithms was known at the time. Cotes's formula reveals a simple, fundamental connection here, and the connection runs through complex numbers. Even at the time this was a very nice result, but the realization of just how profound it was dawned slowly. It took a couple hundred years.

Around this time a totally unrelated question was starting to generate interest. There was a feeling that an equation like this...

x3 + x2 - 3x = 4

...ought to have three solutions. And indeed it does, but the feeling was that more generally every polynomial equation of degree n ought to have n solutions, though not all of them would necessarily be distinct from one another. And this turned out to be true... if you counted complex solutions. This landmark result was finally proved in 1806. It's now called the fundamental theorem of algebra.

But complex numbers were still controversial until 1799, when something ironic happened. A land surveyor, Caspar Wessel, discovered that the complex numbers could be thought of as points (or vectors) on a plane. Each complex number a + bi corresponded to the point (a, b). Adding, subtracting, and multiplying complex numbers could be done with any flat surface, a compass, and a ruler.

This was astonishing, because it made the past two centuries of work involving complex numbers suddenly much easier to visualize, in a totally unexpected way. (Graph the solutions to the equation x13 = 1. You'll see thirteen points arranged in a perfect circle—or regular triskadecagon, if you prefer—around 0. I don't know how to convey how unexpected and beautiful that is.) Ironic, too, because the underlying idea of plane coordinates was first explored by René Descartes, the same man who coined the derogatory term “imaginary number” back in 1637.

Cotes's formula had been independently discovered by Leonhard Euler, and with the discovery of the complex plane, its fame grew. The formula now bears Euler's name; and one particularly nice case (where x) is called Euler's identity:

eiπ + 1 = 0

This is widely considered the most beautiful equation in mathematics. There's certainly something about it. It's as though all the biggest concepts in math came to lunch, and while they were all there together they posed for a photograph.

...So this is why complex numbers are important. It's a little matter of the most beautiful mathematical discovery of all time.

But incredibly enough, the story doesn't stop there either.

(Continued in part 3.)


JJD said...

A small correction: Euler's nice formula is the case where x = pi, not x = 0.

Thanks for the interesting blog about math!

jto said...

Oooops! Thanks. I fixed it.

Anonymous said...

who's using my formula??????



hyunjungsoh said...

Interesting..! :D
You know, if they taught math like it was another story book in school, I would have been really interested. Sadly, it wasn't the case. :(