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Cotes left to the world a handful of unpublished math papers.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.)

A complex story, part 1

For RT, who seemed a little skeptical.

Once upon a time, there was an Italian mathematician named Niccolò Tartaglia.

Niccolò's claim to fame—well, his secondary claim to fame—is that he discovered a formula that you could use to solve any cubic equation. Well, maybe not any cubic equation. It worked for many cubic equations. Unfortunately, for some equations the formula gave results that involved the square root of a negative number. Which was clearly nonsense. But strangely, Tartaglia found that if he just pretended that negative numbers had square roots (numbers that weren't exactly real numbers but followed all the same rules of arithmetic), he could just plow ahead with the math and eventually all the oddities would cancel out, leaving ordinary real numbers: the correct solutions.

(Tartaglia's primary claim to fame is that he spent a decade of his life ruthlessly destroying the career and reputation of his friend, Gerolamo Cardano, after Cardano revealed Tartaglia's secret formula to the world. This was a time when mathematics was the exclusive domain of paranoid madmen. Some were so secretive they managed to leave no surviving written work at all.)

From what I've read, Tartaglia apparently had no idea what he was doing, and nobody else could figure it out, either. It was as though there were a sort of mysterious shadow realm lurking behind the real numbers, and occasionally some errand would force you to travel through it, only to emerge (with a shudder of relief) back into the real numbers in the end. Nobody liked this. When René Descartes called these oddities the imaginary numbers, he meant it to sting.

The stigma persists. Most people hear a little about complex numbers in school, not enough to be comfortable with them or understand why people would think they exist (whatever that means) or why they might be useful.

Descartes probably figured a better method for solving cubic equations would eventually come along, and then the “imaginary” numbers could be quietly forgotten. What actually did happen turned out to be a lot more interesting.

(Continued in part 2.)

23 December 2006

Alice in Puzzle-Land

“How do I know for sure that I'm awake?” asked Alice. “Why can't it be that I'm now asleep and dreaming all this?”

“Ah, that's an interesting question and one quite difficult to answer!” replied the King. “I once had a long philosophical discussion with Humpty Dumpty about this. Do you know him?”

“Oh, yes!” replied Alice.

“Well, Humpty Dumpty is one of the keenest arguers I know—he can convince just about anyone of just about anything when he puts his mind to it! Anyway, he almost had me convinced that I had no valid reason to be sure that I was awake, but I outsmarted him! It took me about three hours, but I finally convinced him that I must be awake, and so he conceded that I had won the argument. And then—”

The King did not finish his sentence but stood lost in thought.

“And then what?” asked Alice.

“And then I woke up!” said the King, a bit sheepishly.

—Raymond Smullyan, Alice in Puzzle-Land. This is a fun book of logic puzzles ranging from cute to outrageously intricate. A fine gift for the mathematician on your list (though I hear The Annotated Alice is even better).

09 December 2006

π in Metamath

Metamath has a definition of π, but it's unused. For all the importance of π in math, the Metamath database contains exactly zero theorems that make use of it.

I started to work on a little something and quickly discovered the reason for this. Metamath defines π as the smallest positive zero of the sine function, but as yet it has no proof that any such number exists!

In a way this reflects well on Metamath. I wanted to prove something, but my reasoning depended on hidden assumptions. Working in Metamath very quickly revealed those assumptions and guided me to an understanding of exactly what they were. You learn by doing.