24 December 2006

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.)

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