The Power Of Near Enough

Now here’s another great tool Chiaroscuro did, in figuring out what number raised to the fifth power would be 1/6000. Besides trying out a variety of numbers which were judged to be a little bit low or a little bit high, he eventually stopped.

Wisely, too. The number he really wanted was the fifth root of 1/6000, and while there is one, it’s not a rational number. It goes on forever without repeating and without falling into any obvious patterns. But neither he nor anyone else is really interested in any but the first couple of these digits. We’d wanted to know whether this number was close to 0.25, and it’s closer to 0.17 instead. What the tenth digit past the decimal was we don’t really care about. It’s fine to be close enough to the right answer.

This runs a little against the stereotype of the mathematician. To the extent that popular culture notices mathematicians at all, it’s as people who have a lot of digits past a decimal point. But a mathematician is, in practice, much more likely to be interested in saying something that’s true, even if it isn’t so very precise, and to say that the fifth root of 1/6000 is somewhere near 0.17, or better, is between 0.17 and 0.18, is certainly true. Probably — and I’m attempting here to read Chiaroscuro’s mind, as the only guidance I’ve gotten from him is the occasional confirmation about what my guesses to his calculation were — he found that 0.17 was a little low, and 0.18 was a little high, and the actual value had to be somewhere between the two. The Intermediate Value Theorem, discussed in the previous non-Gemini-Chronology entry, guarantees that between those two is an exactly correct answer. (It’s conceivable that there would be more than one, in fact, although for this problem there’s not.)

Chiaroscuro specifically judged the fifth root of 1/6000 to be 0.176, or 17.6%, and I doubt anyone would seriously argue with that claim. This is even though the actual number is a little bit less than that: it’s nearer 0.175537, but even that is only an approximation. We are putting one of those big ideas into play, subtly, when we accept saying one number is equal to another in this way.