Zeke (commodore_zeke) wrote,

Compound interest

There are lots of interesting math pages around. Here's an oldie-but-goodie: What's Special About This Number?, a page that's been online about as long as my own. The maintainer, Erich Friedman, is out to tell us something mathematically interesting about as many of the counting numbers as possible. This is a task which in one sense can never end... and in another sense already has.

Find that hard to believe? Suppose we stop right now with the current page. It's finite, so there are omissions from it; let's list them in order. (As I write this, the list starts out 391, 424, 460, 488, 508...) By using this list, we can name the first uninteresting number, the second, the third, and so on.

But wait! First, second, third... those numbers are interesting. Everything less than 391 is. So in fact we can fill in the first 390 of those omissions with "the #th uninteresting number". Now we've found something interesting to say about them!

Okay, but what about the rest of our list? We've only removed 390 out of an infinitude of entries. Let's call the surviving numbers not just uninteresting, but "level-2" uninteresting. Surely they're beyond saving.

But wait! There's a first level-2 uninteresting number, a second, and so on. Up until the 391st, those are interesting things to be! Now we've got facts to cite for another 390 numbers. Go us! What's more, we can continue this process with levels 3, 4, and so on of uninterestingness...

Until we get to level 391, that is. 391 isn't interesting, so neither is that level! There are still infinitely many numbers on our list, but now they're beyond the reach of our level system. The surviving numbers are a whole different kind of uninteresting. They're so uninteresting we don't have the words yet to talk about how uninteresting they are. They're some kind of super-uninteresting.

Say, wouldn't it be interesting to know what the first super-uninteresting number is?

You get the picture. I leave it as an exercise for the reader to show that this process can be carried on forever, assuming we're allowed to come up with a new word for each level of uninterestingness. (This is not the same as just making up words willy-nilly; we're not describing, say, 4637 as "the first vreemish number", where "vreemish" is defined to mean "4637 or higher". Our words come from the process above, so they aren't arbitrary.)

"Hold it!" you may say. "There was nothing special about the number 391 in that proof. It would have worked no matter what the smallest unlisted number was!"

I know, right? Friedman must be some kind of workaholic. He could've just listed a fact for 1 and called it a day.

- Z

(Note: The above is a rephrasing/overcomplication of the old "no boring numbers" joke. I think I first read it in Martin Gardner. Some of you probably recognized it early on, so thanks for not interrupting.)

Tags: math
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