{"id":2980,"date":"2014-05-26T20:56:02","date_gmt":"2014-05-27T00:56:02","guid":{"rendered":"http:\/\/www.goodmath.org\/blog\/?p=2980"},"modified":"2014-05-27T10:53:11","modified_gmt":"2014-05-27T14:53:11","slug":"you-cant-even-describe-most-numbers","status":"publish","type":"post","link":"http:\/\/www.goodmath.org\/blog\/2014\/05\/26\/you-cant-even-describe-most-numbers\/","title":{"rendered":"You can’t even describe most numbers!"},"content":{"rendered":"

(This is a revised version of an old post; you can see the original here<\/a>.)<\/em><\/p>\n

Please read the addendum at the bottom – I got carried away with computability, and ended up screwing up quite badly.<\/b><\/p>\n

In the comments on my last post, a bit of discussion came up about some of the strange properties of real numbers, and that made me want to go back and get this old post about numbers that can’t even be described<\/em>, and update it.<\/p>\n

In those comments, we were talking about whether or not π contains all sorts of interesting subsequences. The fact is, when it comes to π, we just don’t know.<\/p>\n

But… We do know that there are many numbers that do<\/em>. There’s a property called normality<\/em>. Informally, normality just says that taken to infinity, all sequences of digits are equally likely to occur. In an infinitely long normal number, that means that any finite length subsequence will<\/em> occur, given enough time. My name, your name, the complete video of the original version of Star Wars, a photograph of the dinner I just ate and never took a photo of – it’s all there, in any normal number.<\/p>\n

If you think about that, at first, it might seem profound: there are numbers that contain absolutely everything<\/em> that ever did exist, and that ever could<\/em> exist. That seems like it should be amazing. If the numbers were at all special, it might be. But they aren’t. Normality isn’t a rare property that’s only possessed by a special meaningful number like π. It’s a property that’s possessed by almost every<\/em> number. If you could create a randomly selected list of a billion real numbers (you can’t in reality, but you can mathematically, using a finite version of the axiom of choice), odds are, all<\/em> of them would be normal – all of them would have this property.<\/p>\n

But here’s where it gets really strange. Those numbers, those normal numbers that contain everything? Most of them can’t even be described<\/em>.<\/p>\n

It gets stranger than that. We know that we can’t really write down an irrational number. We can write down successively more and more precise approximations, but any finite representation won’t work. So we can’t actually write down the overwhelming majority of real numbers. But in fact, not only can’t we write them down, we can’t describe<\/em> the vast majority of numbers.<\/p>\n

Numbers are something that we deal with every day, and we think we understand them. Over the ages, people have invented a variety of notations that allow us to write those numbers down: the familiar arabic notation, roman numerals, fractions, decimals, continued fractions, algebraic series, etc. I could easily spend months on this blog just writing about different notations that we use to write numbers, and the benefits and weaknesses of each notation.<\/p>\n

But the fact is, when it comes to the vast, overwhelming majority of numbers, all of those notations are utterly useless! That statement seems bizarre at best. As strange as it it seems, though it’s true. In order to understand it, though, we have to start at the very beginning: What does it mean for a number to be describable<\/em>?<\/p>\n

<\/p>\n

The basics are really easy to explain: A describable<\/em> number is a number for which there is some finite representation. An indescribable number is a number for which there is no<\/em> finite notation. <\/p>\n