13 February 2008

Is Liouville's number interesting

Mark Dominus writes about uninteresting numbers. In particular he claims that Liouville's number,
\sum_{i=1}^\infty 10^{-i!} = .110001000000000000000001\ldots
is uninteresting; the only thing that's interesting about it is its transcendentality, and that's not a big deal, because almost all real numbers are transcendental. (In fact, "almost all" seems too weak here, to me, although it is technically correct in the measure-theoretic sense.)

But I think this number is interesting. Why? Because an expression that gives it can be written with a very small number of characters (bytes of TeX code, strokes of a pen, etc.) and most numbers can't be.

Of course, this means every number anyone's ever written down is interesting, by this definition -- even if it took them, say, ten thousand characters to define it. Say we work over a 100-symbol alphabet; then there are at most 10010000 or so numbers which can be defined in less than that many characters! (There are multiple definitions for the same numbers; most things one could write are just monkeys at a typewriter, and so on -- but this is an upper bound.) But this number is finite! The complement of a finite set is still "almost all" of the real numbers.

That last paragraph, I don't actually believe. But any number that can quickly be written down is "interesting" in some (weak) sense.

(By the way, "Liouville" is not pronounced "Loo-ee-vill". And I'm told that the city of Louisville in Kentucky is not pronounced this way either.)

edit, 10:22 pm: Take a look at Cam McLeman's The Ten Coolest Numbers. (The list, in reverse order, is: the golden ratio φ = 1.618..., 691, 78557, π2/6, Feigenbaum's constant δ = 4.669201..., 2, 808017424794512875886459904961710757005754368000000000, the Euler-Mascheroni constant γ = 0.577215..., the Khinchin constant K = 2.685252..., and 163.)

11 comments:

Anonymous said...

lee-yoo-vill

Anonymous said...

Have you heard of Kolmogorov complexity? (sounds like you have!)

Jesse Johnson said...

Have you seen the proof that all positive integers are interesting? It's a proof by contradiction using the well ordering principle: If there were an uninteresting integer then there would be a smallest uninteresting integer. But being the smallest uninteresting integer is kind of interesting. Contradiction.

jweed said...

We can make your last "result" even stronger. Since we can only describe countably many real numbers with any set of symbols, there are actually uncountably many real numbers which we can't even describe. So any number that can be written down or described at all (even if it takes pages) is interesting in some (still weaker) sense.

Isabel said...

anonymous,

you're right -- what I'm basically saying is that the property of having low Kolmogorov complexity is interesting.

Philip said...

A question about the integers - what's the smallest positive integer that gets no hits on Google?

Isabel said...

Philip,

if I knew the answer to that question, I wouldn't tell you here. If I told you here, Google would index it, and it would cease to be the answer.

Philip said...

Technically, I think we could spell it out to avoid the attention of Google :)

Daniel said...

We're sort of asking about sup(S), for S = [x : x a positive integer for which Google will return a result], with sup(S) not a member of S.

Finding an upper bound put me essentially at the limit of how man characters Google allows. I was able to find search results for 10^127 expanded, but 10^128 choked.

Not sure if I have the patience to try 10^127 + i, i in Z, until we find the i that breaks the results.

Certainly, starting with a pattern of "1" with a lot of zeroes beyond it is less likely to result in no hits. I wonder if there is some density function to describe that?

Ben Allen said...

I'd conjecture that randomly generated integers are the least interesting, and thus have the fewest Google hits. If you randomly generate some 10-digit (or so) numbers, you'll probably find one that has no hits.

Isabel said...

Ben,

0275649185 reversed -- which I picked randomly -- has no hits. (I'm wording ambiguously so as to keep it that way.)

Incidentally, there are nine distinct digits there. I wonder, if you ask people to select a 10-digit number at random, how many distinct digits will it have? I suspect "random" 10-digit numbers have all ten digits distinct much more than one would expect if people selected them uniformly at random.