I was having a friendly debate this evening as to whether any finite-length string of numbers must occur at least once in any irrational number.
Example 1: π to any n digits will eventually show up in the square root of 2, and vice versa. Same goes for π and e, e and 2^1/2, and for that matter any pair of irrational numbers (transcendental or not).
Example 2:
The unicode for any written expression will appear at least once within any irrational number, including - if you wait long enough - the complete works of Shakespeare, the Bible, and even this short phrase closely related to my username:
“Nothing happens more than once, but all things must happen some day.”
My thinking on this has been that every string must happen at least once, and if once, how not an infinite number of times? Now, I’ll concede that’s intuition talking, and that means I’m the one with the burden of proof here.
So I did a bit of research. Didn’t have to get any further than Wikipedia to find this (to me) startling observation:
If that means there may be a point in π where, for example, you’ll never see a number 7 again, that’s quite a puncture in my point of view on irrational numbers. It also means I’ve got a number of sizeable holes in my understanding of number theory, but that ain’t news.
So… might there be a finite-length string of numbers that never occur in the decimal form of a given irrational number? And if all finite-length strings must occur once, can they appear a finite number of times in an irrational number?
Closing remark: to keep the discussion simple, let’s keep it in base 10 for now if nobody objects.
The irrational number 0.1010010001000010000010000001000000010000000100000000010000000000100000000000100000000000001… (where every one is separated by an increasing number of zeroes) doesn’t contain the digits 2 through 9 at all.
Rephrasing more restrictively:
Might there be a finite-length string of numbers that never occur in the decimal form of π, e, or the square root of 2? And if all finite-length strings must occur once, can they appear only a finite number of times in any of these numbers?
Baffle basically nailed it. Incidentally, the properties many people erroneously think hold of all irrationals (or to at least be known to hold of the special ones like e, pi, and sqrt(2)) are basically those of the normal numbers, which are those such that, no matter what base you express them in, the asymptotic frequency of any given string is the same as would be expected were the digits chosen randomly; i.e., if K is normal, then, for every base b string S of length n, the number of occurrences of S in the first p digits of K’s base b expansion approaches 1/b^n as p gets large.
It’s fairly easy to prove that “almost all” real numbers are normal (which basically means that, if you pick a random real from [0, 1), then the probability it will be normal is 1; it’s not guaranteed, not at all, but it’s very, very likely), and it’s not hard to give explicit definitions of numbers that are clearly provably normal, but pretty much no “nice” numbers like e, pi, or sqrt(2) are known to be normal, even though most mathematicians suspect they are.
And, yes, concerning our three favorite examples of irrational numbers (e, pi, sqrt(2)), for each, it might well be the case that there is some finite-length string of numbers that never occurs in them. After all, for none of them is it known that every digit occurs infinitely; and if some digit doesn’t occur infinitely, then, clearly, a long enough string composed solely of that digit doesn’t occur at all.
If all finite-length string must occur, then that won’t work. For every string there is an infinite number of longer strings that contain the first one as a substring and have to occur, too.
Make sure you’re clear on the various sizes of infinity before you go too far into this swamp. I don’t know enough to add to the current level of discussion, but to my (rusty) math-intuition it sure smells like the different concepts now in play rely on different levels of infinity.
The guy who used random-number generators (W.R. Bennett. See “How Artificial is Intelligence?” in American Scientist, 1977) once said that there wasn’t enough “randomness” in quasi-random number generators to produce enough noise to produce even a meaningful portion of a work of Shakespeare. Depending on how your random numbers are produced, then, it’s not a given that random noise will produce anything specific – even the very first line from Hamlet, let alone the entire works of Shakespeare.
Maybe I’m misunderstanding the question, but it seems very unlikely to me that the finite-length string “7” only appears a finite number of times in the infinite decimal expansion of e. Though I suspect real mathematicians will have something to say about how to define ‘finite’ in this case, and also the difference between ‘very unlikely’ and ‘provably not’.
Since when does “belief” count for anything in mathematics? Unless you are talking about Bayesians, knowing what other people believe but cannot prove in this field is practically pointless.
Yup… there I go - answers to all my questions. Thanks! Too bad they weren’t thinking of the three-letter lower limit on SDMB searches when pi was named back in the 18th century.
Curiously enough, where I was going with this would have been encoding large datasets by pointing to a start and end point within a normal number (in theory, since you’d need to have an infinitely large hard drive to store all of it). But the point that you’d need even more data to encode the addresses of the start and end digits is well made.
Fair enough on belief not lending the same level of credence proof, though mathematicians are capable of holding strong beliefs with substantial evidence (possibly of the inductive kind) which falls short of proof, same as anyone else (physicists, biologists, historians, average Joes). [And, as Squink mentions, there is substantial inductive evidence for believing in the normality of π]. The only difference is that mathematicians place value in the distinction and strive towards complete proof as the ultimate goal, but if you think there are any situations in which evidence can have epistemic value despite falling short of full proof, then you can understand how mathematicians can make and find value in educated conjectures.
I don’t know what your invocation of Goedel’s Incompleteness Theorem is supposed to demonstrate, and I must warn that almost all invocations of it in a context like this are extremely sloppy; it seems very likely that it doesn’t say what you think it does. If it’s merely meant as an example a particular theorem where mathematicians’ beliefs turned out to be wrong, it’s far from clear that people had an expressed strong belief in its negation before it was discovered (it was a shocking result, to be sure, but probably more along the lines of most people having not thought of the possibility than most people having thought of the possibility and concluded it unlikely on the grounds of some evidence or another). Better examples could be found of inductively grounded mathematical conjecture going wrong; for example, the Pólya conjecture. Nonetheless, as I said before, if you feel there is ever any epistemic value in evidence short of full proof, then presumably the same can hold in mathematics.
This does often come up as a flawed method of compression, but thankfully you’ve realized the problem. You wouldn’t need an infinitely large hard drive, though; just pick a computable normal number and be off on your merry way, computing as many digits as you need whenever you need them. (You wouldn’t even need a normal number, per se, just one in which every string occurred; I imagine the expected starting points will be lower with a random normal number than with a random “Every string occurs” number, but since the method is flawed in this regard anyway…)
I had actually conjectured to myself that you were thinking about the matter because of the whole HD-DVD key kerfuffle, the sort of thing that makes people want to say “Hey, but if this string is merely the <something>th through <something>th digits of pi, then we can’t be stifled from sharing the mathematical information.”, or things like that.
OK, if you don’t like “belief”, then let’s put it this way.
Absolutely normal numbers exist.
The quantity of absolutely normal numbers is infinitely larger than the quantity of non-(absolutely normal) numbers.
Therefore, any number selected based on criteria not relevant to the normality of the number is overwhelmingly likely to be absolutely normal.
pi is not known to not be absolutely normal, and has no known properties which relate to lack of absolute normality.
The conclusion is left as an excercise for the reader.
