Has this theorem been proven yet? It seems to me like a guess in the dark, given how much we know about the digit sequences in irrational numbers.

I concede mathsy is correct, however much I'd like to hate it!
However minimal the possibility, it isn't NIL by any yardstick!
Hence, we'd have to assume it's somehwere possible down the track,
disproving it is almost next to impossible!

Corollary 3:-  No irrational number ever contains a string of 1 million continuos any of the numbers from 0 to 1,000,000 after the decimal.

ganesh, I think you have a misunderstanding what exactly a real number is.  Every sequence of digits corresponds to a unique real number (9 repeating is typically barred by definition so that this holds).   Thus, every finite sequence of digits occurs in some irrational number.  To see this, we construct a number with the sequence in question, and tack on the digits of pi.  This must be an irrational number, otherwise pi*(length of that sequence) would be rational.

Ricky,
Inspite of all what you have stated, I find the concept of finding a million zeros on the trot quite difficult, in the places of decimal of pi after the dot. I shall put it this way. Since because an indieent can occur does not make it occur. Just because the possibility cannot be eleminated, it never means it should happen at some point of time. This brings us back to square one. Infinity cannot be intrpreted to one's convenience. It is, exactly as it is. We have no business to tamper with it. My question was obviously too good one for a computer professional. I am a mathematician. I don't care the way the computer scientists work. We mathematicians show the way to the rest of the world, including the Computer sicientists. You know, who teahces whom. The symbols are the concept of the mathematicians. We mathematicians express things in the shortest possible way. The rest of the world listens to us, learns from us. How on earth do you think, as a mathematician I can accept the fact that there are a million consecutive zeros in the value of pi aftert the deciaml simply because the list is endless? ISN'T THAT ABSURDITY?

ganesh wrote:

Proving this is the most difficult, most frustrating to any mathematician.

Theorem :- The number pi does not ever contain a string of 1 million continuos zeros after the decimal.

Corollary 1 :- No irrational number ever contains a string of 1 million continuos zeros after the decimal.

Corollary 2 :- No irrational number ever contains a string of 1 million continuos any of the numbers from 0 to 9 after the decimal.

Attempt proving or disproving this, by counter-example or otherwise!

Now, I'm quite the amateur, so correct me when I veer off the mathematician's path here, as I most likely will do. If pi is non-repeating, non-terminating, then as many digits of pi that you can calculate is only equal to ∞-n/∞ of its digits, right? So, if we looked at pi and said, "Well, it cannot contain one million zeros. And look, we've calculated it one billion places, and no zero or string of numbers 0 through 9." Then aren't we haunted by the fact that this is only ∞-n/∞ of its digits, and the fact that there are infinite numbers left. So, even if we calculated it to some astronomically high digit place and found nothing, we'd have to calculate even more, and more, and more, forever. Well, interestingly enough, I would think that the probability of 14159265358979 is equal to 111..., or 222.... However, if we accept the fact that it is infinite, then however counter-intuitive it seems, doesn't the same probability go for a million zeroes? This is only my unprofessional opinion. But it would be interesting to discuss.

However, my friend, another armchair mathematician, brought up an interesting point. He said that if pi is a division problem, such as circumference divided by diameter, then in order for it to contain one million continuous zeros, pi would have to be terminating, or at least repeating, in that to contain one million zeros, circumference/diameter = 3.14 ...000..., which means it couldn't possibly jump back to 1, right?

Either way, I think it's another interesting point.

Last edited by iheartmaths (2008-02-05 05:43:40)

Inspite of all what you have stated, I find the concept of finding a million zeros on the trot quite difficult, in the places of decimal of pi after the dot. I shall put it this way. Since because an indieent can occur does not make it occur. Just because the possibility cannot be eleminated, it never means it should happen at some point of time. This brings us back to square one. Infinity cannot be intrpreted to one's convenience. It is, exactly as it is. We have no business to tamper with it. My question was obviously too good one for a computer professional. I am a mathematician. I don't care the way the computer scientists work. We mathematicians show the way to the rest of the world, including the Computer sicientists. You know, who teahces whom. The symbols are the concept of the mathematicians. We mathematicians express things in the shortest possible way. The rest of the world listens to us, learns from us. How on earth do you think, as a mathematician I can accept the fact that there are a million consecutive zeros in the value of pi aftert the deciaml simply because the list is endless? ISN'T THAT ABSURDITY?

I would never ask you to.  I have no idea whether or not there are a million zeros after the decimal point consecutively.  My intuition says no, but I don't wish to research such a question (mostly out of lack of significance and interest).

but at the same time, pi is irrational, it has no pattern to it, so for all intensive purposes, it is random.

Just because you don't know what digit is next does not imply it is random.  Pi is by all means not random.

Well, interestingly enough, I would think that the probability of 14159265358979 is equal to 111..., or 222.... However, if we accept the fact that it is infinite, then however counter-intuitive it seems, doesn't the same probability go for a million zeroes? This is only my unprofessional opinion. But it would be interesting to discuss.

Probability only makes sense when there is a concept of chance.  Pi's digits are determined whether you know them or not.

but it looks like the pattern is roughly linear.

You mean logarithmic.  Any argument which involves the phrase "looks like" is simply a conjecture.

Digits of Pi

Specifically, look for BBP formula.

Besides, the point is that pi's digits have the properties of randomness: unpredictability and uniform distribution. Therefore even if they actually aren't random, it's not unreasonable to analyse them as if they are.

This is also not known to be true.  It is still an open question whether or not pi is normal.

That BBP formula finds the nth digit of π in base 16, which makes it largely irrelevant to this discussion since there's no easy way of converting infinite (or very large) strings of digits between bases. Not that I know or can think of, anyway.

However, they do determine the digits of base 10.  And if the base 16 digits are not random and determine the base 10 digits, then...

So I suppose I'm conjecturing again. I'd say it's almost certainly true though, and the only reason it's not definite is that it's (virtually) impossible to prove.

(Incidentally, I like how those two people get credited for saying they don't know something.)

There are so many weird things in mathematics, I don't know how you could say that.  I do agree that if I was a betting man, all my money would go in on pi's normality.  However, mathematics isn't about betting.

Typically, people get credit for saying they don't know something only once they have shown that there really is no easy way of approaching it.

Great discussion you guys are having guys. Although you lost me at some times I really enjoy all this mathematical talk.

Im just a noob math major, hopefully I can become knowledgeable as all you guys.

Guys, I m sorry to interrupt your beautiful discussion which i just bumped in to while surfing. I m a newbie to this site and dont know the rules n regulations but just got registered to post to this discussion. Please forgive me as i m not a mathematician either.

If this discussion is to take a larger form about Irrational numbers then i think you should check something which is called as the Liouville's number. It is irrational as far as i know and it does contain a million zeroes back to back. it is defined as the summation of 10^(-n!) for n = 1 to infinity.

Simply put 0.1100010000000000000000010000000000000.................

Giving 1's only for the digits which are factorial numbers 1 2 6 24 120 and so on.
so there is bound to be a value of some factorial greater than 1.
It is irrational because it neither terminates nor repeats.
Dont have any idea about the pi part though. And some guy called Mahler proved that pi is not a Liouville number. I could not understand this part though.

please rectify me if i am wrong. Thanks

I think pi/1 million will satisfy ganish's proposal.