A couple of pi questions.

1. Has any study been made on the decimal representation of pi to see if any number occurs more often than any number?

2. If the decimal representation of pi is not random – can it be calculated how long it takes to find a string of n numbers long within that representation?

Dropbear said:

A couple of pi questions.

1. Has any study been made on the decimal representation of pi to see if any number occurs more often than any number?

2. If the decimal representation of pi is not random – can it be calculated how long it takes to find a string of n numbers long within that representation?

I expect that the answer to #1 is yes, and that no deviation from the mean distribution has been found.

#2, have you been reading a novel.? remember it was one of the major themes in a SF novel I read many years ago.

In Carl Sagan’s book Contact? So good!

http://en.wikipedia.org/wiki/Contact_(novel)
Ellie, acting upon a suggestion by the senders of the message, works on a program which computes the digits of pi to record lengths in different bases. Very far from the decimal point (1020) and in base 11, it finds that a special pattern does exist when the numbers stop varying randomly and start producing 1s and 0s in a very long string. The string’s length is the product of 11 prime numbers. The 1s and 0s when organized as a square of specific dimensions form a rasterized circle. The extraterrestrials suggest that this is a signature incorporated into the Universe itself. Yet the extraterrestrials are just as ignorant to its meaning as Ellie, as it could be still some sort of a statistical anomaly. They also make reference to older artifacts built from space time itself (namely the wormhole transit system) abandoned by a prior civilization. A line in the book suggests that the image is a foretaste of deeper marvels hidden even further within pi. This new pursuit becomes analogous to SETI; it is another search for meaningful signals in apparent noise.

That’s the one, pods, well done.

Dropbear said:

A couple of pi questions.

1. Has any study been made on the decimal representation of pi to see if any number occurs more often than any number?

Certainly. People have been looking for such patterns throughout history. Of course, they didn’t have much material to work with before the computer era. I’m not sure what the current record is, but according to Wikipedia 10 trillion (10<SUP>13</SUP>) digits of π were calculated in 2011 by Alexander Yee and Shigeru Kondo. Some cute patterns have been found, but nothing that significantly deviates from statistical expectations.

It’s long been conjectured that π is a normal number , i.e., all possible digit sub-sequences occur with the frequency one would expect from a purely random sequence. But nobody knows how to prove this conjecture. FWIW, if a number is normal in any base it’s normal in all (finite integer) bases. Examining finite sequences of digits may shed some light on this matter, and I guess that if a significant deviation from normalcy were discovered, that might lead to a proof that π isn’t normal; OTOH, even 10 trillion digits is a drop in the bucket compared to infinity. :)

A famous sequence that occurs relatively early in the digits of π is the Feynmann point,
Wiki said:

a sequence of six 9s that begins at the 762nd decimal place of the decimal representation of π. It is named after physicist Richard Feynman, who once stated during a lecture he would like to memorize the digits of π until that point, so he could recite them and quip “nine nine nine nine nine nine and so on”, suggesting, in a tongue-in-cheek manner, that π is rational.

See http://en.wikipedia.org/wiki/Feynman_point

Dropbear said:

2. If the decimal representation of pi is not random – can it be calculated how long it takes to find a string of n numbers long within that representation?

Well, in the absence of evidence to the contrary, we can pretend that it’s normal and use standard statistical techniques to estimate the probability of finding a substring of length n within the first m digits.

If you want to find digit sequences in π, check out the Pi Searcher , by David G. Andersen.

I ought to mention that it’s possible to calculate hexadecimal digits of π without having to calculate all the preceding hex digits, using spigot algorithms . It’s easy to use the algorithms to get individual bits of π, but unfortunately nobody’s come up with a decimal version yet. Still, they’re useful for checking the results of other algorithms, since it’s relatively easy to convert a long decimal number to hex.

>>1. Has any study been made on the decimal representation of pi to see if any number occurs more often than any number?

Pi is an infinite decimal, question is meaningless.

Peak Warming Man said:

>>1. Has any study been made on the decimal representation of pi to see if any number occurs more often than any number?

Pi is an infinite decimal, question is meaningless.

The question is not meaningless, and is an important consideration in generating pseudorandom numbers based on the digits of <font="Symbol">p. The irrationality or even the transcendentality of <font="Symbol">p does not imply that there is no regular pattern in the sequence of digits. For example, the number:

<font="Symbol">S_{k = 1 to <font="Symbol">¥ 10^k! = 0.1100010000000000000000010000…}

known as the Liouville constant, is transcendental even though the pattern of digits follows a definite pattern.

Try again:
Peak Warming Man said:

>>1. Has any study been made on the decimal representation of pi to see if any number occurs more often than any number?

Pi is an infinite decimal, question is meaningless.

The question is not meaningless, and is an important consideration in generating pseudorandom numbers based on the digits of p. The irrationality or even the transcendentality of p does not imply that there is no regular pattern in the sequence of digits. For example, the number:

S_{k = 1 to ¥ 10^k! = 0.1100010000000000000000010000…}

known as the Liouville constant, is transcendental even though the pattern of digits follows a definite pattern.

Once more:

Peak Warming Man said:

>>1. Has any study been made on the decimal representation of pi to see if any number occurs more often than any number?

Pi is an infinite decimal, question is meaningless.

The question is not meaningless, and is an important consideration in generating pseudorandom numbers based on the digits of p. The irrationality or even the transcendentality of p does not imply that there is no regular pattern in the sequence of digits. For example, the number:

S_{k = 1 to ¥} 10^k! = 0.1100010000000000000000010000…

known as the Liouville constant, is transcendental even though the pattern of digits follows a definite pattern.

Hang on Poindexter, we’re not talking about the Louisville Constant we’re talking about pi.
Pi is all over the place and no matter how far you go looking for patterns you will always be looking at an infintisimal part of pi that will never be big enough to extrapolate anything of meaningful value.

Peak Warming Man said:

Hang on Poindexter, we’re not talking about the Louisville Constant we’re talking about pi.
Pi is all over the place and no matter how far you go looking for patterns you will always be looking at an infintisimal part of pi that will never be big enough to extrapolate anything of meaningful value.

Sure, we can’t prove that pi is normal by looking at a finite number of decimal places, but as I said earlier, it might give us some clues in constructing a proof that pi is (or isn’t) normal.

A proper proof that a given real number is normal uses the properties of that number to show that the full infinite sequence of its digits has the proper statistics, generally by showing that all sufficiently large subsequences have the proper statistics.

It can be shown that almost all real numbers are normal. However, the only numbers that have been proven normal are ones with very special properties that make it relatively easy to discern the properties of the sequences of their digits in one or more bases, as mentioned at the Wikipedia link I gave earlier. It’s not even known if the most basic irrational number, the square root of 2, is normal or not.

It may be impossible to prove whether or not pi is normal. But even if that is the case we may be able to prove that a proof of pi’s normalcy is impossible. FWIW, I suspect that a general technique for proving normalcy of an arbitrary real is impossible, but I’m hopefully that proofs for simple classes of irrationals may exist. Eg, the quadratic irrationals (numbers of the form (a + b*sqrt( c ))/d for integers a,b,c,d) may be tractable, since they all have regular continued fraction forms. Also, the continued fractions of e and various powers of e have nice patterns that might help in the proof of normalcy. Still, nobody’s been able to exploit those patterns yet to prove normalcy (or otherwise) for any of those numbers.

Peak Warming Man said:

Hang on Poindexter, we’re not talking about the Louisville Constant we’re talking about pi.

My point is that just because p is irrational or even transcedental, it doesn’t mean that p doesn’t have a definite pattern of digits. The Liouville constant is an example of a transcendental number that does have a definite pattern of digits. As PM 2Ring has said, proving that a number such as p is normal is a difficult task that goes way beyond simply proving that it is transcendental. I made this point to highlight the fact that an infinite sequence of digits can be regular in a way other than that required by the rational numbers.

Hmmm. It looks like my earlier HTML omission of this has affected the entire full thread view.

KJW said:

Hmmm. It looks like my earlier HTML omission of this has affected the entire full thread view.

And don’t think that we’re ever going to forgive you for that.

Never!

:)

sibeen said:

And don’t think that we’re ever going to forgive you for that.

Never!

:)

Thanks :-) It needed that extra end tag which you supplied in your quoting of my post to fully fix the thread (because my omission occurred in two posts)

my DOB occurs at position 16,366,340 if you include the 19, and position 115,868 if you don’t ;)

>>my DOB occurs at position 16,366,340 if you include the 19, and position 115,868 if you don’t ;)

On a normal forum you could post that content in the knowledge that no one will check it.

anyway no , my question did not come from Contact, but from something I found on that peer reviewed journal “facebook.com” which inferred that any length string of digits you were searching for, would eventually be found in Pi..

I was wondering if there was a way of saying, if I was looking for a string of N digits, then I’d expect to be able to find it in M digits of Pi

Peak Warming Man said:

>>my DOB occurs at position 16,366,340 if you include the 19, and position 115,868 if you don’t ;)

On a normal forum you could post that content in the knowledge that no one will check it.

thats ok, I applied a 256 bit triple DES encryption on it first

>which inferred that any length string of digits you were searching for, would eventually be found in Pi..

As pi is infinite, then surely, by a hand waving definition, any length string of digits less than infinite could of course be found.

why not just buy a ball of string and cut bits off to the length you want?

sibeen said:

As pi is infinite, then surely, by a hand waving definition, any length string of digits less than infinite could of course be found.

No. As my posts pointed out, this is not necessarily true (the Liouville constant is an example that doesn’t) .

>>As pi is infinite, then surely, by a hand waving definition, any length string of digits less than infinite could of course be found.

I’m assuming that the terms that KJW and PM refered to are taking into account for an infinite number hitting recurrance, ie it goes for a couple of trillion googleplexes and then starts recurring, I don’t know if it even can.

So KJW, can an infinite decimal number start recurring or does it need to start that way?

>No. As my posts pointed out, this is not necessarily true (the Liouville constant is an example that doesn’t) .

That placates me a bit KJW, otherwise then there will be a ASCII sting of digits that state:

God exists, and jesus christ is the son of god, and the pope is his representative on earth.

Of course there would also be a string of digits that state:

sibeen is god, and he’s a right evil bastard to boot.

God exists, and jesus christ is the son of god, and the pope is his representative on earth.

I checked, that starts at position 666

Peak Warming Man said:

So KJW, can an infinite decimal number start recurring or does it need to start that way?

p can’t be an infinitely repeating sequence of digits of any length. That would make it rational (p has been proven to be transcendental). But it can have regularity in other ways. And it need not have within it any particular sequence of digits.

KJW said:

p can’t be an infinitely repeating sequence of digits of any length.

That includes starting out in some way, then infinitely repeating some other sequence.

KJW said:

KJW said:

p can’t be an infinitely repeating sequence of digits of any length.

That includes starting out in some way, then infinitely repeating some other sequence.

Now I’m confused.

Didn’t you say that the Liouville Constant (which we should call the ki constant here), was transcendental even though it had a regular repeating series of digits?

KJW said:

sibeen said:

As pi is infinite, then surely, by a hand waving definition, any length string of digits less than infinite could of course be found.

No. As my posts pointed out, this is not necessarily true (the Liouville constant is an example that doesn’t) .

But if it is infinite and “normal” it is true.

There will for instance be a string of digits which codes for two statements:
“The following statement is true”
and
“The preceding statement is false”

As soon as a computer computes those the Universe will enter an infinite regression and disappear up its own black hole.

The Rev Dodgson said:

KJW said:

KJW said:

p can’t be an infinitely repeating sequence of digits of any length.

That includes starting out in some way, then infinitely repeating some other sequence.

Now I’m confused.

Didn’t you say that the Liouville Constant (which we should call the ki constant here), was transcendental even though it had a regular repeating series of digits?

The Liouville constant isn’t a repeating sequence of digits. But the sequence of digits do have a definite pattern (the location of the 1s is simply defined).

KJW said:

The Rev Dodgson said:

KJW said:

That includes starting out in some way, then infinitely repeating some other sequence.

Now I’m confused.

Didn’t you say that the Liouville Constant (which we should call the ki constant here), was transcendental even though it had a regular repeating series of digits?

The Liouville constant isn’t a repeating sequence of digits. But the sequence of digits do have a definite pattern (the location of the 1s is simply defined).

If they are not repeating why are there not an infinite number of different strings?

The Rev Dodgson said:

But if it is infinite and “normal” it is true.

Only by definition of normality. The normality of p requires proof and doesn’t follow simply because p is transcendental.

The Rev Dodgson said:

If they are not repeating why are there not an infinite number of different strings?

For one thing, the Liouville constant contains no other digits in its decimal representation than 0 and 1. Also, there is no sequence of 1s longer than two digits.

KJW said:

Also, there is no sequence of 1s longer than two digits.

And even …11… occurs only once at the start and nowhere else.

KJW said:

The Rev Dodgson said:

If they are not repeating why are there not an infinite number of different strings?

For one thing, the Liouville constant contains no other digits in its decimal representation than 0 and 1. Also, there is no sequence of 1s longer than two digits.

But is there a maximum number of 0s in a sequence?

KJW said:

KJW said:

Also, there is no sequence of 1s longer than two digits.

And even …11… occurs only once at the start and nowhere else.

The string 1111 occurs at position 12,700 counting from the first digit after the decimal point. The 3. is not counted.

The string 111111 occurs at position 255,945 counting from the first digit after the decimal point. The 3. is not counted.

The Rev Dodgson said:

But is there a maximum number of 0s in a sequence?

The overall point is that a transcendental number does not require any particular sequence of digits to exist anywhere in its entire sequence.

Dropbear said:

KJW said:

KJW said:

Also, there is no sequence of 1s longer than two digits.

And even …11… occurs only once at the start and nowhere else.

The string 1111 occurs at position 12,700 counting from the first digit after the decimal point. The 3. is not counted.

KJW was talking about the ki constant, not pi.

oops sorry, ill go back in my box

Dropbear said:

The string 1111 occurs at position 12,700 counting from the first digit after the decimal point. The 3. is not counted.

For p, maybe, but not for the Liouville constant.

KJW said:

The Rev Dodgson said:

But is there a maximum number of 0s in a sequence?

The overall point is that a transcendental number does not require any particular sequence of digits to exist anywhere in its entire sequence.

But in the case of the ki constant, either there is a limit on the number of zeros in sequence, or the sequence of zeros could be any length, in which case there is sequence that could be coded for any arrangement of a finite number of symbols from a finite set.

The Rev Dodgson said:

But in the case of the ki constant, either there is a limit on the number of zeros in sequence, or the sequence of zeros could be any length, in which case there is sequence that could be coded for any arrangement of a finite number of symbols from a finite set.

If including sequences within longer sequences, then there are sequences of 0s of all lengths. But if one is only considering the maximum lengths of sequences, then not all lengths exists.

Dropbear said:

anyway no , my question did not come from Contact, but from something I found on that peer reviewed journal “facebook.com” which inferred that any length string of digits you were searching for, would eventually be found in Pi..

I assume it was something like this: We are in Digits of Pi and Live Forever

That link does mention that pi must be a normal number to contain every possible finite string of digits, but it seems that Pickover’s not particularly bothered by the fact that there’s currently no proof of pi’s normalcy; he’s more interested in discussing the implications of mathematical immortality. Of course, if it turns out that pi isn’t normal, there are plenty of other real numbers that are.

Dropbear said:

I was wondering if there was a way of saying, if I was looking for a string of N digits, then I’d expect to be able to find it in M digits of Pi

Sure, as long as we assume that pi is normal. There are 10^Npossible N digit sequences, so a random string of 10^N digits ought to contain each N digit sequence once, on average. So if we let M = 10^N+2 we should get roughly 100 copies of each N digit sequence, and the odds of any given N digit sequence not occurring at all within the M digit sequence is roughly 1%.

PM 2Ring said:

So if we let M = 10^N+2 we should get roughly 100 copies of each N digit sequence, and the odds of any given N digit sequence not occurring at all within the M digit sequence is roughly 1%.

I think. But I Am Not A Statistician. :)