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        riddles >> general problem-solving / chatting / whatever >> Predicting Pi digits
        
(Message started by: jpk2009 on Mar 10^th, 2010, 4:09pm)

Title: Predicting Pi digits
Post by jpk2009 on Mar 10^th, 2010, 4:09pm

I read recently about attempts to prove the decimal expansion of pi to be random. I found one thing to be very strange. Apparently a proof for one number base does not evidently imply a proof for other bases! Can you imagine proving that predicting the next digit in the expansion, say, in base 10, can only be done with a probability of 1/10 would NOT imply that doing the same thing in base 9 would have a probability of 1/9?

Title: Re: Predicting Pi digits
Post by jpk2009 on Mar 10^th, 2010, 4:15pm

Let's say that in base 9, pi repeats after 100 digits - it's not clear to me that that should mean it would repeat in base 10. I believe it might repeat in a base that is a multiple or divisor of the repeating base, but if the two bases are mutually prime (say 9 and 10), it seems that it could repeat in one without repeating in the other.

Title: Re: Predicting Pi digits
Post by Obob on Mar 10^th, 2010, 4:18pm

Pi will not repeat in any base, because it is an irrational number. If it were to repeat in any base, it would repeat in all of them, because repeating numbers in any base are rational and all rational numbers are repeating in every base.

It seems unlikely that the digits would be randomly distributed in one base but not in another, but there's no a priori reason for this.

Title: Re: Predicting Pi digits
Post by towr on Mar 11^th, 2010, 12:33am

I think there is a method to calculate the nth digit of pi in base 16 without having to consider the preceding digits. This doesn't help you to get the digits in bases other than 2, 4 and 8.

Title: Re: Predicting Pi digits
Post by Grimbal on Mar 12^th, 2010, 4:42am

But if there is such a thing as base Pi, we get
Pi = 10.000...
And ... tadaaa ... it is periodic!
::)

Title: Re: Predicting Pi digits
Post by BenVitale on Mar 13^th, 2010, 10:15pm

This paper shows that the digits of Pi and Golden Number (http://pagesperso-orange.fr/jean-yves.boulay/pi/eng1.htm) are not random occurrences.

Title: Re: Predicting Pi digits
Post by towr on Mar 14^th, 2010, 1:41am

on 03/13/10 at 22:15:07, BenVitale wrote:

This paper shows that the digits of Pi and Golden Number (http://pagesperso-orange.fr/jean-yves.boulay/pi/eng1.htm) are not random occurrences.

That isn't even what the paper is about. It only talks about the order of the first occurrence of the ten digits.

Title: Re: Predicting Pi digits
Post by Obob on Mar 14^th, 2010, 9:21am

Wow, that paper is such complete bullsh*t it's not even funny.

As far as I can tell, the "phenomenon" described in the paper is this: take any real number between 0 and 1 which has all digits in its base 10 expansion. Write out the order in which each digit first appears. Now add the first six digits and add the last four. You get two numbers whose sum is 45. His "remarkable" observation is that for some "important" numbers, the first six sum to 27 and the last 4 sum to 18.

For example, with pi, the digits occur in the order 1 4 5 9 2 6 3 8 7 0. The first six sum to 1+4+5+9+2+6 = 27, while the last four sum to 3+8+7+0 = 18.

Somewhat surprisingly, the golden ratio and e also have the same property. However, the author also presents a list of far less common numbers which satisfy the same property.

The fact of the matter is, if the "named numbers" (important constants such as pi, e, etc.) have randomly distributed digits, then the sum of six of the digits which occur first will be a "random" number between 0+1+2+3+4+5 = 15 and 4+5+6+7+8+9 = 39. Notice that 27 is exactly half way between 15 and 39. Consequently, (I think) the number 27 can also be realized as the sum of six distinct numbers between 0 and 9 in more ways than any other number between 15 and 39 (I haven't checked this, but it's certainly plausible). Thus 27 is the most common possible sum of the first six occurring digits.

It wouldn't surprise me if 10% or more of all numbers have the property that the six digits which occur first sum to 27. So you can just chalk it up to coincidence that pi, the golden ratio, and e all have this property.

Another (slightly less) funny thing is that the paper doesn't consider at all what happens in other bases. If this is supposed to be some natural property satisfied by these mathematical constants, then starting off by looking in base 10 is a bit silly.

Title: Re: Predicting Pi digits
Post by towr on Mar 14^th, 2010, 10:07am

on 03/14/10 at 09:21:24, Obob wrote:

Consequently, (I think) the number 27 can also be realized as the sum of six distinct numbers between 0 and 9 in more ways than any other number between 15 and 39 (I haven't checked this, but it's certainly plausible).

You're right. A 27-18 split is the most probable, with a probability of 3 in 35.

Title: Re: Predicting Pi digits
Post by BenVitale on Mar 14^th, 2010, 12:36pm

I agree ... Maybe I should have created another thread for this. I apologize. ... But I thought it was still interesting because of its originality.

At any rate, since it was Pi Day, I posted this paper. I found another one that I could post a link to, that is if you care to see it.

Title: Re: Predicting Pi digits
Post by ThudanBlunder on Mar 14^th, 2010, 6:44pm

on 03/12/10 at 04:42:39, Grimbal wrote:

But if there is such a thing as base Pi, we get
Pi = 10.000...
And ... tadaaa ... it is periodic!
::)

And Base http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/phi.gif allows for more than one (http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/phigits.html#reduce1) representation. :o

Title: Re: Predicting Pi digits
Post by towr on Mar 15^th, 2010, 1:46am

on 03/14/10 at 18:44:40, ThudanBlunder wrote:

And Base http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/phi.gif allows for more than one (http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/phigits.html#reduce1) representation. :o

So does base pi; ~~pi = 0.999... = 1~~ :P

Title: Re: Predicting Pi digits
Post by Grimbal on Mar 15^th, 2010, 3:23am

on 03/15/10 at 01:46:53, towr wrote:

So does base pi; pi = 0.999... = 1 :P

I take it that in base PI the digit '9' is worth 2.14159...₁₀, then.

Title: Re: Predicting Pi digits
Post by towr on Mar 15^th, 2010, 3:38am

on 03/15/10 at 03:23:22, Grimbal wrote:

I take it that in base PI the digit '9' is worth 2.14159...₁₀, then.

Err, yeah. 9 = 10-1, after all ;D

[edit]
Additionally, pi = 10 = 3.01102111... = 2.31220002... in base pi.
[/edit]

Title: Re: Predicting Pi digits
Post by BenVitale on Mar 15^th, 2010, 5:23pm

Obob, Towr

I received a private message from the author of the paper: Pi and Golden Number are not random occurrences

He thanked me for my interest in this.
He would like to emphasize the fact that the study of the constants is only on the first appearance of the ten digits.
In conclusion, he proposes to consider the existence of a new family of numbers having the characteristics described in this article. Family of numbers which the number Pi and the Golden Mean are the most significant representatives. Also, the author recalls and insists that this new field of research investigates, in numbers, only the first appearance of the ten digits of the decimal system and suggests that it is not fruitful to extend the investigations to the following appearances. The fact, not presented here but experienced by the author, that these investigations are sterile paradoxically reinforces the idea that the phenomena introduced in this study must be subject to greater attention.

Title: Re: Predicting Pi digits
Post by Obob on Mar 15^th, 2010, 6:50pm

What you just said can more or less be found word for word in the paper.

The guy's a crackpot.

Title: Re: Predicting Pi digits
Post by Michael Dagg on Mar 15^th, 2010, 6:51pm

Here is a good read on this topic (attached).

Enjoy.

Title: Re: Predicting Pi digits
Post by BenVitale on Mar 16^th, 2010, 12:46pm

Thanks Michael.

I'll Tweet him the link ... Do you know if there's a good tool on the web to translate this PDF file into French?

Title: Re: Predicting Pi digits
Post by Obob on Mar 16^th, 2010, 1:37pm

The paper Michael linked to isn't related to the paper you linked to, Ben. It's concerned with the randomness of the whole sequence of digits, not just the initial digits. (In fact, the initial digits don't even matter at all in the paper Michael posted). The paper Michael posted is related to the original poster's question.