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Topic: Predicting Pi digits (Read 3179 times) |
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jpk2009
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Predicting Pi digits
« on: Mar 10th, 2010, 4:09pm » |
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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?
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jpk2009
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Re: Predicting Pi digits
« Reply #1 on: Mar 10th, 2010, 4:15pm » |
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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.
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Obob
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Re: Predicting Pi digits
« Reply #2 on: Mar 10th, 2010, 4:18pm » |
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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.
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| « Last Edit: Mar 10th, 2010, 4:19pm by Obob » |
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towr
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Re: Predicting Pi digits
« Reply #3 on: Mar 11th, 2010, 12:33am » |
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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.
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Grimbal
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Re: Predicting Pi digits
« Reply #4 on: Mar 12th, 2010, 4:42am » |
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But if there is such a thing as base Pi, we get
Pi = 10.000...
And ... tadaaa ... it is periodic!
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Benny
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Re: Predicting Pi digits
« Reply #5 on: Mar 13th, 2010, 10:15pm » |
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This paper shows that the digits of Pi and Golden Number are not random occurrences.
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towr
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Re: Predicting Pi digits
« Reply #6 on: Mar 14th, 2010, 1:41am » |
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on Mar 13th, 2010, 10:15pm, BenVitale wrote:That isn't even what the paper is about. It only talks about the order of the first occurrence of the ten digits.
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Obob
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Re: Predicting Pi digits
« Reply #7 on: Mar 14th, 2010, 9:21am » |
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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.
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| « Last Edit: Mar 14th, 2010, 9:32am by Obob » |
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towr
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Re: Predicting Pi digits
« Reply #8 on: Mar 14th, 2010, 10:07am » |
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on Mar 14th, 2010, 9:21am, 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). |
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You're right. A 27-18 split is the most probable, with a probability of 3 in 35.
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Benny
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Re: Predicting Pi digits
« Reply #9 on: Mar 14th, 2010, 12:36pm » |
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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.
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ThudnBlunder
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Re: Predicting Pi digits
« Reply #10 on: Mar 14th, 2010, 6:44pm » |
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on Mar 12th, 2010, 4:42am, Grimbal wrote:But if there is such a thing as base Pi, we get
Pi = 10.000...
And ... tadaaa ... it is periodic!
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And Base  allows for more than one representation. 
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towr
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Re: Predicting Pi digits
« Reply #11 on: Mar 15th, 2010, 1:46am » |
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on Mar 14th, 2010, 6:44pm, ThudanBlunder wrote:So does base pi; pi = 0.999... = 1 
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| « Last Edit: Mar 15th, 2010, 4:25am by towr » |
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Grimbal
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Re: Predicting Pi digits
« Reply #12 on: Mar 15th, 2010, 3:23am » |
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on Mar 15th, 2010, 1:46am, towr wrote:
So does base pi; pi = 0.999... = 1
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I take it that in base PI the digit '9' is worth 2.14159...10, then.
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towr
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Re: Predicting Pi digits
« Reply #13 on: Mar 15th, 2010, 3:38am » |
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on Mar 15th, 2010, 3:23am, Grimbal wrote:| I take it that in base PI the digit '9' is worth 2.14159...10, then. |
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Err, yeah. 9 = 10-1, after all 
[edit]
Additionally, pi = 10 = 3.01102111... = 2.31220002... in base pi.
[/edit]
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| « Last Edit: Mar 15th, 2010, 4:43am by towr » |
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Benny
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Re: Predicting Pi digits
« Reply #14 on: Mar 15th, 2010, 5:23pm » |
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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.
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| « Last Edit: Mar 15th, 2010, 5:24pm by Benny » |
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If we want to understand our world — or how to change it — we must first understand the rational choices that shape it.
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Obob
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Re: Predicting Pi digits
« Reply #15 on: Mar 15th, 2010, 6:50pm » |
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What you just said can more or less be found word for word in the paper.
The guy's a crackpot.
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Michael Dagg
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Here is a good read on this topic (attached).
Enjoy.
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Regards,
Michael Dagg
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Benny
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Re: Predicting Pi digits
« Reply #17 on: Mar 16th, 2010, 12:46pm » |
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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?
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If we want to understand our world — or how to change it — we must first understand the rational choices that shape it.
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Obob
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Re: Predicting Pi digits
« Reply #18 on: Mar 16th, 2010, 1:37pm » |
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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.
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