March 14: Pi day; five tasty facts about the famous ratio

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I looked up my and my boyfriend's dates of birth with the Pi search website.

My date of birth in our Dutch notation is 04-12-1975; in the US notation it is 12-04-1975.
My boyfriend's date of birth in our Dutch notation is 13-03-1986; in the US notation it is 03-13-1986.

The Pi search website came up with these results:

The string 04121975 occurs at position 97,409,220 counting from the first digit after the decimal point.

The string 12041975 occurs at position 60,644,972 counting from the first digit after the decimal point.

The string 13031986 did not occur in the first 200000000 digits of pi after position 0.
(Sorry! Don't give up, Pi contains lots of other cool strings.)

The string 03131986 occurs at position 6,161,283 counting from the first digit after the decimal point.

So not all strings of birth date numbers will be found in the first 200,000,000 digits of π after position 0.
But maybe they would if you would search an infinite π ...

sorry... there's a little glitch in there... it should be a 2 not a 3

3.141592653589... etc

I meant an infinite string of digits taken from pi, sorry I didn't clarify that.

e should also produce the same result, and the square root of 2 is another endless non-recursive decimal last I checked.

I think you may need to supply a reference for a proof of each of these assertions.

I can easily supply a definition of a series of digits which will pass many tests for randomness, but does NOT have probability one of matching some given strings of digits.

E.g. I suggest the number which is the digits of pi, but with the digits of e being used to select how many digits of pi to skip, before inserting a digit taken in rotation from 0 to 9. This would be:

pi = 3.14159265358979323846264338327950288...

e = 2.718281...

pie = 3.104159265132589793232843626433834257950288...

Now, I pretty sure that one could find the string "1111111111" in there, but you will not find the string "11111111111".

(PS. Wikipedia is your friend.... for alpha.)

Assuming pi is an infinite string of digits without significant repeating patterns, then there should be an infinite number of arbitrarily ordered strings within it, so it would be more surprising to NOT find a given string.

The number of strings that can be found in just a couple hundred million places already make it certain to find n=5 or n=6 digit strings doesn't it?

Expanding to infinity, I'd be pleasantly surprised to find that it does have a repeating structure and/or avoids producing certain types of strings.


I don't have proof other than intuition, though if you apply things like diagonal arguments, or insertion of other infinite length non-recursive strings, the likelihood of not finding a specific ordering of digits of a given length has to be so low as to make such discoveries worthy of study in themselves, I'd think?


As for alpha, I've long exhausted the info wiki has on it.

Though, I'm wondering if the discrepancy between 1/(4pi^3+pi^2+pi) and α is related both to the current age of the Universe, and the local gravity well.

Just going to 11 digits, α - 1/(4pi^3+pi^2+pi) = 0.000017196080168, so close it kinda has to drive you crazy wondering why it isn't exact, huh?

Oh, wait, whoops: that had an error in the data input, which skews my number there, lemme fix it.



That is currently the best estimate of the value, until they get the error corrected in CODATA 2010 or 2011.

Define what this means.


If this is what you defined the above as, then it would be true.

I.e. if your assumption about pi is that it accords with the unspecified definition of "an infinite string of digits without significant repeating patterns", and that definition happened to somehow preclude the series I gave earlier (which so far as I can see, it doesn't), then you might be able to prove your conclusion, that "there should be an infinite number of arbitrarily ordered strings within it".

So... I don't think I can see why the assumption about pi shpould be made, in any case; and I don't even understand what you are assuming; and I don't think whatever you define it as can give you your conclusion.


Yes indeed. If you define the source string as random, then the conclusion may be proven. If the source string is not random (e.g. it is the decimal expansion of pi), then you must supply a proof of this.


If it were a random string of merely "a couple hundred million places" digits, the expected number of missing 5-digit strings will be low, I guess. The expected number of missing 6-digit numbers would be rather higher. Offhand, I'm not sure what the values would be. For 7-digit strings, I seem to have come up with the answer 0.0206, but that seems highly suspect.

Anyway... you can find this out for yourself. Just use the web search and type in the strings to find.


What is "a repeating structure"?

There are certainly very straightforward ways of generating the digits of pi. If this is what you mean by "a repeating structure", then pi has one.


... and you are back to the confusion between a random series of digits and the digits of pi.


... and neither does anyone else, so far.


I wouldn't, because now you have started talking about "likelihood", which in the context of a non-random sequence of digits, become meaningless.

As to the "discoveries" you suggest, I don't understand what they would be.

And now I look, I'm not sure what "insertion of other infinite length non-recursive strings" can mean.




.. I answered your question.

If you hit like 300 million digits and it starts over at 14159265358979323... then it isn't what I am talking about.


If this is what you defined the above as, then it would be true.[/quote]

I clarified the above definition.



As I stated, I just said it oddly, if pi turns out to go into a cycle of repetition after n digits, then naturally you could only find strings of a certain length or shorter.



If it is infinitely long, and non-repeating on any significant scale (i.e. there is never a point where it starts over at 1415925636897932384626433) then I would be surprised to find it never produces certain strings of certain lengths, in particular if you took the longest digit strings it does produce, and then found a gap between that and n+2 digit strings where it doesn't contain any, that would strike me as odd.

I said it weird, sorry.



That site listed a 100% probability of 5 digit strings in the first 200 million places.



I mean on the scale of the current known digits, if it were found to contain only a repeating string of 300 million digits, for example, that would be fascinating, and not what I was expecting. As far we know there are no such repetitions on that scale.



As I said, I didn't explain what I meant properly.



Assuming that e and pi are both infinitely long, and don't cycle through arbitrarily long strings of digits, simply inserting the digits of each between the other would be expected to produce a vast range of n-digit strings. Gaps in said strings would strike me as odd, and interesting, just as the structures that show up in the divisorplot.com site images do.

... then it would be a rational number. pi is not a rational number.

Again I didn't explain properly, say it hit 300 million digits, and repeated a portion of the first n digits, then went on for another few million digits, cycling through a few types of strings in an irrational manner.

Not that it would be 1/(first 300 million digits).