Monday, 11 August 2014

Another question about the distribution of pi digits (and other famous irrationals).

Fiddling around I came across something I'd like to toss out there. I researched here, and google, but found very little on my specific idea/question. I have the idea, and some empirical evidence, but lack the symbols and math knowledge required to express and prove the idea. Have any of you come across this? What you think overall?



Let p be some decimal place of pi, and let q be the frequency of x e (0,1,2,3,4,5,6,7,8,9) up to p. Then as p -> infinity, q/p -> 1/9 = .1.



It seems to be the case up to 10 million digits of pi. I don't have the resources to test much more. But it also seems to be true for Euler's number, phi, and sqrt2. I know enough about math to be dangerous, I guess. Excuse me if I don't use terms and ideas appropriately as it has been quite some time for me.




Further, consider the set P = {q/p,...} as p -> infinity. Then it seems that P is bounded above and 1/9 is the not only the least upper bound for P, but that the sequence of p/q's converges to 1/9.



That is about all I can say about this. I do have further inquiries however.




  1. Is this true for all irrational numbers? I am back and forth on this.

  2. Is this important if it is true?

  3. Has this been looked or is it established already?

  4. What does it mean that pi (and others) adhere to this sort of probabilistic distribution?

  5. If it does apply to all irrationals, if we consider the irrationals from say 0 and 1, would there be a uniform shift in how the numbers are distributed?




Ok, I'm spent. Thanks for any replies!

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