The value of PI in Decimal units to 20 places is 3.14159 26535 89793 23846... and the occurrence of digits 0 through 9 in the decimal part of the sequence is: 0 (none) 1,1 2,2 3,3,3 4,4 5,5,5 6,6 7 8,8 9,9,9

As we can see 0 doesn't appear at all, followed by 7 (One occurrence), then 1,2,4,6,8 (Two occurrences each) and then 3,5,9 (Three occurrences each).

Does this distribution smoothen out at some point? Say will PI() to xx million places produce an equal occurrences of all digits (as would happen if these were random digits)? Or is there something inherently non random in PI() which creates a weighted average in favor of certain digits?

The question is also w.r.t PI() in Binary units. At the end of xx million places of PI(), is the total number of 0's and 1's at 50% each?

Just curious.

ps: This is not merely theoretical question, but asked from practical perspective -- i.e. since PI() is calculated to Billion+ p[laces, I was wondering is someone has actually carried out such an exercise; and if so, what is the result?

To further clarify, if PI () was being spit out by a PI producing machine and we do not know the value of PI(), and every day the world waits for the NEXT PI() digit, then I am sure that no one can predict the next PI value, and so it is random. But in a different world, the PI producing machine always knows the answer and so produces the next digit in the series consistently. In other words, the PI producer always knows what the next digit is going to be and is not spitting out the numbers randomly. My question is that over millions of places, can we spot a pattern of digit distribution that would tell on the inherent nature of PI? Or have we found that after a billion places, we end up getting 10 piles of equal weights suggesting that the PI producer is after all random?

  • 665
  • 2
  • 13
  • 7
    A pedantic note: you're not asking if the digits of $\pi$ are random, you are asking if they are uniformly distributed. – Duncan Ramage Aug 12 '19 at 01:17
  • 6
    Possible duplicate of [Distribution of the digits of Pi](https://math.stackexchange.com/questions/51829/distribution-of-the-digits-of-pi) also [this one](https://math.stackexchange.com/questions/2826520/can-we-find-every-finite-sequence-of-pi-within-pi) You can search the site for pi normal and find many references. – Ross Millikan Aug 12 '19 at 01:21
  • 2
    If a number has equal distribution of all digits it is called simply normal. It is unknown whether $\pi$ is normal. See https://en.wikipedia.org/wiki/Normal_number – M. Nestor Aug 12 '19 at 01:23
  • If the digits in $\pi$ were "random," then sometimes the first digit would be $3$, sometimes it would be $4$, etc. – Joel Reyes Noche Aug 12 '19 at 01:25
  • Would you consider the number ["0.1234567891011121314..."](https://en.wikipedia.org/wiki/Champernowne_constant) random? – Joel Reyes Noche Aug 12 '19 at 01:40
  • @joel. I would not consider it to be random since I can, with a "reasonable" sense predict the next few digits. – Lemur Aug 12 '19 at 01:45
  • Maybe check this out? https://math.stackexchange.com/questions/216343/does-pi-contain-all-possible-number-combinations –  Aug 12 '19 at 01:53

0 Answers0