Do All Digit Sequences in Pi Occur Equally Often?

[saddlestone-man]

TL;DR

Is any sequence of digits equally likely in the value of Pi?

As the value of Pi is taken to more decimal places, does any sequence of digits become equally likely?

I'm thinking of sequences like ...123456789.... ...333333... and so on.

best regards ... Stef

 

[PeroK]

TL;DR Summary: Is any sequence of digits equally likely in the value of Pi?

As the value of Pi is taken to more decimal places, does any sequence of digits become equally likely?

I'm thinking of sequences like ...123456789.... ...333333... and so on.

best regards ... Stef

It is supposed that $\pi$ is a normal number, where the digits are uniformly distributed. This is, however, unproven.

https://en.wikipedia.org/wiki/Normal_number

 

[phinds]

It is supposed that $\pi$ is a normal number, where the digits are uniformly distributed. This is, however, unproven.

https://en.wikipedia.org/wiki/Normal_number

Not only that, pi has been calculated out to an enormous number of decimal places and no special repetition has been observed. As @PeroK said, nothing is proven so It may well happen at some point but there's no reason to believe that it will and the evidence so far is against it.

 

[jedishrfu]

Carl Sagan used something like that as a plot device in his novel Contact. The protagonist, Ellie Arroway finds proof of her trip only in the digits of PI in some base where they discover a hidden message in the image of a circle embedded in the sequence.

Pi is a transcendental number and has a non-repeating decimal sequence. It is assumed to be evenly distributed across all the digits but there's no proof as yet.

https://en.wikipedia.org/wiki/Pi

I imagine it would be a fun exercise to explore.

 

[FactChecker]

View attachment 342466

I would assume that in the first 105 trillion known digits with a hypothesized uniform distribution, a short sequence like '123456789' or '333333333' already appears somewhere in the 105 trillion. I haven't calculated the probability, assuming a uniform distribution.

 

[saddlestone-man]

Many thanks for the answers.

I wondered that there may be some sequences of digits (say 1,000 repetitions of the same digit, or something like 123123123123123123 ....) which would indicate that the value had converged, which of course in the case of Pi is impossible.

 

[FactChecker]

Many thanks for the answers.

I wondered that there may be some sequences of digits (say 1,000 repetitions of the same digit, or something like 123123123123123123 ....) which would indicate that the value had converged, which of course in the case of Pi is impossible.

By 'indicate that the value had converged', do you mean 'fool you into thinking that the value had converged'?

 

[saddlestone-man]

I think what I'm saying is .... say you had two long sequences of numbers, one of which was extracted from the value of Pi and the other which was the tail end of a large long division (whose answer did not have an infinite number of digits): would you be able to tell the difference?

 

[FactChecker]

No. Any finite sequence of digits within $\pi$ could be from the division of the integer represented by that sequence by the appropriate power of 10.

 

[WWGD]

Since this is in the Statistics section, maybe you can look up tabulated data and do a $\chi^2$ goodness of fit for the uniform distribution on $\{0,1,2,..., 9\}$?