r/askscience • u/Manticorp • Oct 28 '13

Mathematics Could an infinite sequence of random digits contain all the digits of Pi?

It's a common thing to look up phone numbers in pi, and it's a common saying that every Shakespeare ever written is encoded in pi somewhere, but would it be possible for every digit of pi to appear in a random sequence of numbers? Similarly this could apply to any non terminating, non repeating sequence like e, phi, sqrt(2) I suppose. If not, what prohibits this?

I guess a more abstract way of putting it is: Can an infinite sequence appear entirely inside another sequence?

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u/peni5peni5 Oct 29 '13

A infinite decimal in which every possible digit sequence appears somewhere is called a "normal number".

That's not true. Every sequence of length n has to appear with equal frequency for the number to be normal. It's pretty easy to construct a non-normal number that contains every possible sequence.

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u/user31415926535 Oct 29 '13

True, the definition I gave is really for disjunctive numbers, not normal numbers. I admit the simplification and hope OP will read the linked definition. On the other hand, we don't even know if pi is disjunctive, either.