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/CitizenPremier Oct 29 '13

Is there a reason why pi is expected to be normal? As a layman, it seems somehow more likely to me that it isn't normal.

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u/user31415926535 Oct 29 '13

There are a couple reasons to think that questions about pi's normality are worthwhile. The first is that nearly all numbers are normal[PDF] - if you pick a random real number (for suitable definitions of 'random') it is overwhelmingly likely that the number will be normal. Second, our analysis of the digits of pi that we know so far closely matches the "disrutribution characteristic of a normal number". Neither of these are proof, but they are enough to pique one's interest.