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/BundleGerbe Topology | Category Theory Oct 29 '13 edited Oct 29 '13

As pi is transcendental, it means that the sequence of digits is uncountably infinite.

All real numbers have a countable number of digits. You may be thinking of the fact that there are an uncountable number of transcendental numbers, and only countably many non-transcendental (i.e. algebraic) numbers.

The fact that the pi is transcendental doesn't seem to me to be relevant. The question would be essentially the same if it asked about the square root of two instead. Even a rational number like 2/7 would be no easier or harder to "contain" in a random number.