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i believe it has been proven that not only is Pi irrational, but that the decimal expansion of Pi contains all possible finites sequences of digits.
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Anyone able to provide a citation for this? It seems wrong to me.
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a quick googling gave me this:
http://sprott.physics.wisc.edu/pickover/pimatrix.html
doesn't look like the best source. however, it does point at out that such numbers are called transcendental numbers. so even if you don't believe pi is transcendental, then just imagine some other transcendental number.
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It seems to me that he is basing his claim on the fact that the expansion of pi is infinite, non-repeating with all digits occuring with equal frequency. He seems to make an intuitive leap that this means all finite sequences will eventually occur which is just plain wrong (eq 0.123456789112233445566778899111... has the above properties but never contains the finite sequence 28).
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If every "next digit" of pi was truly random then I think a probability argument could be made that any Fixed Sequence of length N has a 1/10^N chance of coming up next. Thus, since you have infinitely many 1/10^N chances of it coming up next it almost surely comes up over and over again.
It would be like flipping a coin infinitely many times. If you did, you would almost surely see 1 googolplex of heads flipped in a row somewhere in the infinite sequence of flips. Not only that, but you would see it infinitely many times, with probability 1.
However, I don't think it's clear that every next digit of pi behaves as if it is completely random. The conjecture might still be provable but I don't think it's obvious.
PairTheBoard