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What are numbers in which we can find arbitrary sequence of digits (in a certain base-$n$ expansion)? I know that $0.123456789101112131415\cdots$ does (and its analogues in other bases), but does this property hold for some more familiar numbers like algebraic integers or $e$, $\gamma$ or $\pi$?

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This is definitely related to normal numbers. AFAIK, no one knows whether the familiar numbers you've listed is normal. –  lhf Aug 27 '12 at 0:56
    
indeed. but interestingly normality seems independent of the property I mentioned. –  progressiveforest Aug 27 '12 at 0:59
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If a number is normal, it has any arbitrary sequence of digits included. Normality is a stronger condition-it requires that the asymptotic density of any $n$ digit string is $10^{-n}$. We know that most numbers are normal, but it is hard to prove individual ones normal unless they are carefully constructed. –  Ross Millikan Aug 27 '12 at 1:11
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... and Euler's constant $\gamma$ is not known to be irrational at all (but of course it is irrational) –  GEdgar Aug 27 '12 at 2:17
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up vote 2 down vote accepted

Nobody knows. For all we know, the decimal expansions of $\sqrt2$, $e$, and $\pi$ could all have nothing but zeros and ones from some point on.

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