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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](http://en.wikipedia.org/wiki/Normal_number). AFAIK, no one knows whether the familiar numbers you've listed is normal.2012-08-27
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    indeed. but interestingly normality seems independent of the property I mentioned.2012-08-27
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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.2012-08-27
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    ... and Euler's constant $\gamma$ is not known to be irrational at all (but of course it is irrational)2012-08-27

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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.