A question about decimal representation of irrational numbers.

Question

Is this true that any finite word of the alphabet $\mathcal{A_9}=\{0,1,2, \ldots,8,9\}$ appears somewhere in the decimal representation of $\sqrt{2}$ ?

Thanks !

Answer 1

What you are trying to ask is if $\sqrt{2}$ is a normal number. But while it is widely believed to be so, a proof has yet to be found.

Fun fact: Same goes for $\pi$, and that if true, everything in the universe can be expressed in the decimal digits of $\pi$ or $\sqrt{2}$.

Answer 2

Your question is the same, or close to, the question of whether $\sqrt 2$ is normal :http://en.wikipedia.org/wiki/Normal_number This is still open.