I know that pi contains every finite number sequence, but i am not sure why. Since you can take any irrational number and change every $9$ to an $8$, then clearly not all irrational numbers are the same way, since the resulting number won't have any sequence with $9$ in it. Then what nunbers are this way, and how can you show that its true for pi (or any other number)?

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    $\begingroup$ It isn't known that $\pi$ has this property. So far as I am aware, it is perfectly possible (albeit highly unlikely) that from some point on, $\pi$ only uses two digits. $\endgroup$ – lulu Jun 11 '18 at 11:46
  • $\begingroup$ Besides from specially constructed numbers we do now know the normality of any number, in particualr for $e$, $\pi$ and every irrational algebraic number, the normality is an open problem. It is conjectured however that they are all normal in every base, but not for a single base this has been proven. $\endgroup$ – Peter Jun 11 '18 at 12:17
  • $\begingroup$ It has been shown that every digit string of length $11$ occurs somewhere in $\pi$. I have even heard of a museum with a computer that shows the first occurence of some date with length $4$, for example $3112$. Apparently, $\pi$ is the irrational number that fascinates people more than every other irrational constant. Memory contests are only made with $\pi$ , for example. $\endgroup$ – Peter Jun 11 '18 at 12:20

The numbers with that property are called normal numbers in base $10$. And nobody knows whether or not $\pi$ is normal in base $10$ (or in any other base). A number which is normal in base $10$ is$$0.12345678910111213\ldots$$


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