I read somewhere that any finite number sequence must be found in $\pi$. For example, $0998975645455$ must be somewhere in the digits of $\pi$.

The reason for this was that $\pi$ is irrational, meaning it's infinite and irregular.

But $\sum_{k=0}^n \frac{1}{10^{k!}}$ is transcendental and its digits are composed of $0$ and $1$ only.

So irrationality does not guarantee the property I mentioned.

Is it really true that one can find any number sequence from $\pi$?

Mike Park
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    See http://math.stackexchange.com/questions/216343/does-pi-contain-all-possible-number-combinations – Klint Qinami Jul 21 '16 at 23:55
  • For finite sequences [here](http://math.stackexchange.com/questions/96632/do-the-digits-of-pi-contain-every-possible-finite-length-digit-sequence). It appears to be a *very open* problem. –  Jul 21 '16 at 23:58
  • Irrationality is not enough as you correctly spotted. It is not actually proven that $\pi$ has this property, but it is believed by most to be true. See the linked post by @TnilkImaniq for a similar question with good answers. – Eff Jul 22 '16 at 00:07
  • So do normal numbers have the property? – Mike Park Jul 22 '16 at 00:11
  • @MikePark: Yes; the _definition_ of a number being normal is that not only does every digit sequence _appear_ in the decimal expansion; every digit sequence appears _infinitely many times_ with the same (limiting) frequency we would expect to find it with in a string of random digits. – hmakholm left over Monica Jul 22 '16 at 00:15
  • @HenningMakholm Thank you! – Mike Park Jul 22 '16 at 11:56

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