# Can any number sequence be found in $\pi$? [duplicate]

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 $\pi$?

## marked as duplicate by Elliot G, user228113, Hayden, Joffan, Henning MakholmJul 22 '16 at 0:12

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

• – Klint Qinami Jul 21 '16 at 23:55
• For finite sequences here. It appears to be a very open problem. – user228113 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 0:07
• So do normal numbers have the property? – Mike Park Jul 22 '16 at 0:11
• @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. – Henning Makholm Jul 22 '16 at 0:15