In the decimal form of an irrational number like:
$$\pi=3.141592653589\ldots$$
Do we have all the numbers from $0$ to $9$. I verified $\pi$ and all the numbers are there. Is this true in general for irrational numbers ?
In other words, for an irrational number
$$x=\sum_{n\in \mathbb{Z}} a_n 10^n$$
Does $a_n$ takes all the numbers between $0$ and $9$ ?
Best Answer
Certainly not, since rational numbers are exactly the ones with an eventually periodic expansion. So for example, the number $0.01001100011100001111\ldots$ is irrational.