Is it rational or not? I have two interesting question :
Is this number rational or not:
$$0.F_{1}F_{2}F_{3}...$$, where $F_{i}$ - Fibonacci number.
And is this number rational or not:
$$0.p_{1}p_{2}...$$
 A: Both numbers are irrational. It's clear that neither number has the decimal expansion eventually consisting of only zeroes, so if we prove that they have arbitrarily long trails of zeroes, we will get our result.
Let's start with the latter number, known as, as Watson mentioned in a comment, Copeland–Erdős constant. We have $\gcd(1,10^{n+1})=1$, so by Dirichlet's theorem, there are infinitely many primes $p$ satisfying $p\equiv 1\pmod{10^{n+1}}$. But this prime has $n$ consecutive zeroes in its decimal expansion, namely second, third,..., $n+1$-th.
As for the former number, I will use the following lemma, which I leave for you to try to prove it:

For any number $k$, there is a number $m$ such that $k\mid F_m$.

(hint for proof: consider sequence $(F_m\mod k,F_{m+1}\mod k)$. Using pigeonhole principle, show that this sequence eventually repeats, and hence it's periodic (not only eventually periodic). Lastly recall $F_0=0$)
Now let's use the lemma with $k=10^n$. Then $F_m$ will contain a trail of $n$ zeroes, namely its last $n$ digits.
