TITLE: Is the number $0.112358132134...$ rational or irrational?
QUESTION [24 upvotes]: Just out of curiosity, is the number $0.112358132134...$ a rational or irrational number?
$...$ stands for Fibonacci sequence not repeating decimals!

REPLY [2 votes]: Nonrigorous argument: As $n \to \infty$ $F_n \to \varphi F_{n-1}$. So given a very high N we can say $F_n \approx \lfloor(\varphi^{n-N}F_N)\rfloor$ $\forall n > N$. Since $\varphi$ is irrational, whenever $F_n$ has one more digit than $F_{n-1}$ there won't be any pattern to the extra digit, so $0.F_0F_1F_2...$ won't ever repeat.