• A real number is rational (i.e. a ratio of two integers) if and only if its decimal expansion is eventually periodic. “Eventually periodic” means that, if we denote the $n^{\rm th}$ decimal place by $d_n\text{,}$ then there are two positive integers $k$ and $p$ such that $d_{n+p}=d_n$ whenever $n \gt k\text{.}$ So the part of the decimal expansion after the decimal point looks like \begin{gather*} .\underbrace{a_1 a_2 a_3 \cdots a_k} \underbrace{b_1 b_2\cdots b_p} \underbrace{b_1 b_2\cdots b_p} \underbrace{b_1 b_2\cdots b_p} \cdots \end{gather*} It is possible that a finite number of decimal places right after the decimal point do not participate in the periodicity. It is also possible that $p=1$ and $b_1=0\text{,}$ so that the decimal expansion ends with an infinite string of zeros.
• $e$ is irrational.
• $\pi$ is irrational.