Se $\frac{\pi}{e}$ o $\frac{e}{\pi}$ irracional?

Question

Es claro que si $\displaystyle \frac{\pi}{e}$ o $\displaystyle \frac{e}{\pi}$ son irracionales o no?

Si no, entonces no existiría $q,p\in \mathbb{Z}$ tal que $$p\cdot \pi = q\cdot e$$

Answer 1

Una MUY SIMPLE COMENTARIO $$\begin{cases}\frac{e}{\pi}=t\\e+\pi=s\end{cases}\qquad (*)$$ would imply $$\pi=\frac{s}{t+1}\\e=\frac{st}{t+1}$$ Consequently and least one of $t$ and $s$ en (*) deben ser trascendental.