Prove that $Ae + B/e$ is irrational. Here A and B are non-zero integers.
We know that the sum of two irrational numbers need not be an irrational number.
For example $\sqrt2+1$ and $1-\sqrt2$ are irrational numbers. But there sum $(\sqrt2+1)+ (1-\sqrt2)=2\enspace (2 \in \mathbb{Q})$ is not an irrational number.
But for this problem, I am finding difficulty even how to begin with, which will slowly lead me to the proof.
Best Answer
Suppose that $Ae + B/e$ is rational. Then $Ae + B/e = p/q$ for some $p,q \in \mathbb{Z}$. Then $Ae^2 - \frac{p}{q} e + B = 0$. Clearing denominators gives us $e$ as a root of $Aqx^2 - px + Bq$ which is impossible since $e$ is transcendental.