TITLE: $\pi + e$ is rational or $\pi-e$ is rational
QUESTION [10 upvotes]: I was asked to find the truth value of the statement: 

$$
\pi + e \; \text{ is rational or } \pi - e\; \text{ is rational }
$$

I am only allowed to use the fact that $\pi, e $ are irrational numbers and cannot use the theory of transcendental numbers.
Cannot proceed. any help would be appreciated.

REPLY [11 votes]: $2\sqrt2$ and $\sqrt2$ are two distinct irrational numbers s.t. the statement: '$2\sqrt2+\sqrt2$ is rational or $2\sqrt2-\sqrt2$ is rational' is not true.
$\sqrt2$ and $2-\sqrt2$ are two distinct irrational numbers s.t. the statement: '$\sqrt2+(2-\sqrt2)$ is rational or $\sqrt2-(2-\sqrt2)$ is rational' is true.
This illustrates that the statement cannot be proved purely based on the fact that $\pi$ and $e$ are irrational.