TITLE: Have any long-suspected irrational numbers turned out to be rational?
QUESTION [219 upvotes]: The history of proving numbers irrational is full of interesting stories, from the ancient proofs for $\sqrt{2}$, to Lambert's irrationality proof for $\pi$, to Roger Apéry's surprise demonstration that $\zeta(3)$ is irrational in 1979.
There are many numbers that seem to be waiting in the wings to have their irrationality status resolved. Famous examples are $\pi+e$, $2^e$, $\pi^{\sqrt 2}$, and the Euler–Mascheroni constant $\gamma$. Correct me if I'm wrong, but wouldn't most mathematicians find it a great deal more surprising if any of these numbers turned out to be rational rather than irrational?
Are there examples of numbers that, while their status was unknown, were "assumed" to be irrational, but eventually shown to be rational?

REPLY [2 votes]: Another constant that has "no business being rational" I think, although a bit elementary:  Choose a point at random in the unit disk $D=\left\{x^2+y^2\leq 1\right\}$.  Then the expected value $E$ of its distance to the origin is a rational number!  (click below for solution).

 $E=\frac{2}{3}$