Most QA9 readers will know the nonconstructive proof that there are two irrational numbers a and b such that a to the power b is rational. There is interesting discussion about this here. I recently came across a 2014 note by Roger Hindley about the history of how this example came to be more broadly known. It is interesting reading. Hindley credits a 1953 paper by Dov Jarden in Scripta Mathematica (volume 19, page 229) with the proof. In case you do not have access to this journal in a local library, I have decided to include a scanned copy of the article below:
It’s a Math tweet from 1953.