A rational number of the form a~a with a irrational

摘要

Here is a very well known elementary proof that there are irrational numbers a and b so that a~b is rational. We know 2~(1/2) is irrational. If 2~(1/2) is rational, then we are done, while if it is not rational, then(2~(1/2)~(2~(1/2)))~(2~(1/2))=2 and we are done. This elementary but non-constructive argument motivates, but does not quite answer, the question of whether there is a single irrational number a so that a~a is rational. The answer to that question is a resounding affirmative and is also elementary.