All real numbers are obviously either rational, (algebraic and irrational) or transcendental. So these descriptors partition the reals into three disjoint sets.
From what I've seen, all the difficult results placing a real number into one of these classes have been of the form "x is transcendental".
Does anyone know of any non-trivial results which show that a specific number is rational or algebraic? In other words, does anyone know any non-trivial results (advanced undergraduate or higher) which define a specific x and then prove a statement of the type "x is rational" or "x is algebraic"?
I mean to exclude results that are proved simply by translating the real number into a simpler form.
For example, there are integrals which can only be solved by non-elementary means and which happens to equal 2. That's not the type of thing I mean.
I mean a result like "The sum of n^(-3) from n = 1 to infinity is rational".