Date: Apr 3, 2013 1:35 PM
Author: Frederick Williams
Subject: Re: Proofs that numbers are rational, algebraic, or transcendental.

Frederick Williams wrote:
>
> Paul wrote:

> >
> > 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".
> >
> > Except that the result should be true.

>
> (The sum of n^{-k} from n = 1 to infinity)pi^{-k} is rational for k = 2,
> 3, ...
> For even k the result was known to Bernoulli, for odd it is more recent.


"More recent" as in "not yet proven". :-)

--
When a true genius appears in the world, you may know him by
this sign, that the dunces are all in confederacy against him.
Jonathan Swift: Thoughts on Various Subjects, Moral and Diverting