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Re: Proofs that numbers are rational, algebraic, or transcendental.
Posted:
Apr 3, 2013 1:27 PM


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 nontrivial results which show that a specific number is rational or algebraic? In other words, does anyone know any nontrivial 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 nonelementary 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.
 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



