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Topic: Proofs that numbers are rational, algebraic, or transcendental.
Replies: 6   Last Post: Apr 3, 2013 2:07 PM

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Posts: 419
Registered: 10/7/06
Re: Proofs that numbers are rational, algebraic, or transcendental.
Posted: Apr 3, 2013 1:48 PM
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On Apr 3, 5:50 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 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"?

Let S(n) = sum (1 / (pi*k)^n) over 1 <= k < infinity.

For some n, S (n) is proven rational, and for others it isn't. (It is
related to Bernoulli numbers when n is even).

For example, the infinite sum of 1/k^2 is pi^2 / 6, so S (2) = 1/6.

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