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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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David Bernier

Posts: 3,892
Registered: 12/13/04
Re: Proofs that numbers are rational, algebraic, or transcendental.
Posted: Apr 3, 2013 1:16 PM
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On 04/03/2013 12:50 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"?
> 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.
> Thank you,
> Paul Epstein

It all depends on the formula given for the number.
Some series are exactly summable, such as the geometric series.

With definite integrals, it often happens that the result
can be put in closed form, with standard calculus functions
applied to "simple numbers". But if the result in closed form
is a sum of known irrationals, and some of them just
might be transcendental (or are transcendental), then it
can be really hard.

sum_{n=1, oo} 1/n^3 also known as zeta(3), was shown
to be irrational by a very ingenious method (from what
I've read) by the Frenchman Apery maybe 25 years ago
or thereabouts.


at least one of pi*e , pi+e is irrational.


Look at the polynomial:
(x-pi)*(x-e) = x^2 + x*(-pi-e) + pi*e .

What would follow if both pi*e and
pi+e were rational numbers ?
What are the roots of
p(x) = x^2 + x*(-pi-e) + pi*e ?

BTW: It's long known that pi is irrational, and so is e.



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