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Re: Proofs that numbers are rational, algebraic, or transcendental.
Posted:
Apr 3, 2013 1:16 PM


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 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. > > 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.
Problem:
at least one of pi*e , pi+e is irrational.
Hint:
Look at the polynomial: (xpi)*(xe) = x^2 + x*(pie) + 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*(pie) + pi*e ?
BTW: It's long known that pi is irrational, and so is e.
dave
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