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Re: Simple question about irrationals, with a short note in the margin. :)
Posted:
Jan 14, 2014 2:05 AM


On Tue, 14 Jan 2014, Port563 wrote:
> "William Elliot" <marsh@panix.com> wrote... > > On Tue, 14 Jan 2014, Port563 wrote: > > > >> CAN AN ALGEBRAIC IRRATIONAL RAISED TO (the power of) AN ALGEBRAIC > >> IRRATIONAL > >> (not necessarily the same one) BE RATIONAL? > > > >> Prove the answer. > >> This proof must be _very_ short. > > > > Gelfand and Schneider. > > The Gelfond (not Gelfand)  Schneider result is that > a^b = transcendental > where a is either a nonzero rational or an algebraic irrational > and b is an algebraic irrational > > Which makes the answer: > "No"
It also makes (sqr 2)^sqr 2) transcendetal
> CAN AN IRRATIONAL RAISED TO AN IRRATIONAL BE RATIONAL? > > then the answer is "Yes", with what I think is an attractive proof that is > not reliant on GelfondSchneider : > > > SQRT(2) is trivially proven to be irrational. > > Consider SQRT(2)^SQRT(2) > > It is either rational or not (we ignore Gelfond  Schneider which proves it > irrational and furthermore transcendental) > > If that expression is rational, then the answer to the question is "Yes", by > example. > > OTOH, if it is irrational, then consider: > > (SQRT(2)^SQRT(2))^SQRT(2) > > which then is an example of an irrational raised to an irrational. > > That expression evaluates to 2, which is rational, meaning the answer to the > question in this case too is "Yes". > > So, without the need to establish the nature of SQRT(2)^SQRT(2) itself, > we've established that an irrational power of an irrational may be > irrational. > > > STOP YOUR RUDE SHOUTING; REST IGNORED. > > It was not shouting, it was highlighting. :)
It comes across as shouting and is consider such by Usenet.
_This_ is highlighting.
> On to the followup questions: > Gelfond  Schneider also disposes of the first of them: > > [Can an algebraic irrational raised to an algebraic irrational be > transcendental?] > > Answer: Yes (furthermore, it _must_ be transcendental) > > However, GelfondSchneider may not resolve the following questions  what > are the answers to them? > > Can an algebraic irrational raised to a transcendental be an algebraic > irrational? > > Can an algebraic irrational raised to a transcendental be rational? > > Can a transcendental raised to an algebraic irrational be an algebraic > irrational?
((sqr sqr 2)^sqr 2)^sqr 2
> Can a transcendental raised to an algebraic irrational be rational?
((sqr 2)^sqr 2)^sqr 2
> Can a transcendental raised to a transcendental be an algebraic irrational? > > Can a transcendental raised to a transcendental be rational? > > > > Reminders: > Reals only, everywhere > Where types are alike, there's no requirement the power and base must be the > same number > > > > >



