Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

Topic: Simple question about irrationals, with a short note in the margin. :-)
Replies: 30   Last Post: Jan 15, 2014 6:17 PM

 Messages: [ Previous | Next ]
 William Elliot Posts: 2,637 Registered: 1/8/12
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?

> >
> >> 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 non-zero rational or an algebraic irrational
> and b is an algebraic irrational
>
> "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 Gelfond-Schneider :
>
>
> 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 follow-up 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, Gelfond-Schneider 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
>
>
>
>
>

Date Subject Author
1/13/14 Port563
1/13/14 William Elliot
1/14/14 Port563
1/14/14 William Elliot
1/14/14 Port563
1/14/14 William Elliot
1/14/14 Port563
1/15/14 William Elliot
1/15/14 Port563
1/15/14 Port563
1/15/14 Port563
1/15/14 William Elliot
1/15/14 Port563
1/15/14 albrecht
1/15/14 Port563
1/15/14 Port563
1/15/14 Port563
1/15/14 quasi
1/15/14 Port563
1/15/14 quasi
1/15/14 Port563
1/15/14 quasi
1/15/14 quasi
1/15/14 Port563
1/15/14 quasi
1/15/14 Port563
1/15/14 quasi
1/15/14 quasi
1/15/14 Port563
1/15/14 quasi
1/15/14 quasi