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Proof Involving Rational and Irrational Numbers

Date: 09/13/2004 at 10:23:43
From: John
Subject: Proving existence of irrational a, b, where a^b is rational

The question I'm stuck on is:

Prove the existence of 2 irrational numbers, a and b, where a^b is
rational.

Thanks for any help!



Date: 09/13/2004 at 10:50:35
From: Doctor Luis
Subject: Re: Proving existence of irrational a, b, where a^b is rational

Hi John,

Good question!  This has been considered before, and there's a very 
interesting (if not famous) proof of this result.  The proof is 
interesting in the sense that it is non-constructive, i.e. we never 
actually find specific irrational values a,b that are both decidedly
irrational.  Nonetheless, we can still prove the theorem.

We start by noting that sqrt(2) is irrational and that 2 is rational.

Next, we define the number x = sqrt(2)^sqrt(2).

By logic (law of excluded middle), we can confidently say that our 
number x is either rational or it is irrational.

So, we consider each of the two possibilities:

 1) Say x is rational.  Then we are done!  We have found
    two values a = sqrt(2), b = sqrt(2) such that a^b is rational.
    The theorem is proven, at least for this possibility.

 2) Say x is irrational.  Then we are done, too!  Take 
    a = x = sqrt(2)^sqrt(2), and b = sqrt(2).  Then a^b = 2
    is rational.  The theorem is proven for the second possibility.

Therefore the theorem is true because we have exhausted the two
possibilities.  The interesting part is that it doesn't matter whether 
x is rational or irrational!  Either way, the theorem is proven.

Incidentally, you can actually PROVE that x = sqrt(2)^sqrt(2) is 
irrational by using a result known as Gelfond's theorem.  You can read 
more about it if you follow these URLs:

  Gelfond's Theorem
    http://mathworld.wolfram.com/GelfondsTheorem.html 

  Gelfond-Schneider Constant: 2^sqrt(2)
    http://mathworld.wolfram.com/Gelfond-SchneiderConstant.html 

However, our little proof is quite elegant, albeit indirect.

Let us know if you have any other thought-provoking questions.

- Doctor Luis, The Math Forum
  http://mathforum.org/dr.math/ 
Associated Topics:
High School Analysis
High School Sets

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