Results 1 to 5 of 5
Like Tree3Thanks
  • 2 Post By ebaines
  • 1 Post By SlipEternal

Math Help - Proof that there exists irrational x,y such that x^y is rational. Valid or not?

  1. #1
    Newbie
    Joined
    Sep 2013
    From
    USA
    Posts
    2

    Proof that there exists irrational x,y such that x^y is rational. Valid or not?

    Hi all. I was asked the following question in class and came up with a proof.
    The question is as to whether there exists irrational x, y such that xy ​is rational.
    I know the classic proof to show that it exists is to find irrational x, y such that it works.
    That involves a lot of guess and check, although it is obviously a valid proof.

    The proof I came up with is as follows:
    Let x=y=(1/2). Then for xx, (1/2)^(1/2) is not rational. Therefore, there are rational numbers x that map to irrational numbers xx.
    There are some irrational xx that are mapped to by rational x, and therefore, there cannot be a 1-1 correspondence between irrational x and irrational xx, since the size of the set of irrational numbers x is the same size as the set of irrational numbers xx.
    Therefore, there must be some irrational x that map to non-irrational xx. We are closed to the reals here, so numbers that are not irrational are rational.

    The professor says this is wrong, but she cannot really explain why. She just says "it doesn't work because we are dealing with infinite sets". I am in my first year of undergraduate studies, so perhaps my understanding is not developed fully yet.

    Is this proof valid? If not, can someone provide a better explanation as to why not?

    Thanks.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Member
    Joined
    Sep 2012
    From
    Planet Earth
    Posts
    195
    Thanks
    49

    Re: Proof that there exists irrational x,y such that x^y is rational. Valid or not?

    No, it isn't valid, unless I'm missing something here. But by your logic, there exists an irrational number whose square root is rational, just because there exists a rational number whose square root is irrational. Think about it, the logic is analogous, yet it is clearly false.

    You assumed that just because A is a proper subset of B, there cannot be a 1-1 correspondence between A and B, but this is only true for finite sets. For example, there IS a one-one correspondence between {all odd positive integers} and {all positive integers}. You can get it by listing out all the odd positive integers in order by size, and then each of them has an index, which is a positive integer.

    You should check out this article
    Hilbert's paradox of the Grand Hotel - Wikipedia, the free encyclopedia
    Last edited by SworD; September 26th 2013 at 10:23 PM.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Super Member ebaines's Avatar
    Joined
    Jun 2008
    From
    Illinois
    Posts
    901
    Thanks
    241

    Re: Proof that there exists irrational x,y such that x^y is rational. Valid or not?

    Why not simply use an existence proof - find an example of irrational values for x and y that satisfy the criteria that x^y is rational. One that comes to mind right away is

    e^{\ln (2)} = 2

    You would have to show that both e and  \ln (2) are irrational, but that's not too difficult.
    Thanks from topsquark and Shakarri
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Joined
    Nov 2010
    Posts
    1,378
    Thanks
    506

    Re: Proof that there exists irrational x,y such that x^y is rational. Valid or not?

    Quote Originally Posted by thomasmgill View Post
    Hi all. I was asked the following question in class and came up with a proof.
    The question is as to whether there exists irrational x, y such that xy ​is rational.
    I know the classic proof to show that it exists is to find irrational x, y such that it works.
    That involves a lot of guess and check, although it is obviously a valid proof.

    The proof I came up with is as follows:
    Let x=y=(1/2). Then for xx, (1/2)^(1/2) is not rational. Therefore, there are rational numbers x that map to irrational numbers xx.
    There are some irrational xx that are mapped to by rational x, and therefore, there cannot be a 1-1 correspondence between irrational x and irrational xx, since the size of the set of irrational numbers x is the same size as the set of irrational numbers xx.
    Therefore, there must be some irrational x that map to non-irrational xx. We are closed to the reals here, so numbers that are not irrational are rational.

    The professor says this is wrong, but she cannot really explain why. She just says "it doesn't work because we are dealing with infinite sets". I am in my first year of undergraduate studies, so perhaps my understanding is not developed fully yet.

    Is this proof valid? If not, can someone provide a better explanation as to why not?

    Thanks.
    You don't even have 1-1 correspondence among the rationals: 1^1 = 0^0 = 1. See the wikipedia article 0^0.
    Thanks from topsquark
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor

    Joined
    Apr 2005
    Posts
    14,973
    Thanks
    1121

    Re: Proof that there exists irrational x,y such that x^y is rational. Valid or not?

    You have been told why your proof is not valid. Consider, instead:

    Look at \sqrt{3}^{\sqrt{2}}. IF it is rational we are done! If it is NOT rational then what about that to the \sqrt{2} power?
    Last edited by HallsofIvy; September 30th 2013 at 02:38 PM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 3
    Last Post: July 19th 2010, 05:04 PM
  2. example of irrational + irrational = rational
    Posted in the Algebra Forum
    Replies: 3
    Last Post: April 4th 2010, 03:44 AM
  3. Replies: 2
    Last Post: January 31st 2010, 05:40 AM
  4. proof rational + irrational
    Posted in the Discrete Math Forum
    Replies: 6
    Last Post: May 25th 2009, 05:27 AM
  5. irrational between any two rational proof
    Posted in the Calculus Forum
    Replies: 2
    Last Post: March 15th 2008, 08:45 PM

Search Tags


/mathhelpforum @mathhelpforum