1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Irrational number to an irrational power

  1. Jun 13, 2013 #1
    1. The problem statement, all variables and given/known data


    if a and b are irrational numbers, is a^b necessarily an irrational number ? prove it.




    3. The attempt at a solution
    this is an question i got from my first maths(real analysis) class (college) , and have to say, i have only little knowledge about rational number, i would like to give it a try to find a solution but need some help, i know that a^b is not necessarily an irrational number cause if a = √2^√2 and b = √2 it becomes rational.

    but i don't think its the proof that the question is expecting as its on real analysis. any help would be greatly appreciated and sorry for my English too.
     
  2. jcsd
  3. Jun 13, 2013 #2

    CAF123

    User Avatar
    Gold Member

    You cannot prove that statement since, as you have shown via a counterexample, it is false. The question might have said:
    'If a and b are irrational numbers, is ab necessarily irrational? Find a counterexample or prove it.​
     
  4. Jun 13, 2013 #3
    sorry for that mate it actually says to prove my answer , yes or no. i know its no but i have no idea how to prove it. :confused:
     
  5. Jun 13, 2013 #4

    Borek

    User Avatar

    Staff: Mentor

    Showing the counterexample you have proven the statement to be false. So the answer is "no" and the counterexample is the proof.
     
  6. Jun 13, 2013 #5

    collinsmark

    User Avatar
    Homework Helper
    Gold Member

    Assuming that one also starts with the proof that (√2)√2 is irrational.
     
  7. Jun 13, 2013 #6

    Borek

    User Avatar

    Staff: Mentor

    Good point :blushing:
     
  8. Jun 13, 2013 #7

    epenguin

    User Avatar
    Homework Helper
    Gold Member

    Don't need to prove that, since if it is rational we have found our counterexample. However if it is irrational, I don't see the proof that (√2)(√2)(√2) is rational.
     
  9. Jun 13, 2013 #8

    dextercioby

    User Avatar
    Science Advisor
    Homework Helper

    It's not the tower power of sqrt(2), rather

    $$\left(\sqrt{2}^{\sqrt{2}}\right) ~ ^{\sqrt{2}} $$

    which gives you the desired counterexample to the question in the problem, iff you manage to prove that the nr. inside the bracket is irrational.
     
    Last edited: Jun 13, 2013
  10. Jun 13, 2013 #9

    collinsmark

    User Avatar
    Homework Helper
    Gold Member

    Also a good point. :blushing:
     
  11. Jun 13, 2013 #10

    epenguin

    User Avatar
    Homework Helper
    Gold Member

    Independently of whether or not (√2)(√2)(√2) and ((√2)(√2))(√2) are the same thing. :biggrin:

    If you did, how would it follow that your expression here is rational?
     
    Last edited: Jun 13, 2013
  12. Jun 14, 2013 #11
    thanks guys , proving sqrt2 ^sqrt2 is irrational , need to try on that one . thanks for the help.
     
  13. Jun 14, 2013 #12

    Mentallic

    User Avatar
    Homework Helper

    [tex](\sqrt{2}^{\sqrt{2}})^{\sqrt{2}}=(\sqrt{2})^{\sqrt{2}\cdot \sqrt{2}}=(\sqrt{2})^2=2[/tex]
     
  14. Jun 14, 2013 #13

    Curious3141

    User Avatar
    Homework Helper

    That's a toughie. Look up the Gelfond-Schneider theorem. With the theorem, it's trivial, but proving that theorem is very hard.

    But as for the original question, this is a well-known proof that's often cited as a classic non-constructive proof. The proposition is that an irrational raised to an irrational power can be rational.

    So we consider ##x = \sqrt{2}^\sqrt{2}##. Either ##x## is rational or irrational. If it's the former, our work is done. If the latter, then let ##y = x^\sqrt{2} = \sqrt{2}^2 = 2##. In this case, we've taken an irrational (##x##), raised it to an irrational power (##\sqrt{2}##) and the result is rational. QED.

    It's a non-constructive proof because we didn't actually commit ourselves as to whether ##x## was rational or irrational. But we "covered our bases" either way.
     
  15. Jun 15, 2013 #14
    thanks mate, can any one recommend me some books for additional reading on rational numbers ( from the basics ) , it would be great .
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Irrational number to an irrational power
Loading...