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Math Help - rational number well ordering principle.

  1. #1
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    rational number well ordering principle.

    rational number well ordering principle. thanks.
    Last edited by lemon721; February 8th 2010 at 09:27 PM.
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  2. #2
    MHF Contributor Bruno J.'s Avatar
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    For the first one, consider u=\sqrt{2}^{\sqrt{2}}. If u is rational, we are done. Otherwise, what can you say about u^{\sqrt{2}}?

    For the second one, where are you stuck?
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  3. #3
    MHF Contributor Drexel28's Avatar
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    For the first one, what about e^{\ln(2)}?
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    MHF Contributor Bruno J.'s Avatar
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    Sure, but assuming e is irrational is assuming much more than assuming \sqrt 2 is. Moreover, while I'm sure \ln 2 is irrational, I wouldn't know how to prove it. (Though I can show e is irrational!)
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  5. #5
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Bruno J. View Post
    Sure, but assuming e is irrational is assuming much more than assuming \sqrt 2 is. Moreover, while I'm sure \ln 2 is irrational, I wouldn't know how to prove it. (Though I can show e is irrational!)
    Suppose that \ln(2)=\frac{p}{q} then 2=e^{\frac{p}{q}}. You said you can prove that e^x is irrational for rational values, yes?
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  6. #6
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    just have no clue to start it ?
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  7. #7
    MHF Contributor Bruno J.'s Avatar
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    Quote Originally Posted by Drexel28 View Post
    Suppose that \ln(2)=\frac{p}{q} then 2=e^{\frac{p}{q}}. You said you can prove that e^x is irrational for rational values, yes?
    Yeah. I'm that tired!

    But, in any case, the OP is certainly not expected to come up with the proof of e's irrationality. The proof I outlined is perfectly good (and the archetype of a non-constructive proof)!
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    MHF Contributor Bruno J.'s Avatar
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    Quote Originally Posted by lemon721 View Post
    just have no clue to start it ?
    Did you try what I suggested?
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  9. #9
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    The first one I think u r right...
    I mean the second one...cannot handle the well ordering principle..
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