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Math Help - TA’s Challenge Problem #7

  1. #1
    Senior Member TheAbstractionist's Avatar
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    TA’s Challenge Problem #7

    “Prove” that 1 is transcendental.

    “Proof”:

    The Gelfond–Schneider theorem states that if \alpha and \beta are algebraic numbers such that \alpha\ne0,\,1 and \beta\notin\mathbb Q, then \alpha^\beta is a transcendental number.

    Hence, taking \alpha=-1 and \beta=\sqrt2, we have that (-1)^{\sqrt2} is transcendental.

    But (-1)^{\sqrt2}=(-1)^{2\frac{\sqrt2}2}=\left((-1)^2\right)^{\frac{\sqrt2}2}=1^{\frac{\sqrt2}2}=1.

    Hence 1 is transcendental.

    Explain what is wrong with the above “proof”.
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  2. #2
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    I guess maybe a couple of things are wrong with it, but for one, the proof manipulates things so that \alpha=1, and then attempts to apply Gel'fond-Schneider, contrary to hypothesis.
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  3. #3
    Senior Member TheAbstractionist's Avatar
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    You are on the right track, but you need to be more specific about what things are being “manipulated”.
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  4. #4
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    Okay, well... the correct way to evaluate (-1)^{\sqrt{2}} would probably be to use the rule z^\alpha=\exp(\alpha\log z), which I believe has infinitely many values when \alpha is not a rational real number.... So the expression is multi-valued, and so we can't apply Gel'fond-Schneider?
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  5. #5
    Senior Member TheAbstractionist's Avatar
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    Gelfond–Schneider certainly applies in this case.

    But you’ve the nail on the head.
    Spoiler:
    (-1)^{\sqrt2} is not a real number. It is complex-valued, and for complex multiplication, the exponent laws for real numbers break down – e.g. \left[(-1)(-1)\right]^{1/2}\ne(-1)^{1/2}(-1)^{1/2}.
    Last edited by TheAbstractionist; August 7th 2009 at 05:39 AM.
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  6. #6
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    Quote Originally Posted by TheAbstractionist View Post
    Gelfond–Schneider certainly applies in this case.

    But you’ve the nail on the head.
    Spoiler:
    (-1)^{\sqrt2} is not a real number. It is complex-valued, and for complex multiplication, the exponent laws for real numbers break down – e.g. \left[(-1)(-1)\right]^{1/2}\ne(-1)^{1/2}(-1)^{1/2}.
    Heh... I think you've been far too generous in crediting me with providing a solution to this. My answers were clumsy all along. What I ought to have said was that Gel'fond-Schneider cannot be applied in the way that it was being applied in the OP. I could tell that something was wrong with the way the exponent was being taken, but I wasn't sure exactly what, because I've always done these "by the book" using logs, and so to be honest, I wasn't really sure where the problem was in the OP.

    Which is sort of embarrassing to me, because I am supposed to be a specialist in transcendental number theory....
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