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Math Help - Proving the irrationality of the sum of non perfect square roots?

  1. #1
    Member mfetch22's Avatar
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    Proving the irrationality of the sum of non perfect square roots?

    I know how to prove that the \sqrt{2} is irrational, and that \sqrt{3} is irrational. But how does one go about using the tricks involved in the \sqrt{2} proof on something like proving that P is irrational if:

    P = \sqrt{2} + \sqrt{3}

    ?

    Thanks in advance
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  2. #2
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    You can prove this by contradiction. Assume P is rational.

    then try squaring the equation

     P = \sqrt2  + \sqrt 3

    Use the facts that a square of a rational is a rational, and rational number addition and scalar multiplication is closed under the rational field.

    Fiddle around with the squared equation and you'll just have to prove  \sqrt 6 is irrational using those tricks you mentioned, which leads to a contradiction (a rational number = an irrational number)
    Last edited by Gusbob; September 30th 2010 at 04:44 PM.
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  3. #3
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    If I'm not missing something you have posted this before, and it has received a considerable attention in here.
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  4. #4
    Member mfetch22's Avatar
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    My bad, yes I must've posted it before. I had thought it was different problem. I wasn't sure. Wont happen again
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  5. #5
    Member mfetch22's Avatar
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    Oh, and thanks for the help Gusbob, I figured out the polynomial that represented the squared version and had no trouble proving the irrationality of 6^(1/2). That was a clever idea to use the square of the sum of the square roots. I hadn't thought of that before, I know I used it once along time ago, but I forgot about it. Thanks for the help.
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