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Math Help - Need help proving by contraposition

  1. #1
    Dko
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    Need help proving by contraposition

    Hello,

    I have a home work question that asks me to prove by contraposition that for all positive real numbers that if x is irrational, then x^1/2 is also irrational.

    So far I have the Contrapositive of for all positive real numbers, x^1/2 is rational then x is rational. Which means (x^1/2)/1. But this is where I am stuck. I don't know what to do with it beyond that.

    Can anyone give me even just a hint? Thank you in advance.
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  2. #2
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    Quote Originally Posted by Dko View Post
    Hello,

    I have a home work question that asks me to prove by contraposition that for all positive real numbers that if x is irrational, then x^1/2 is also irrational.

    So far I have the Contrapositive of for all positive real numbers, x^1/2 is rational then x is rational. Which means (x^1/2)/1. But this is where I am stuck. I don't know what to do with it beyond that.

    Can anyone give me even just a hint? Thank you in advance.
    The contrapositive of the statement is:

    "If \sqrt{x} is rational then x is rational."


    Now set up your arguments.

    If \sqrt{x} is rational, it can be written as a fraction of integers.

    \sqrt{x} = \frac{a}{b}

    (\sqrt{x})^2 = \left(\frac{a}{b}\right)^2

    x = \frac{a^2}{b^2}, another rational number.
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