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Math Help - Analysis

  1. #1
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    Analysis

    1. If  r is rational (  r \neq 0 ) and  x is irrational, prove that  r+x and  rx are irrational. So  r = \frac{p}{q} where  p,q \in \bold{Z} . So then  \frac{p}{q} + x and  \frac{p}{q}x somehow have to be irrational.

    2. Prove that there is no rational number whose square is  12 . So maybe use a proof by contradiction (i.e. assume that  \frac{p^{2}}{q^{2}} = 12 )?
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  2. #2
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    Say that r + x = s where s is rational then x = s - r.
    Do you see the contradiction?
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  3. #3
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    Because than  x is rational which is a contradiction.
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  4. #4
    Super Member wingless's Avatar
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    Quote Originally Posted by heathrowjohnny View Post
    2. Prove that there is no rational number whose square is  12 . So maybe use a proof by contradiction (i.e. assume that  \frac{p^{2}}{q^{2}} = 12 )?
    You're right about \frac{p^{2}}{q^{2}} = 12

    \frac{p^{2}}{q^{2}} = 12

    \left| \frac{p}{q} \right| = \sqrt{12}

    \frac{p}{q} = \sqrt{12},~~ \frac{p}{q} = -\sqrt{12}

    \sqrt{12} and -\sqrt{12} are irrational. Irrational numbers cannot be expressed as rational numbers. So there's no such \frac{p}{q}

    Well, we could also start with " a is a rational number" instead of \frac{p}{q}.
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