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Math Help - If 'a' is rational and 'b' is irrational is 'ab' neccesarrily irrational?

  1. #1
    Member mfetch22's Avatar
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    If 'a' is rational and 'b' is irrational is 'ab' neccesarrily irrational?

    So, this is a question in my textbook. I'm sure it's a very simple answer, that's right under my nose, but that I'm just missing for some reason. Anyway, if we have two numbers a and b and a is rational, but b is irrational, then is ab neccesarily irrational? Please provide a proof with your answer, thanks in advance.
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  2. #2
    MHF Contributor Also sprach Zarathustra's Avatar
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    Suppose a*b=t (rational!)

    a rational so a=p/q

    t is rational so t=m/n

    Hence, ab=m/n ==> b=m/n * q/p ==> b rational! A contradiction!
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    Unless a = 0...
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    And a single counterexample is sufficient to prove a general statement false!
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