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Math Help - rational irrational expressions

  1. #1
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    rational irrational expressions

    Okay I feal so stupid because I cannot figure this out but I am just brain dead and this homework is due in a couple hours so some help would be greatly appreciated...

    Note: A real number m is rational iff m=p/q, were p and q are integers and q is not equal to zero.

    problem:

    Suppose x and y are real numbers, prove or give a counterexample.

    a) if x is irrational and y is irrational, then x + y is irrational
    b) if x + y is irrational, then x is irrational or y is irrational
    c) if x is irrational and y is irrational, then xy is irrational
    d) if xy is irrational, then x is irrational or y is irrational.
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  2. #2
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    Quote Originally Posted by luckyc1423 View Post

    a) if x is irrational and y is irrational, then x + y is irrational
    No consider. x=\pi and y=-\pi.
    b) if x + y is irrational, then x is irrational or y is irrational
    Assume that "x is irrational or y is irrational" is false. Meaning "x is rational and y is rational" (if you learned logic this is called de'Morgan's law of negation).
    Then, x+y is rational. A contradiction because x+y is irrational. Thus, this statement is true.
    c) if x is irrational and y is irrational, then xy is irrational
    No. Consider x=\sqrt{2} and y=\frac{1}{\sqrt{2}}.
    d) if xy is irrational, then x is irrational or y is irrational.
    I will leave this as an excerise.
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