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Math Help - Proof by Contradiction

  1. #1
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    Proof by Contradiction

    I'm on one my last Math problems of the assignment, and I can't seem to figure out where to start or where to go with it. The problem is:
    -Use Proof by contradition to prove the following:
    Assume p is rational and q is irrational. Prove if p =/= 0, then p*q is irrational.

    Any help would be appreciated, thanks
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  2. #2
    Super Member Matt Westwood's Avatar
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    Quote Originally Posted by kallivis View Post
    I'm on one my last Math problems of the assignment, and I can't seem to figure out where to start or where to go with it. The problem is:
    -Use Proof by contradition to prove the following:
    Assume p is rational and q is irrational. Prove if p =/= 0, then p*q is irrational.

    Any help would be appreciated, thanks
    The way I'd look at it is:

    If pq were rational it could be expressed in the form s/t where s and t are integers.

    Now express p in the same form (m/n, say) and multiply pq by 1/p.

    Continue from there.
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  3. #3
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    I appreciate the assistance, so I put in what you said and now have
    q= s/tq which should mean p*q= s^2/t^2q, but wouldn't that mean since that is a ratio that p*q is a rational nymber would be true, therefore not giving a contradition? Or am I missing something still?
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  4. #4
    Super Member Matt Westwood's Avatar
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    Quote Originally Posted by kallivis View Post
    I appreciate the assistance, so I put in what you said and now have
    q= s/tq which should mean p*q= s^2/t^2q, but wouldn't that mean since that is a ratio that p*q is a rational nymber would be true, therefore not giving a contradition? Or am I missing something still?
    I don't think you read my post correctly.

    I said: "... express pq in the form s/t" not "express p in the form s/t".

    Look, it goes a bit like this.

    You want to show that pq is irrational if p is rational and q irrational. So what we're going to do is say: if pq were rational, then we would show that q also has to be rational, which contradicts our premise that q is irrational, hence demonstrating a contradiction.

    So, suppose pq = s/t, p = m/n.

    Then q = 1/p (s/t) = (n/m)(s/t) = (ns) / (mt).

    But m,n,s,t are all integers so ns and mt are integers ...

    ... which means ...?
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  5. #5
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    Have you already proved that the rational numbers are closed under multiplication? If so, you can use a proof by contradiction. Suppose, with p rational, not equal to 0 and q irrational, pq = r is a rational number, then p= r(1/q) is a product of rational numbers.
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