Indirect ProofsDate: 01/30/97 at 20:21:52 From: M.Quinn Subject: proof problems For the following statement, give a proof if the statement is true, or a counterexample (with explanation) if the statement is false: If r is any nonzero rational number, and s is any irrational number, then r/s is irrational. I think this is true, but I can't prove it. I know s must be an integer and an integer isn't irrational Am I going the right way? Date: 01/31/97 at 11:15:50 From: Doctor Wilkinson Subject: Re: proof problems Well, so far so good. You're correct that the statement is true. Let's try to figure out a proof. "Irrational" is a negative concept. That is, a number is irrational if it's NOT the quotient of two integers, so you typically have to use an "indirect" proof. That means, assume the number is rational and show that that assumption leads you to something you know is false. So suppose r/s is rational. That means r/s = m/n, where m and n are integers. Let's multiply by ns to get rid of the fractions. That gives us rn = ms. But now what we're really interested in is s. So let's divide both sides by m. (We know we can do this because if m were zero, r would be zero: that's what that extra hypothesis was for!). This gives us: s = rn/m r is rational, n and m are integers, so that makes s rational. But we know it isn't. Contradiction! So our original assumption was wrong, and r/s is irrational. Do you see how this works? This is a typical indirect proof. I hope this helps a little. You seem to be on the right track. -Doctor Wilkinson, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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