Date: 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. B