The Math Forum

Ask Dr. Math - Questions and Answers from our Archives
Associated Topics || Dr. Math Home || Search Dr. Math

Indirect Proofs

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.  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!   
Associated Topics:
High School Number Theory

Search the Dr. Math Library:

Find items containing (put spaces between keywords):
Click only once for faster results:

[ Choose "whole words" when searching for a word like age.]

all keywords, in any order at least one, that exact phrase
parts of words whole words

Submit your own question to Dr. Math

[Privacy Policy] [Terms of Use]

Math Forum Home || Math Library || Quick Reference || Math Forum Search

Ask Dr. MathTM
© 1994- The Math Forum at NCTM. All rights reserved.