Proof That Product is Irrational

Date: 03/28/2001 at 15:55:10
From: Ellen
Subject: number theory

How can I prove that the product of a rational number and an 
irrational number is irrational without using specific examples?

Date: 03/28/2001 at 17:56:08
From: Doctor Douglas
Subject: Re: number theory

Hi Ellen, and thanks for writing to the Math Forum.

Actually, you need to specify that the rational number is nonzero, 
because in that case, the product of a zero and any number, irrational 
or not, is zero, which is rational.

Now, with this restriction, we want to show that the product of any 
nonzero rational and any irrational number is also irrational.

We can attempt to do this using a proof by contradiction:

Let R be any given rational number and S be any given irrational 
number. Because R is rational, R = p/q for some integers p,q. Then the 
product R*S = p*S/q. Is there any possibility that this product could 
be rational? If so, then

     p*S/q = u/v

for some pair of integers u and v. Then this equation says that 

     S = u*q / (v*p)

as long as v and p are nonzero, which means that S is rational 
(because it is the quotient of the two integers u*q and v*p). This 
contradicts a known assumption (S is in fact irrational). Thus we 
conclude that it is impossible that p*S/q is rational (again, assuming 
that p is nonzero), and therefore the product R*S is irrational.

I hope this helps. Please write back if you have more questions about 
this.

- Doctor Douglas, The Math Forum
  http://mathforum.org/dr.math/