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### 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/
```
Associated Topics:
College Number Theory
High School Number Theory

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