Proof That Product is IrrationalDate: 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/ |
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