Proof by Contradiction: A Monkey's UncleDate: 12/09/98 at 01:36:43 From: Paul Schmidt Subject: Proof by contradiction I don't completely understand how to verify a mathematical fact by using a proof by contradiction. I think that this has something to do with proving that an assumed fact is not true, but I'm not exactly sure how to do it. Can you please explain the various steps involved in this type of proof, and if possible provide a simple example? Thank you very much! - Paul Schmidt Date: 12/09/98 at 02:48:26 From: Doctor Mike Subject: Re: Proof by contradiction Hi Paul, Have you ever heard something like "If ... is true, then I'm a monkey's uncle"? That is like proof by contradiction. For this technique, you (1) Assume the opposite of what you want to prove is true, (2) logically use that to come up with something that is obviously false, and (3) say "I must have been wrong about what I assumed above." Put another way, you assume the fact you are trying to prove is false, come to the conclusion that you are the uncle of a monkey (or something equally absurd), which shows that the fact really was true in the first place. The most well known fact or theorem that us usually used to illustrate this kind of proof is that the square root of a prime number (like 2 or 3 or 5) is irrational, i.e., that it cannot be written as a fraction of 2 integers M/N. You can find those proofs in the Dr. Math archives at http://mathforum.org/dr.math/ Select High School, and then look for Square and Cube Roots. There are several answers for you to read; for example, you could try: Proof that Sqrt(3) is Irrational http://mathforum.org/dr.math/problems/gardner8.14.97.html Proof that Sqrt(2) is Irrational http://mathforum.org/dr.math/problems/tim.8.29.96.html I hope this helps. - Doctor Mike, The Math Forum http://mathforum.org/dr.math/ |
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