Proof by Contradiction: A Monkey's Uncle

Date: 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/