Proof by Contradiction: Prove It by Ruling Out the Opposite

Here is a detective trick for proving something is true: pretend the OPPOSITE is true, then show it leads to something impossible. To prove Sam did not eat the last cookie, suppose he DID. But Sam was at school all day, far from the kitchen. That is impossible! So Sam could not have eaten it. Mathematicians call this trick proof by contradiction.

You will learn the three steps of a proof by contradiction. You will see how finding an impossible result proves your point. You will follow a famous example: showing there is no biggest number.

Step 1: Assume the opposite of what you want to prove. Step 2: Follow that assumption carefully until it crashes into something impossible or silly. Step 3: Because the opposite cannot be true, the thing you wanted to prove must be true after all.

Some people think there must be a biggest number somewhere. Let's prove there is not. Step 1: Assume there IS a biggest number, and call it B. Step 2: Now add one to it: B plus 1. That is even bigger than B! So B was not really the biggest after all. That is our contradiction. Step 3: Since assuming a biggest number led to nonsense, there is no biggest number. You can always add one more.

A clear statement is either true or false, with nothing in between. If assuming it is false leads straight to something impossible, then it cannot be false. The only choice left is that it is true. People have used this powerful trick for more than 2,000 years, all the way back to the ancient Greek mathematicians.