Proof By Contradiction Explained in 5 Minutes

Proof By Contradiction Explained in 5 Minutes

Proof by contradiction is a method of proving mathematical statements. It's not the only method of course, but it's a very powerful technique.

Let's use it to prove a famous mathematical statement..

"There are an infinite amount of prime numbers."

Here's how proof by contradiction works..

First assume the statement is false. Therefore there must be a finite number of primes, say m of them, and we can write them as..


Now consider the number n..


This number cannot be divided by any of our primes, so it must be a new prime which is a contradiction. In fact it's absurd!

So assuming the original statement was false led to a contradiction, so the original statement must be true. In other words we've proved it.

Proof by contradiction is often used in cases where a regular proof would be extremely difficult. In fact, try this simple test the next time you're talking to a mathematician..

Say to them, "Do you know how to prove that the square root of 2 is irrational?" They will smile knowingly, because there's a famous proof of this that uses proof by contradiction. Allow them to smile and then add, "Of course, not using proof by contradiction". Their smile will immediately vanish!

Sometimes proof by contradiction is referred to by its Latin name "reductio ad absurdum" which has the wonderful translation "reduction to absurdity".

Content written and posted by Ken Abbott