Proof by Contradiction

A contradiction is where one statement is the opposite of another.

Alex: "You were at the beach yesterday."

Sam: "I was not at the beach yesterday!"

Alex and Sam's statements contradict each other.

Proof by Contradiction

In a proof by contradiction, the contrary (opposite) is assumed to be true at the start of the proof.

After logical reasoning at each step, the assumption is shown not to be true.

Example: Prove that you can't always win at chess


Let us start with the contrary: you can always win at chess.

I now set up a board, let you take the first move, then turn the board around and let you take the opponent's move.

Both sides cannot win, and you are playing both sides!

So the idea that you can always win is absurd, and I have proven that you cannot always win.

 OK, a little bit tricky, but you get the idea.

A proof by contradiction is also known as "reductio ad absurdum" which is the Latin phrase for reducing something to an absurd (silly or foolish) conclusion.

Proof That √2 is an Irrational Number

Euclid proved that √2 (the square root of 2) is an irrational number by first assuming the opposite. This is one of the most famous proofs by contradiction. Let's take a look at the steps.

First Euclid assumed √2 was a rational number.

A rational number  is a number that can be in the form p/q where p and q are integers and q is not zero.

He then went on to show that in the form p/q it can always be simplified.

But we can't go on simplifying an integer ratio forever, so there is a contradiction.

So √2 must be an irrational number.

Find out more at Euclid's Proof that √2 is Irrational.

Real Numbers

Another famous proof by contradiction was given by Georg Cantor when he showed the Real Numbers to be countable.