Assume the opposite

The idea

In a proof by contradiction, we assume the opposite of what we want to prove and show that this assumption leads to a contradiction, so the assumption must be wrong.

A brilliant technique

Around 400 BCE, Hippasus of the Pythagorean school wanted to show that √2 is irrational. He used a brilliant technique called proof by contradiction.

What do you think “proof by contradiction” means?

• Assume what you want to prove and check it
• Assume the opposite and show that it leads to nonsense
• Argue with someone until they agree

Rain or no rain

Try the method on a simple claim. Put the steps in the right order.

Assume, contradict, conclude

In a proof by contradiction, we assume the opposite of what we want to prove and show that this assumption leads to a contradiction, so the assumption must be wrong.

To prove that √2 is irrational we assume the opposite: that √2 is rational. The next lesson fills in the steps.

Notes

In a proof by contradiction we assume the opposite of what we want to prove, show that the assumption leads to a contradiction, and conclude that it must be wrong.

Check yourself

In a proof by contradiction, what do we assume at the start?

The assumption leads to a contradiction. Our logical steps were flawless. What follows?

To prove by contradiction that √2 is irrational, what do we assume?

Which pair of statements is a contradiction?

• The opposite of what we want to prove — correct. Yes. If we can show the opposite is impossible, the claim must be true.
• What we want to prove. That would be assuming the answer. We assume the opposite instead.
• Nothing at all. The whole method starts by assuming the opposite.
• The claim we wanted to prove is false. The claim is what we wanted to prove. It is the opposite that was wrong.
• The assumption must be wrong — correct. Yes. The steps are fine, so the only thing that can be wrong is the assumption.
• We learn nothing. We learn a lot: the assumption is impossible.
• √2 is irrational. That is what we want to prove. We assume the opposite.
• √2 is a whole number. The opposite of irrational is rational, not whole.
• √2 is a rational number — correct. Yes. We assume the opposite of irrational.
• “p/q is in simplest form” and “p and q are both even” — correct. Yes. If both are even they share the factor 2, so p/q is not in simplest form.
• “p is even” and “p = 2k”. These agree with each other. An even number can be written as 2k.
• “q is an integer” and “q ≠ 0”. Both can be true at the same time, for example q = 5.