It is often possible to prove a statement is false by assuming that it is true. If it is true, then it should be consistent with all the other mathematical theorems which have been proved to be true. If we can show that it is not consistent with one or more of these other statements then we have shown that it is false.

Disproving a statement in this way is called 'proof by contradiction'. Proof by contradiction can be a tricky skill to learn.

Example: Prove that if then is irrational.

Suppose that and that is rational.

We can then write where and are integers, so the original equation becomes Raising both sides to the power of gives us This means that amongs all the powers of 3, there is at least one power of 5, and amongst all the powers of 5 there is at least one power of 3. Both of these statements are false, so the statement, ' and is rational' is false.

Example: Prove that is irrational.

Suppose that is rational.

Write Raise 2 to the power of both sides to give then as before and amongst all the powers of 5 there is a power of 2 and amongst all the powers of 2 there is a power of 2.

Both these statements are false, so the statement ' is rational' is false. 