Proof By Contradiction
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 thatand thatis rational.
We can then write
where
and
are integers, so the original equation becomes
Raising both sides to the power ofgives 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, 'andis rational' is false.
Example: Prove that
is irrational.
Suppose thatis 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.