Proofs
It is never enough to give lots of examples in maths if you want to prove something. To disprove something you need only give a counterexample, but a proof must be proof positive, conclusive, decisive.
For example, suppose you want to prove something simple and intuitive – that the product of two even numbers is an even number. You cannot say simply that 2*4=8 therefore proved. A proper proof would go like:
Take two even numbers. Since they are even they can be written as
and
where
and
are whole numbers. Then
Sinceandare whole numbers so is
hence the product of any two even numbers is even.
Often you are asked to give either proofs or counter examples. For example:
Prove that the difference between two prime numbers is even.
Numbers can be odd or even There are 4 cases to consider
|
|
|
odd | odd | even |
odd | Even | odd |
even | odd | odd |
even | even | even |
Suppose then that we pick an odd prime number -7 – and an even prime number – the only possibility being 2. 7-2=5 which is odd therefore the statement is disproved.
Prove that the difference of any two square numbers is odd:
The counterexample is easy. 25 and 9 are both square numbers but 25-9=16 which is even.
If the question had been, “Prove that the difference of any two old square numbers is even”, then the proof would have been much trickier:
Since the square numbers are odd, we can write them as
and
since if the original number were even, the square number would be even.
We recognise this as a difference of squares which factorises in this way:
which is divisible by 4 hence even.