Let’s start by being sceptical – this is a method of proving a statement is
incorrect. We’ll take a specific statement as an example. Suppose you hear a
claim that any number multiplied by itself results in an even number. Do you
believe this? Before jumping in with an answer we should try a few examples. If
we have a number, say 6, and multiply it by itself to get 6 × 6 = 36 we find that
indeed 36 is an even number. But one swallow does not make a summer. The
claim was for any number, and there are an infinity of these. To get a feel for the
problem we should try some more examples. Trying 9, say, we find that 9 × 9 =
- But 81 is an odd number. This means that the statement that all numbers
when multiplied by themselves give an even number is false. Such an example
runs counter to the original claim and is called a counterexample. A
counterexample to the claim that ‘all swans are white’, would be to see one black
swan. Part of the fun of mathematics is seeking out a counterexample to shoot
down a would-be theorem.
If we fail to find a counterexample we might feel that the statement is correct.
Then the mathematician has to play a different game. A proof has to be
constructed and the most straightforward kind is the direct method of proof.
The direct method
In the direct method we march forward with logical argument from what is
already established, or has been assumed, to the conclusion. If we can do this we
have a theorem. We cannot prove that multiplying any number by itself results in
an even number because we have already disproved it. But we may be able to
salvage something. The difference between our first example, 6, and the
counterexample, 9, is that the first number is even and the counterexample is
odd. Changing the hypothesis is something we can do. Our new statement is: if
we multiply an even number by itself the result is an even number.
First we try some other numerical examples and we find this statement
verified every time and we just cannot find a counterexample. Changing tack we
try to prove it, but how can we start? We could begin with a general even
number n, but as this looks a bit abstract we’ll see how a proof might go by
looking at a concrete number, say 6. As you know, an even number is one which
is a multiple of 2, that is 6 = 2 ×3. As 6 × 6 = 6 + 6 + 6 + 6 + 6 + 6 or,
written another way, 6 × 6 = 2 × 3 + 2 × 3 + 2 × 3 + 2 × 3 + 2 × 3 + 2 × 3