or, rewriting using brackets,
6 × 6 = 2 × (3 + 3 + 3 + 3 + 3 + 3)
This means 6 × 6 is a multiple of 2 and, as such, is an even number. But in
this argument there is nothing which is particular to 6, and we could have started
with n = 2 × k to obtain
n × n = 2 × (k + k +... + k)
and conclude that n × n is even. Our proof is now complete. In translating
Euclid’s Elements, latter-day mathematicians wrote ‘QED’ at the end of a proof to
say job done – it’s an abbreviation for the Latin quod erat demonstrandum
(which was to be demonstrated). Nowadays they use a filled-in square. This is
called a halmos after Paul Halmos who introduced it.
The indirect method
In this method we pretend the conclusion is false and by a logical argument
demonstrate that this contradicts the hypothesis. Let’s prove the previous result
by this method.
Our hypothesis is that n is even and we’ll pretend n × n is odd. We can write n
× n = n + n +... + n and there are n of these. This means n cannot be even
(because if it were n × n would be even). Thus n is odd, which contradicts the
hypothesis.
This is actually a mild form of the indirect method. The full-strength indirect
method is known as the method of reductio ad absurdum (reduction to the
absurd), and was much loved by the Greeks. In the academy in Athens, Socrates
and Plato loved to prove a debating point by wrapping up their opponents in a
mesh of contradiction and out of it would be the point they were trying to prove.
The classical proof that the square root of 2 is an irrational number is one of this
form where we start off by assuming the square root of 2 is a rational number
and deriving a contradiction to this assumption.
The method of mathematical induction
Mathematical induction is powerful way of demonstrating that a sequence of
statements P 1 , P 2 , P 3 ,... are all true. This was recognized by Augustus De
Morgan in the 1830s who formalized what had been known for hundreds of