Fermat's Last Theorem (40 page)

The brackets can be expanded and simplified. Therefore,

The 2
xy
can be cancelled from both sides. So we have

which is Pythagoras' theorem!

The argument is based on the fact that the area of the large square must be the same no matter what method is used to calculate it. We then logically derive two expressions for the same area, make them equivalent, and eventually the inevitable conclusion is that
x
²
+
y
²
=
z
²
, i.e. the square on the hypotenuse,
z
²
, is equal to the sum of the squares on the other two sides,
x
²
+
y
²
.

This argument holds true for all right-angled triangles. The sides of the triangle in our argument are represented by
x
,
y
and
z
, and can therefore represent the sides of any right-angled triangle.

Euclid's aim was to prove that √2 could not be written as a fraction. Because he was using proof by contradiction, the first step was to assume that the opposite was true, that is to say, that √2 could be written as some unknown fraction. This hypothetical fraction is represented by
p
⁄
q
, where
p
and
q
are two whole numbers.

Before embarking on the proof itself, all that is required is a basic understanding of some properties of fractions and even numbers.

(1) If you take any number and multiply it by 2, then the new number must be even. This is virtually the definition of an even number.

(2) If you know that the square of a number is even, then the number itself must also be even.

(3) Finally, fractions can be simplified:
¹⁶
⁄
₂₄
is the same as
⁸
⁄
₁₂
; just divide the top and bottom of
¹⁶
⁄
₂₄
by the common factor 2. Furthermore,
⁸
⁄
₁₂
is the same as
⁴
⁄
₆
, and in turn
⁴
⁄
₆
is the same as
²
⁄
₃
. However,
²
⁄
₃
, cannot be simplified any further because 2 and 3 have no common factors. It is impossible to keep on simplifying a fraction forever.

Now, remember that Euclid believes that √2 cannot be written as a fraction. However, because he adopts the method of proof by contradiction, he works on the assumption that the fraction
p
⁄
q
does exist and then he explores the consequences of its existence:

If we square both sides, then

This equation can easily be rearranged to give

Now from point (1) we know that
p
²
must be even. Furthermore, from point (2) we know
p
itself must also be even. But if
p
is even, then it can be written as 2
m
, where
m
is some other whole number. This follows from point (1). Plug this back into the equation and we get

Divide both sides by 2, and we get

But by the same arguments we used before, we know that
q
²
must be even, and so
q
itself must also be even. If this is the case, then
q
can be written as 2
n
, where
n
is some other whole number. If we go back to the beginning, then

The 2
m
⁄2
n
can be simplified by dividing top and bottom by 2, and we get

We now have a fraction
m
⁄
n
, which is simpler than
p
⁄
q
.

However, we now find ourselves in a position whereby we can repeat exactly the same process on
m
⁄
n
, and at the end of it we will generate an even simpler fraction, say
g
⁄
h.
This fraction can then be put through the mill again, and the new fraction, say
e
⁄
f
, will be simpler still. We can put this through the mill again, and repeat the process over and over again, with no end. But we know from point (3) that fractions cannot be simplified forever. There must always be a simplest fraction, but our original hypothetical fraction
p
⁄
q
does not seem to obey this rule. Therefore, we can justifiably say that we have reached a contradiction. If √2 could be written as a fraction the consequence would be absurd, and so it is true to say that √2 cannot be written as a fraction. Therefore √2 is an irrational number.

Let us call the length of Diophantus' life
L.
From the riddle we have a complete account of Diophantus' life which is as follows:

¹
⁄
₆
of his life,
L
⁄
₆
, was spent as a boy,

L
⁄
₁₂
was spent as a youth,

L
⁄
₇
was then spent prior to marriage,