Chapter 1
The Real Numbers
1.1
Discussion: The Irrationality of
√
2
Toward the end of his distinguished career, the renowned British mathematician
G.H. Hardy eloquently laid out a justification for a life of studying mathematics
in
A Mathematician’s Apology
, an essay first published in 1940. At the center
of Hardy’s defense is the thesis that mathematics is an aesthetic discipline. For
Hardy, the applied mathematics of engineers and economists held little charm.
“Real mathematics,” as he referred to it, “must be justified as art if it can be
justified at all.”
To help make his point, Hardy includes two theorems from classical Greek
mathematics, which, in his opinion, possess an elusive kind of beauty that,
although di
ﬃ
cult to define, is easy to recognize.
The first of these results is
Euclid’s proof that there are an infinite number of prime numbers. The second
result is the discovery, attributed to the school of Pythagoras from around 500
B.C., that
√
2 is irrational. It is this second theorem that demands our attention.
(A course in number theory would focus on the first.) The argument uses only
arithmetic, but its depth and importance cannot be overstated. As Hardy says,
“[It] is a ‘simple’ theorem, simple both in idea and execution, but there is no
doubt at all about [it being] of the highest class. [It] is as fresh and significant
as when it was discovered—two thousand years have not written a wrinkle on
[it].”
Theorem 1.1.1.
There is no rational number whose square is 2.
Proof.
A rational number is any number that can be expressed in the form
p/q
,
where
p
and
q
are integers. Thus, what the theorem asserts is that no matter
how
p
and
q
are chosen, it is never the case that (
p/q
)
2
= 2. The line of attack
is indirect, using a type of argument referred to as a proof by contradiction.
The idea is to assume that there
is
a rational number whose square is 2 and
then proceed along logical lines until we reach a conclusion that is unacceptable.
At this point, we will be forced to retrace our steps and reject the erroneous
1

2
Chapter 1.
The Real Numbers
assumption that some rational number squared is equal to 2. In short, we will
prove that the theorem is true by demonstrating that it cannot be false.
And so assume, for contradiction, that there exist integers
p
and
q
satisfying
(1)
p
q
2
= 2
.
We may also assume that
p
and
q
have no common factor, because, if they had
one, we could simply cancel it out and rewrite the fraction in lowest terms. Now,
equation (1) implies
(2)
p
2
= 2
q
2
.
From this, we can see that the integer
p
2
is an even number (it is divisible by
2), and hence
p
must be even as well because the square of an odd number is
odd. This allows us to write
p
= 2
r
, where
r
is also an integer. If we substitute
2
r
for
p
in equation (2), then a little algebra yields the relationship
2
r
2
=
q
2
.

#### You've reached the end of your free preview.

Want to read all 34 pages?

- Spring '18
- Real Numbers, G.H. Hardy