18 I The Expanding Universe of Numbers
But it was proved, probably by a disciple of Pythagoras, that there is no rational
numberxsuch thatx^2 =2. (A more general result is proved in Book X, Propo-
sition 9 of Euclid’sElements.) We give here a somewhat different proof from the
classical one.
Assume that such a rational numberxexists. Sincexmay be replaced by−x,we
may suppose thatx=m/n,wherem,n∈N.Thenm^2 = 2 n^2. Among all pairsm,n
of positive integers with this property, there exists one for whichnis least. If we put
p= 2 n−m, q=m−n,thenpandqare positive integers, since clearlyn<m< 2 n.But
p^2 = 4 n^2 − 4 mn+m^2 = 2 (m^2 − 2 mn+n^2 )= 2 q^2.Sinceq<n, this contradicts the minimality ofn.
If we think of the rational numbers as measuring distances of points on a line from
a given originOon the line (with distances on one side ofOpositive and distances on
the other side negative), this means that, even though a dense set of points is obtained
in this way, not all points of the line are accounted for. In order to fill in the gaps the
concept of number will now be extendedfrom ‘rational number’ to ‘real number’.
It is possible to define real numbers as infinite decimal expansions, the rational
numbers being those whose decimal expansions are eventually periodic. However, the
choice of base 10 is arbitrary and carrying through this approach is awkward.
There are two other commonly used approaches, one based onorderand the other
ondistance. The first was proposed by Dedekind (1872), the second by M ́eray (1869)
and Cantor (1872). We will follow Dedekind’s approach, since it is conceptually sim-
pler. However, the second method is also important and in a sense more general. In
Chapter VI we will use it to extend the rational numbers to thep-adic numbers.
It is convenient to carry out Dedekind’s construction in two stages. We will first
define ‘cuts’ (which are just the positive real numbers), and then pass from cuts to
arbitrary real numbers in the same way that we passed from the natural numbers to the
integers.
Intuitively, a cut is the set of all rational numbers which represent points of the line
between the originOand some other point. More formally, we define acutto be a
nonempty proper subsetAof the setPof all positive rational numbers such that
(i)if a∈A,b∈P and b<a,then b∈A;
(ii)if a∈A,then there exists a′∈A such that a<a′.
For example, the setIof all positive rational numbersa<1 is a cut. Similarly, the
setTof all positive rational numbersasuch thata^2 <2 is a cut. We will denote the
set of all cuts byP.
For anyA,B∈Pwe writeA<BifAis a proper subset ofB. We will show that
this induces atotal orderonP.
It is clear that ifA<BandB<C,thenA<C. It remains to show that, for any
A,B∈P, one and only one of the following alternatives holds:
A<B, A=B, B<A.