The aim of this chapter is to introduce axiomatically the set of Real numbers.W. Rudin Principles of Mathematical Analysis.Z+ with ba. In order to make this equation soluble we have to enlarge the set Z+
by introducing negative integers as unique solutions of the equations
a + x = 0 (existence of additive inverse)
28
CHAPTER 2.N. This necessitates adding {0}
to N, declaring 01, thereby obtaining the set of non-negative integers Z+.(Fundamental theorem of arithmetic) Every positive integer except 1
can be expressed uniquely as a product of primes.N. Then
(a
b
>
c
d
)
 (
adbc)
. The following theorem provides a very important property of rationals. Theorem 2.1.2. Between any two rational numbers there is another (and, hence, infinitely
many others).Z. In order to solve (2.1.1) (for a = 0) we have to enlarge our
system of numbers again so that it includes fractions b/a (existence of multiplicative inverse
in Z - {0}).Those of you familiar with basic concepts of algebra will find that axioms A.1 - A.11 charac terize R as an algebraic field.N. Our extended system, which is denoted by Z, now contains all integers and
can be arranged in order
Z = {.Hence, appealing to the Fundamental Theorem of
Arithmetic, p
2
is even, and hence p is even.Multiplying this equality by n
2 we obtain
m3
n = mn + 7n
2
, which is impossible since the right-hand side is an integer and the left-hand
side is not.2.2 The Field of Real Numbers
In the previous sections we discussed the need to extend N to Z, and Z to Q. The rigorous
construction of N can be found in a standard course on Set Theory.The last axiom links the operations of summation and multiplication. The set of rationals Q also forms an algebraic field (that is, the rational
numbers satisfy axioms A.1 - A.11).(order)
The first difficulty occurs when we try to come up with the additive analogue of a.1 = 1.a = a
for a ?Indeed, since b, d and m are positive we have
[a(b + md)b(a + mc)]  [madmbc]  (adbc),
and
[d(a + mc)c(b + md)]  (adbc).Suppose for a contradiction that the rational number p
q
(p ? Z, q ? N, in lowest terms)
is such that ( p
q
)
2 = 2.In this course we postulate the existence of the set of real numbers R as well as basic
properties summarized in a collection of axioms.R) [(ab)c = a(bc)] (associativity of multiplication).We also mention at this point the Fundamental theorem of arithmetic.Z. The equation
(2.1.1) ax = b
need not have a solution x ?Here hcf(p, q) stands for the highest common factor of p and q, so when writing p/q for a
rational we often assume that the numbers p and q have no common factor greater than 1.All the arithmetical operations in Q are straightforward.Q and consider the equation
(2.1.2) x
2 = a.
In general (2.1.2) does not have rational solutions.The last statement contradicts our assumption that p and q have
no common factor.The last theorem provides an example of a number which is not rational.No rational x satisfies the equation x
3 = x + 7.First we show that there are no integers satisfying the equation x
3 = x + 7.For a
contradiction suppose that there is. Then x(x + 1)(x - 1) = 7 from which it follows that x
divides 7.Direct verification shows that these numbers do not
satisfy the equation.Second, show that there are no fractions satisfying the equation x
3 = x+7.Theorem 2.1.1..1 = 1 ....{0} ?!!!!