ELEMENTARY MATHEMATICS
W W L CHEN and X T DUONG
c
°
W W L Chen, X T Duong and Macquarie University, 1999.
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Chapter 1
BASIC ALGEBRA
1.1. The Real Number System
We shall assume that the reader has a knowledge of the real numbers. Some examples of real numbers
are 3, 1
/
2,
π
,
√
23 and
−
3
√
5. We shall denote the collection of all real numbers by
R
, and write
x
∈
R
to denote that
x
is a real number.
Among the real numbers are the collection
N
of all natural numbers and the collection
Z
of all
integers. These are given by
N
=
{
1
,
2
,
3
,...
}
and
Z
=
{
...,
−
3
,
−
2
,
−
1
,
0
,
1
,
2
,
3
}
.
Another subcollection of the real numbers is the collection
Q
of all rational numbers. To put it simply,
this is the collection of all fractions. Clearly we can write any fraction if we allow the numerator to
be any integer (positive, negative or zero) and insist that the denominator must be a positive integer.
Hence we have
Q
=
½
p
q
:
p
∈
Z
and
q
∈
N
¾
.
It can be shown that the collection
Q
contains precisely those numbers which have terminating or
repeating decimals in their decimal notations. For example,
3
4
=0
.
75
and
−
18
13
=
−
1
.
384615
are rational numbers. Here the digits with the overline repeat.
It can be shown that the number
√
2 is not a rational number. This is an example of a real number
which is not a rational number. Indeed, any real number which is not a rational number is called an
irrational number. It can be shown that any irrational number, when expressed in decimal notation, has
†
This chapter was written at Macquarie University in 1999.