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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.
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W W L Chen and X T Duong : Elementary Mathematics
nonterminating and nonrepeating decimals. The number
π
is an example of an irrational number. It
is known that
π
=3
.
14159
....
The digits do not terminate or repeat. In fact, a book was published some years ago giving the value of
the number
π
to many digits and nothing else–avery uninteresting book indeed! On the other extreme,
one of the states in the USA has a law which decrees that
π
= 3, no doubt causing a lot of problems for
those who have to measure the size of your block of land.
1.2. Arithmetic
In mathematics, we often have to perform some or all of the four major operations of arithmetic on
real numbers. These are addition (+), subtraction (
−
), multiplication (
×
) and division (
÷
). There are
simple rules and conventions which we need to observe.
SOME RULES OF ARITHMETIC.
(a) Operations within brackets are performed Frst.
(b) If there are no brackets to indicate priority, then multiplication and division take precedence over
addition and subtraction.
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