FIRST YEAR CALCULUS
W W L CHEN
c
W W L Chen, 1982, 2005.
This chapter originates from material used by the author at Imperial College, University of London, between 1981 and 1990.
It is available free to all individuals, on the understanding that it is not to be used for financial gain,
and may be downloaded and/or photocopied, with or without permission from the author.
However, this document may not be kept on any information storage and retrieval system without permission
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Chapter 1
THE NUMBER SYSTEM
1.1. The Real Numbers
The purpose of the first four sections of this chapter is to discuss a number of the properties of the
real numbers. Most readers will be familiar with these properties, or have at least used most of them,
perhaps sometimes unaware of their generality. We do not propose to discuss here these properties in
great detail, and shall only give a brief introduction. Throughout, we denote the set of all real numbers
by
R
, and write
a
∈
R
to indicate that
a
is a real number.
The first collection of properties of
R
is generally known as the Field axioms. We offer no proof of
these properties, and simply treat and accept them as given.
FIELD AXIOMS.
(A1) For every
a, b
∈
R
, we have
a
+
b
∈
R
.
(A2) For every
a, b, c
∈
R
, we have
a
+ (
b
+
c
) = (
a
+
b
) +
c
.
(A3) For every
a
∈
R
, we have
a
+ 0 =
a
.
(A4) For every
a
∈
R
, there exists
−
a
∈
R
such that
a
+ (
−
a
) = 0
.
(A5) For every
a, b
∈
R
, we have
a
+
b
=
b
+
a
.
(M1) For every
a, b
∈
R
, we have
ab
∈
R
.
(M2) For every
a, b, c
∈
R
, we have
a
(
bc
) = (
ab
)
c
.
(M3) For every
a
∈
R
, we have
a
1 =
a
.
(M4) For every
a
∈
R
such that
a
= 0
, there exists
a
−
1
∈
R
such that
aa
−
1
= 1
.
(M5) For every
a, b
∈
R
, we have
ab
=
ba
.
(D) For every
a, b, c
∈
R
, we have
a
(
b
+
c
) =
ab
+
ac
.
Remark.
The properties (A1)–(A5) concern the operation addition, while the properties (M1)–(M5)
concern the operation multiplication. In the terminology of group theory, not usually covered in first
Chapter 1 : The Number System
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First Year Calculus
c
W W L Chen, 1982, 2005
year mathematics, we say that the set
R
forms an abelian group under addition, and that the set of all
nonzero real numbers forms an abelian group under multiplication. We also say that the set
R
forms a
field under addition and multiplication. The property (D) is called the Distributive law.
The set of all real numbers also possesses an ordering relation, so we have the Order axioms.
ORDER AXIOMS.
(O1) For every
a, b
∈
R
, exactly one of
a < b
,
a
=
b
,
a > b
holds.
(O2) For every
a, b, c
∈
R
satisfying
a > b
and
b > c
, we have
a > c
.
(O3) For every
a, b, c
∈
R
satisfying
a > b
, we have
a
+
c > b
+
c
.
(O4) For every
a, b, c
∈
R
satisfying
a > b
and
c >
0
, we have
ac > bc
.
Remark.
Clearly the Order axioms as given do not appear to include many other properties of the
real numbers. However, these can be deduced from the Field axioms and Order axioms. For example,
suppose that
x >
0. Then by (A4), we have
−
x
∈
R
and
x
+ (
−
x
) = 0. It follows from (O3) and (A3)
that 0 =
x
+ (
−
x
)
>
0 + (
−
x
) =
−
x
, giving
−
x <
0.
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 Math, Calculus, Real Numbers, Cos, Complex number, W W L Chen

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