1.2. The Real Number System as a Number Field
In this section, we will attempt to present the set of real numbers together with the defined
operations on the set as a
mathematical system
. By this, we mean, we are going to construct the
real numbers by use a minimum number of axioms (generally accepted propositions or
statements) and proceeding to deduce more statements (generally called theorems) that will
characterize the system. Since most statements will be about algebraic properties, it is natural to
call this system an
algebraic system
(or
algebraic structure
).
Numbers have several uses; naming, ordering, counting, and measuring, to name a few. In this
course, we shall be using numbers for counting and measuring. We have seen numbers used to
count the elements of a set, hence the term “counting numbers”. In some instances, we will be
using numbers to measure the length of a line segment or the area of a region; thus, the counting
numbers will not suffice for these purposes. These numbers will be collective called
real
numbers
and we shall denote by
R
the set of all real numbers.
First let us state the axioms that deal with equality of real numbers. These properties are so basic
that we sometimes take them for granted.
Axiom E1
.
Reflexive Property of Equality
For any a
∈
R,
a = a.
Axiom E2
.
Symmetric Property of Equality
For a, b
∈
R, if a = b then b = a.
Axiom E3
.
Transitive Property of Equality
For a, b, c
∈
R
, if a = b and b = c, then a = c.
In elementary mathematics, particularly in arithmetic, we learned two basic operations on real
numbers, namely
addition
and
multiplication
.
We will assume we all know how to perform
these operations
. Addition of real numbers is denoted by “+” while multiplication is denoted by
“
⋅
”. Thus we write, 2 + 3 = 5 and 2
⋅
3 = 6. The result of addition is called the
sum
, while
performing multiplication results to a
product
.
In the axioms that follow, there will be one form for addition and one for multiplication and we
will indicate this by placing an “A” or an “M” after the number of the axiom.
The next axioms are so familiar that reference to them is often neglected but they are important
in the study of algebraic systems.
Axiom 1A
.
Closure Property for Addition
For any a, b
∈
R
, a + b
∈
R
. (The sum of two real numbers is a real number.)
Axiom 1M
.
Closure Property for Multiplication
For any a, b
∈
R
, a
⋅
b
∈
R
. (The product of two real numbers is a real number.)