294 PROPERTIES OF NUMBER SYSTEMS Chapter 9
"Write down what you know about the rational number system,... the in-
tegers,... the complex numbers"? As one example, the three algebraic
properties listed previously are just as true, when applied to Z or Q, as
they are about R. The question that begins to arise when we begin to
list descriptive properties of R is, "What properties characterize R?"
Stated differently, "What description can we give of R that, among other
things, distinguishes it from its familiar subsets N, Z, and Q and its super-
set C?'Answering that question is the goal of this chapter. In the course
of developing this answer we will justify the use of the real numbers, as
opposed to, say, the rationals, as the "universal set" for calculus of a single
variable, by showing that properties specific to R are crucial for the proofs
of certain basic theorems of calculus.
A more direct, and more open-ended question, in line with the preceding
paragraph, is, "What are the real numbers?" In this text we provide two
types of answer. As indicated, the material in this chapter provides a
descriptive answer. In Chapter 10 we will give a constructive answer. With
the positive integers as our starting point, we "build" first the integers,
then the rationals, then the reals, using a process based on equivalence
relations and equivalence classes, a process of considerable importance in
a number of areas of upper-level mathematics.
9.1 Fields
We can answer descriptively the question, "What are the real numbers?"
in just a few sentences. To do so, however, we must use terms that require
the better part of this chapter for their definition and elaboration. An
important theorem of advanced mathematics states that there is at most
one complete ordered Jield. Accepting the truth of that theorem (whose
proof is beyond the scope of this text), and assuming the existence of a
"complete ordered field," we give it the name "the real numbers." The
real numbers are the unique complete ordered field! Now we only need
to answer the questions: "What is a field? What is an ordered field?
What does it mean for an ordered field to be complete?" This article is
dedicated to answering the first question.
The definition of field requires first that we define binary operation on
a set.
DEFINITION 1
Given a nonempty set S, we define a binary operation *on S to be any mapping of
S x S into S. We denote by a * b the value of the function * at the ordered pair
(a, b).EXAMPLE 1 Familiar operations include addition and multiplication on
the set Z of integers, union and intersection on the collection of all
subsets of any given universal set U, addition of n-dimensional vectors,