# The Real Numbers

What are the real numbers? As explained in the
introduction to Chapter 3, there are two

approaches to answering this question. One approach would be to construct the
set R of real

numbers starting with the set N of natural numbers. Along the way, we would need
to construct

the set Z of integers and the set Q of rational numbers.

The other approach—the one followed here—is not to say what the real numbers
are, but

only to describe how they behave. In other words, our approach is to state, in
the form of

axioms, the fundamental properties of the set of real numbers.

Our axiomatic approach does not, of course, establish that any set having these
properties

actually exists! To establish that one needs to take a constructive approach.
But the axiomatic

approach does uniquely characterize the set of real numbers in the sense that
any two sets

having these properties are "essentially" the same (in technical terms, they are
"order isomorphic"

as ordered fields). This essential uniqueness is established in Section 3.4.
However, in

this course we shall not look into the matter of uniqueness of R.

The axioms for R can be summarized as follows:

**R is an ordered field that includes N, has the Archimedean Ordering Property,
and has the Nested Interval Property.**

To say that R is a

**field**means that there are operations of addition and multiplication in

R with "the usual" algebraic properties. These algebraic properties are listed in Axioms 3.1.4.

In particular, each x ∈ R has a negative -x for which x + (-x) = 0, and each nonzero y ∈ R

has a multiplicative inverse y

^{ -1}for which yy

^{ -1}= 1. Then subtraction and division of real

numbers may be defined by x - y = x + (-y) and, for y ≠ 0, x/y = xy

^{ -1}.

Beyond those basic properties we shall assume without proof all the familiar algebraic properties

listed in Proposition 3.1.12.

To say that the field R is ordered means that there is a relation < in R having "the usual"

algebraic properties such as those listed in Proposition 3.2.4. Among these properties are the

following:

• For each x ∈ R, exactly one of the relations 0 < x, x = 0, and x < 0 holds.

• For all x,y ∈ R, x < y 0 < y - x.

• For all x,y ∈ R, if 0 < x and 0 < y, then 0 < x + y and 0 < xy.

From just these three properties all the other familiar properties of order for real numbers

(as listed in Proposition 3.2.4) can be deduced. Here’s an example: For all x,y, z ∈ R,

x < y =>x + z < y + z.

In fact, assume x < y. This means 0 < y - x. But y - x = (y + z) - (x + z). Thus 0 <

(y + z) - (x + z). This means x + z < y + z.

**Using just the properties of ordering listed above together with the usual algebraic**

Exercise 1.

Exercise 1.

properties of addition and multiplication, deduce:

(a) If 0 < x, then -x < 0.

(b) If x < y, then -y < -x.

(c) 0 < 1

When we say here that R "includes" N, we do not mean merely that N R. Rather, we mean

also that:

• if m,n ∈ N, then their sum m+ n and product m· n as
elements of R are the same as

their sum and product, respectively, as natural numbers [as defined in Example
1.2.10 (2)

and Exercise 1.2.11 (3)];

• the number 0 ∈ N is the identity element for addition in the field R and the
number 1 ∈ N

is the identity element for multiplication in the field R; and

• if m,n ∈ N, then the relation m < n holds for m and n as elements of R if and
only if it

holds for them as elements of N [as defined in Exercise 1.2.11 (2)].

With N as a subset of R, the set Z of all integers and the set Q of all rational
numbers may

be defined as:

The** Archimedean Ordering Property** of the ordered field R is the following:

For every ε ∈ R with ε > 0 and for each c ∈ R, there exists a positive integer n
such

that nε > c.

This property may be expressed by saying that if you take a real number c, no
matter how

large, and a positive real number ε, no matter how small, then if you add ε to
itself enough

times, the sum will be greater than c.

For more about the Archimedean Ordering Property, see Section 3.2.

**Proposition 2. **Let c ∈ R. Then there exists a unique n ∈ Z for which n
≤ c < n +
1.

Proof. Existence. Case (i): c ≥ 0. By the Archimedean Ordering Property, there
exists some

k ∈ N such that k · 1 > c. By the Well-Ordering Principle for N, there is a
least such k; call it

. Let . Then n ≤ c < n + 1 (why?).

Case (ii): c < 0. Then -c > 0 and so, by what was just proved, there exists an
integer m

with m ≤ -c < m+ 1. Then …(finish the existence proof in this case.)

Uniqueness. Exercise.

**Theorem 3** (Order Density of Q in R). If a, b ∈ R with a < b, there exists some q
∈ Q for which

a < q < b.

Proof. See the proof of Theorem 3.2.27. In that proof, just change F to R, Q(F)
to Q, and N(F)

to N.

For real numbers a and b, as usual the closed interval [a, b] is the set { x ∈ R
: a ≤ x ≤ b}.

The **Nested Interval Property **of R is the following:

If is a sequence of closed intervals in R that is
decreasing in the

sense that or each n = 0, 1, 2, . . . , then
.

For more about the Nested Interval Property, see Section 3.3.

We already know there is no rational number c for which c^{2} = 2. One of the
consequences

of the Nested Interval Property is that there is some real number c for which c^{2}
= 2 or, as we

shall say, that exists.

With the theory of calculus at our disposal this would be easy to prove: see
Exercise 4

**Exercise 4. **Prove that exists by applying the Intermediate Value Theorem to
the function

f : R → R given by f (x) = x^{2}.

For the method in Exercise 4, you need to know that the
function f is continuous (and

so you need to know a precise definition for "continuous"); you also need to
know have the

Intermediate Value Theorem at your disposal. So we shall not use such a proof.

That exists is proved by an indirect method in Section 3.3, namely, by first
deducing

from the Archimedean Ordering Property and the Nested Interval Property that
each nonempty

subset of R that is bounded above has a least upper bound.

In this note we give a different proof that directly invokes the Archimedean
Ordering Property

and the Nested Interval Property. In following the proof it may help if you draw
a diagram

with a line representing R and mark on it the points and intervals constructed.
Be sure to

supply any missing justifications for steps.