1.4. The Real Numbers 23

all rational numbers to the left of the location x. We next need to define an order

relation and operations of addition and multiplication in R.

The order relation on R is simple: we say x ≤ y if Lx ⊂ Ly. An element x ∈ R

is, then, non-negative if L0 ⊂ Lx. With this definition of order on R we can assert

that

Lx = {r ∈ Q : r x}

for all x ∈ R (not just for x ∈ Q).

Addition of real numbers is defined as follows: if x, y ∈ R, then we set

Lx + Ly = {r + s : r ∈ Lx,s ∈ Ly}.

It is easily verified that this is also a Dedekind cut (Exercise 1.4.10) and, hence, it

corresponds to an element of R. We define x + y to be this element.

The product of two non-negative numbers x and y is defined as follows: we set

K = {rs : r ∈ Lx,r ≥ 0,s ∈ Ly,s ≥ 0} ∪ {t ∈ Q : t 0}.

This is a Dedekind cut (Exercise 1.4.11), and we define xy to be the corresponding

element of R. For pairs of numbers where one or both is negative, the definition

of product is more complicated due to the fact that multiplication by a negative

number reverses order.

Of course Q ⊂ R, since each rational number was already the cut number of a

Dedekind cut. It is easily checked that the definitions of addition, multiplication,

and order given above agree with the usual ones in the case that the numbers are

rational.

The numbers in R that are not in Q are called irrational numbers. It turns

out that there are many more irrational numbers than there are rational numbers.

To make sense of this statement requires a discussion of finite sets and infinite sets

and how some infinite sets are larger than others. We present such a discussion in

the appendix.

The Completeness Axiom. This is the property of the real number system

that distinguishes it from the rational number system. Without it, most of the

theorems of calculus would not be true.

A subset A of an ordered field F is said to be bounded above if there is an

element m ∈ F such that x ≤ m for every x ∈ A. The element m is called an upper

bound for A. If, among all upper bounds for A, there is one which is smallest (less

than all the others), then we say that A has a least upper bound.

Definition 1.4.3. An ordered field F is said to be complete if it satisfies:

C. Each non-empty subset of F which is bounded above has a least upper bound.

If one defines the real number system R in terms of Dedekind cuts of the

rationals and defines addition, multiplication, and order as above, then one can

prove that the resulting system is an ordered field. To carry out all the details of

this proof is a long and tedious process and it will not be done here. However,

it is quite easy to prove that R, as defined in this way, satisfies the completeness

axiom C.