1.4. The Real Numbers 21

where a and b are integers. This set has a smallest element by Exercise 1.2.19.

Use the division algorithm (Exercise 1.2.20) to show that this smallest element

divides both m and n.

14. Use the result of the preceding exercise to prove that if a prime p divides the

product nm of two positive integers, then it divides n or it divides m.

1.4. The Real Numbers

As pointed out in the previous section, the set of rational numbers is riddled with

“holes” where there ought to be numbers. Here we will try to make this statement

more precise and then indicate how these holes can be “filled”, resulting in the

system of real numbers. In addition to the ordered field axioms, the real number

system satisfies a new axiom C – the completeness axiom. Later in the section we

will state it and explore its consequences.

The construction of the real numbers that we outline below is motivated by the

idea that a “hole” in the rational numbers is a location along the rational number

line where there should be a number but there is no rational number. What do we

mean by a “location” along the rational number line? Well, if this has meaning,

then it should make sense to talk about the rational numbers that are to the left

of this location and those that are to the right of this location. This should lead to

a separation of the rational numbers into two sets – one to the left and one to the

right of the given location. In fact, we can define a location on the rational line to

be such a separation. This leads to the notion of a Dedekind cut.

Dedekind Cuts. If r is a rational number, consider the infinite interval Lr

consisting of all rational numbers to the left of r. That is,

(1.4.1) Lr = {x ∈ Q : x r}.

This set is a non-empty, proper subset of Q. It has no largest element, since, for

each x r, there are rational numbers larger than x that are also less than r (for

example, (x+r)/2 is one such number). It also has the property that if x ∈ Lr, then

so is any rational number less than x. It turns out that there are also subsets of Q

with these three properties that are not of the form Lr for some rational number.

A subset of Q with these three properties is called a Dedekind cut. That is,

Definition 1.4.1. A subset L of Q is called a Dedekind cut, or simply a cut in the

rationals, if it satisfies the following three conditions:

(a) L = ∅ and L = Q;

(b) L has no largest element;

(c) if x ∈ L, then so is every y ∈ Q with y x.

The reason for calling such a set L a “cut” is that, if R is the complement of L,

then each number in L is to the left of each number in R. Thus, the rational line

is separated or cut into left and right halves. Since each half determines the other,

we choose to focus on just the left half in this discussion.