18 1. The Real Numbers

A fraction of the form

n

1

is identified with the integer n. This makes the set of

integers Z a subset of Q.

The above construction yields a system that satisfies A1 through A4, M1

through M4, and D. It is therefore a field. We call it the field of rational numbers

and denote it by Q. We won’t prove here that Q satisfies all of the field axioms,

but a few of them will be verified in the examples and exercises of this section. We

will also use the examples and exercises to show how the field axioms can be used

to prove other standard facts about arithmetic in fields such as Q.

Example 1.3.4. Assuming that Z satisfies the axioms of a commutative ring,

verify that Q satisfies A3 and M3.

Solution: The additive identity in Z is the integer 0, which is identified with

the fraction

0

1

. If we add this to another fraction

n

m

, the result is

0

1

+

n

m

=

0 · m + 1 · n

1 · m

=

n

m

.

Thus, 0 =

0

1

is an additive identity for Q and axiom A3 is satisfied.

The multiplicative identity in Z is the integer 1, which is identified with the

fraction

1

1

. If we multiply this by another fraction

n

m

, the result is

1

1

·

n

m

=

1 · n

1 · m

=

n

m

.

Thus, 1 =

1

1

is a multiplicative identity for Q and axiom M3 is satisfied.

Example 1.3.5. Verify that Q satisfies M4.

Solution: We know that the elements of Q of the form

0

m

represent the zero

element of Q. Thus, each non-zero element is represented by a fraction

n

m

in which

n = 0. Then

m

n

is also a fraction, and

m

n

·

n

m

=

nm

nm

=

1

1

= 1.

Thus,

m

n

is a multiplicative inverse for

n

m

. This proves that M4 is satisfied in Q.

The Ordered Field of Rational Numbers. Using the order relation on the

integers, it is easy to define an order relation on Q. If r is an element of Q, then we

declare r ≥ 0 if r can be represented in the form

n

m

for integers n ≥ 0 and m 0.

The order relation is then defined by declaring

p

q

≤

n

m

if and only if

n

m

−

p

q

≥ 0.

With the order relation defined this way, Q becomes an ordered field. That is, it

satisfies the axioms in the following definition.

Definition 1.3.6. A field F is called an ordered field if it has an order relation

“≤” such that the following are satisfied for all x, y, z ∈ F:

O1. Either x ≤ y or y ≤ x.

O2. If x ≤ y and y ≤ x, then x = y.

O3. If x ≤ y and y ≤ z, then x ≤ z.