1.3. Integers and Rational Numbers 17

Solution: Suppose x + z = y + z. On adding −z to both sides, this becomes

(x + z) + (−z) = (y + z) + (−z).

Applying the associative law of addition (A2) yields

x + (z + (−z)) = y + (z + (−z)).

But (z + (−z)) = 0 by A4 and x +0 = x by A3 and A1. Similarly, y + 0 = y. We

conclude that x = y. This proves (a).

By A3, 0 + 0 = 0. By D and A3,

x · 0 + x · 0 = x · (0 + 0) = x · 0 = 0 + x · 0.

Using (a) above, we conclude that x · 0 = 0.

To prove (c), we first note that, by definition, −xy is the additive inverse of

xy (it follows from (a) that there is only one of these). We will show that (−x)y is

also an additive inverse for xy. By D, (b), and A1,

xy + (−x)y = (x + (−x))y = 0 · y = 0.

This proves that (−x)y is an additive inverse for xy and, hence, it must be −xy.

Subtraction in a commutative ring is defined in terms of addition and the

additive inverse by setting

x − y = x + (−y).

The system of integers satisfies all the laws of Definition 1.3.1, and so it is a

commutative ring. In fact, it is a commutative ring with an order relation, since

the order relation on N can be used to define a compatible order relation on Z.

However, Z is still inadequate as a number system. This is due to our need to talk

about fractional parts of things. This defect is fixed by passing from the integers

to the rational numbers.

The Field of Rational Numbers. A field is a commutative ring in which divi-

sion is possible as long as the divisor is not 0. That is,

Definition 1.3.3. A field is a commutative ring satisfying the additional axiom:

M4. (Multiplicative Inverses) For each non-zero element x there is an element

x−1

such that

x−1x

= 1.

In a field, an element y can be divided by any non-zero element x. The result

is x−1y, which can also be written as y/x or

y

x

.

The rational number system Q is a field that is constructed directly from the

integers. The construction begins by considering all symbols of the form

n

m

, with

n, m ∈ Z and m = 0. We identify two such symbols

n

m

and

p

q

whenever nq = mp.

The resulting object is called a fraction. Thus,

4

6

and

2

3

represent the same fraction

because 4 · 3 = 6 · 2. The set Q is then the set of all fractions.

Addition and multiplication in Q are defined in the familiar way:

n

m

+

p

q

=

nq + mp

mq

and

n

m

·

p

q

=

np

mq

.