10.2 DEVELOPMENT OF THE INTEGERS AND RATIONAL NUMBERS 349a contradiction of Theorem 7(d). If x is negative and xy = 1 for some y E Z,
then necessarily y is negative. But then -x and -y are both positive in-
tegers not equal to 1 and ( - x)( - y) = xy = 1, again, contradicting Theorem
7(d). We conclude that the system (Z, +, .) of integers fails to satisfy the
multiplicative inverse property and, in fact, that all integers not equal to + 1
fail to have multiplicative inverses. It is primarily for this reason that we
wish to expand Z to a system Q, the rationals, in which the favorable prop-
erties of Z are retained and the algebraic shortcoming just illustrated is
remedied.
The construction of Q from Z is very similar to the construction of Z
from N, just accomplished. You will recall that the elements of Z are
equivalence classes of ordered pairs of positive integers; elements of Q will
soon be seen to be equivalence classes of certain ordered pairs of integers.
Once constructed, Z was proved to contain a substructure Z+ "isomorphic
to" N (so that, essentially, N E Z). We will see that Q contains a substruc-
ture isomorphic to Z, so that, for all intents and purposes, Z is a subset
of Q. As indicated earlier, the primary advantage of Z, compared to N, is
that it makes up for algebraic deficiencies of N (i.e., lack of additive identity
and inverse properties). The corresponding primary advantage of Q is to
make up for the lack of multiplicative inverses of most nonzero elements
of Z.
Because of all this familiarity, we will provide in the text only the bare
outlines of the construction of Q, leaving the details to you. To start, we
consider the set of all ordered pairs of integers (x, y) such that y # 0, in
symbols, Z x (Z - (0)). For the sake of familiarity, you should think of
(x, y) as the fraction x/y. Define a relation a on this set by the rule
(x, y) z (w, z) if and only if xz = yw. It is easy to verify that x is an equiv-
alence relation on Z x (Z - (0)). Denote by Q the set of all equivalence
classes corresponding to w, denoting the equivalence class determined by
an ordered pair (x, y) by the symbol [[(x, y)]]. Note that if a, b, x, y E Z
with b # 0 and y # 0, we have (a, b) E [[(x, y)]] if and only if ay = bx; as
a specific example, we have (1,2) E [[(7, 14)]].
Next, we define operations of addition and multiplication on Q. Guided
by familiar rules of arithmetic, we are led to formulate the following
definition.
DEFINITION- 3
Let r = [[(w, x)]] and s = [[(y, z)]] be elements of Q. We define:
(a) The sum r + s of r and s by the rule r + s = [[(wz + xy, xz)]], and
(b) The product rs by rs = [[(wy, xz)]]Clearly our goal at this stage should be to prove that the set Q, endowed
with the two operations just defined, satisfies a number of familiar algebraic
properties. Of primary interest should be the multiplicative inverse axiom
(Axiom 10 of the field axioms) if, as we expect, Q is to prove to be a field.
But before attempting to prove any such properties, we must address an