350 CONSTRUCTION OF NUMBER SYSTEMS Chapter 10issue that must always be attended to whenever we define algebraic op-
erations on a set whose elements are equivalence classes-prove that the
operations are well defined. It is left to you to formulate the meaning
of this statement (using Theorem 2 as a guide) and write out the proofs
(Exercise 7).
Once it has been verified that addition and multiplication, as described
in Definition 3, are legitimate operations, it remains to show that (Q, +, -) is
a field. To do this, you should verify the 11 field axioms. It should be noted
that [[(0, I)]] and [[(I, I)]] are additive and multiplicative identities for Q,
respectively, while [[(-a, b)]] and [[(b, a)]] are, respectively, the additive
and multiplicative inverses of a given [[(a, b)]] E Q, with the assumption
of a nonzero value of a made in the multiplicative case.
After verifying the 11 axioms, you might next look to Article 9.1, where
number of general field properties are proved, for additional algebraic prop-
erties of Q.
Let us consider next how Z might be regarded as a subset of Q. Given
integers x, a, and b, with b # 0, we say that x is associated with the rational
number [[(a, b)]] if and only if xb = a. You should verify that if x is as-
sociated with [[(a, b)]], then x is also associated with a rational number
[[(c, d)]] if and only if (c, d) E [[(a, b)]]. That is, an integer is associated
with at most one rational number. Note also that a rational number
[[(a, b)]] has an integer n associated with it if and only if b divides a, in
the sense of Example 2, Article 6.1. Noting that any integer x is surely as-
sociated with the rational number [[(x, I)]], we conclude that any integer
x is associated with a unique equivalence class in Q, a class we will denote
by [[x]], so that the mapping x + [[x]] of Z into Q is well defined. Fur-
thermore, this mapping is one to one and not onto (you should supply a
precise description of the subset of Q constituting the image of this map-
ping). Finally, the mapping preserves both addition and multiplication;
that is, [[xi]] + Kx2II = [[XI + x2II and [[xIII. CCX~II = CCXIXJI for
any integers x, and x,. Thus the image of the mapping x 4 [[x]] is asub-
set of Q (closed under addition and multiplication, as you can verify; see
Exercise 8), which, for all intents and purpbses, is identical to Z. ~e&e Z
may be regarded as a subset of Q in exactly the same sense in which we
previously observed N to be a subset of Z (recall the lengthy discussion
between Theorems 2 and 3, earlier in this article).
We are left with the final problem of defining an ordering on Q. Before
addressing this matter, let us first make a remark pertaining to generaliza-
tion of the part of the construction of Q we've seen thus far. Recall that
the (Z, +, .) is a particular example of an algebraic structure known as an
integral domain, that is, a commutative ring with unity having no zero di-
visors (recall the discussions following Theorems 3 and 5). From a course
in abstract algebra you will learn that the process of passing from integers
to quotients of integers can be generalized. In fact, any integral domain D
can be "embedded" in a unique smallest field F containing D as a subdo-