28 Chapter 1 • The Real Number System1.4 Rational Numbers
We have seen in Section 1.3 that every ordered field contains the natural num-
bers. Since fields are closed under subtraction, one would expect every ordered
field to contain all the "integers" (natural numbers, negatives of natural num-
bers, and zero). Further, since fields are closed under division, except by 0,
one would expect every ordered field to contain all the "rational numbers"
(quotients of integers). We begin by defining integers and rational numbers.Definition 1.4.1 The set of integers of an ordered field F is the set'llF = {x E F : x EN, or -x EN, or x = 0}.
Thus, the set of integers of F consists of natural numbers, additive inverses
of natural numbers, and 0.Definition 1.4.2 The set of rational numbers of an ordered field F is the
setQF = {x E F: 3m,n E 'll,F 3 n-/= 0, and x = ~}.
That is, the rational numbers of Fare quotients of integers of F, commonly
called "fractions." One of the most significant features of the rational numbers
of F is that they form a field within the field F; i.e., a "subfield" of F. This is
summarized in the following theorem.Theorem 1.4.3 For any ordered field F with positive subset P, the set QF
of rational elements of F is an ordered field relative to the same operations
+ and · used in F and the positive set p' = P n QF. (QF will be called the
rational subfield of F.)*Proof. Let F be an ordered field. We must show that QF, with the
same + and · used in F and the positive set p', satisfies axioms (AO)-(A4),
(MO)-(M4), (D), and (01)-(03).
(AO) Let x,y E QF. Then 3m,n,m',n' E 'll,F 3 n , n'-/= 0, x = ~ and
y = m' -:;;;. Th us,
m m' mn' +m'n
x + y = - +----, = , E QF. (Justify)
n n nn(Al) This property is "inherited" from F. (Explain what that means)
(A2) Inherited from F. (Explain)
(A3) 0 E QF, since 0 = ~· This element of QF satisfies the condition
specified in (A3).
(A4) Let x E QF. Then 3m, n E ZF 3 n-/= 0 and x = ~·Then -x =
-nm E QF (Justify). This element -x satisfies the property required by (A4).