## The System Q of Rational Numbers

DEFINITION: Let Q = {[p, q] (p, q) E F} and call Q the set of rational numbers. Here [p, q] = { (x, y) I (x, y) (p, q) A (X, y) E F}.

Thus in a very similar manner to equality of integers, equality of rational numbers is determined by [p, q] = [r, s] iff ps = qr. . Also in a manner analogous to the development of + and times for Z, we have the following lemma to ensure these operations are well defined LEMMA: If [a, b] , [c, d] , [e, f] E Q and [c, d] = [e, f ] then

1. [ad + bc, bd] = [a f + be, b f] 2. [ac, bd] = [ae, b f] The lemma guarantees that + and are well defined, i.e., that a

rational number may be replaced by its equal in a sum or product. The proof is quite similar to the proof of the corresponding lemma for Z. Examples 20.1a,b,e, and f illustrate that the lemma works.