16 I The Expanding Universe of Numbers
Addition of rational numbers is defined by(a,b)+(c,d)=(ad+cb,bd),wherebd=0sinceb=0andd=0. To justify the definition we must show that
(a,b)∼(a′,b′)and(c,d)∼(c′,d′)imply(ad+cb,bd)∼(a′d′+c′b′,b′d′).But ifab′=a′bandcd′=c′d,then
(ad+cb)(b′d′)=(ab′)(dd′)+(cd′)(bb′)
=(a′b)(dd′)+(c′d)(bb′)=(a′d′+c′b′)(bd).It is easily verified that, also inQ, addition satisfies the commutative law(A2)
and the associative law(A3).Moreover(A4)and(A5)also hold, the equivalence class
0 containing( 0 , 1 )being an identity element for addition and the equivalence class
containing(−b,c)being the additive inverse of the equivalence class containing(b,c).
Multiplication of rational numbers is defined componentwise:
(a,b)·(c,d)=(ac,bd).To justify the definition we must show that
(a,b)∼(a′,b′)and(c,d)∼(c′,d′)imply(ac,bd)∼(a′c′,b′d′).But ifab′=a′bandcd′=c′d,then
(ac)(b′d′)=(ab′)(cd′)=(a′b)(c′d)=(a′c′)(bd).It is easily verified that, also inQ, multiplication satisfies the commutative law
(M2)and the associative law(M3). Moreover(M4)also holds, the equivalence class 1
containing( 1 , 1 )being an identity element for multiplication. Furthermore, addition
and multiplication are connected by the distributive law(AM1),and(AM2)also holds
since( 0 , 1 )is not equivalent to( 1 , 1 ).
Unlike the situation forZ, however, every nonzero element ofQhas amultiplica-
tive inverse:
(M5)for each a=0,there exists a−^1 such that aa−^1 =1.
In fact, ifais represented by (b,c), thena−^1 is represented by (c,b).
It follows that, for alla,b∈Qwitha=0, the equationax=bhas a unique
solutionx ∈Q, namelyx=a−^1 b. Hence, ifa=0, then 1 is the only solution of
ax=a,a−^1 is uniquely determined bya,anda=(a−^1 )−^1.
We will say that a rational numberaispositiveif it is represented by an ordered
pair (b,c) of integers for whichbc>0. This definition does not depend on the choice
of representative. For suppose 0<bcandbc′=b′c.Thenbc′=0, sinceb=0and
c′=0, and hence 0<(bc′)^2 .Since(bc′)^2 =(bc)(b′c′)and 0<bc, it follows that
0 <b′c′.
Our previous use ofPhaving been abandoned in favour ofN, we will now denote
byPthe set of all positive rational numbers and by−Pthe set of all rational numbers