3 Real Numbers 17−a,wherea∈P. From the corresponding result forZ, it follows that(P1)continues
to hold inQ. We will show that(P2)and(P3)also hold.
To see that the sum of two positive rational numbers is again positive, we observe
that ifa,b,c,dare integers such that 0<aband 0<cd,thenalso
0 <(ab)d^2 +(cd)b^2 =(ad+cb)(bd).To see that the product of two positive rational numbers is again positive, we observe
that ifa,b,c,dare integers such that 0<aband 0<cd,thenalso
0 <(ab)(cd)=(ac)(bd).Since(P1)–(P3)all hold, it follows as before that Propositions 12 and 13 also hold
inQ. Hence 1∈Pand(O4)′now implies thata−^1 ∈Pifa∈P.Ifa,b∈Pand
a<b,thenb−^1 <a−^1 ,sincebb−^1 = 1 =aa−^1 <ba−^1.
The setPof positive elements now induces an order relation onQ. We writea<b
ifb−a∈P,sothata∈Pif and only if 0<a. Then the order relations(O1)–(O3)
and(O4)′continue to hold inQ.
Unlike the situation forZ, however, the ordering ofQisdense,i.e.ifa,b∈Qand
a<b, then there existsc∈Qsuch thata<c<b. For example, we can takecto be
the solution of( 1 + 1 )c=a+b.
LetZ′denote the set of all rational numbersa′which can be represented by(a, 1 )
for somea∈ Z.Foreveryc ∈ Q,thereexista′,b′∈ Z′withb′ =0 such that
c=a′b′−^1 .Infact,ifcis represented by(a,b), we can takea′to be represented by
(a, 1 )andb′by(b, 1 ). Instead ofc=a′b′−^1 , we also writec=a′/b′.
For anya∈Z,leta′be the rational number represented by(a, 1 ).Themapa→a′
ofZintoZ′is clearly bijective. Moreover, it preserves sums and products:
(a+b)′=a′+b′,(ab)′=a′b′.Furthermore,
a′<b′ if and only ifa<b.Thus the mapa→a′establishes an ‘isomorphism’ ofZwithZ′,andZ′is a copy
ofZsituated withinQ. By identifyingawitha′, we may regardZitself as a subset of
Q. Then any rational number is the ratio of two integers.
By way of illustration, we show that ifaandbare positive rational numbers, then
there exists a positive integerlsuch thatla>b.Forifa=m/nandb=p/q,where
m,n,p,qare positive integers, then
(np+ 1 )a>pm≥p≥b.3 RealNumbers
It was discovered by the ancient Greeks that even rational numbers do not suffice for
the measurement of lengths. Ifxis the length of the hypotenuse of a right-angled tri-
angle whose other two sides have unit length then, by Pythagoras’ theorem,x^2 =2.