2 Integers and Rational Numbers 15Proposition 14For any integers a,b with a> 0 , there exist unique integers q,rsuch
that
b=qa+r, 0 ≤r<a.Proof We consider first uniqueness. Suppose
qa+r=q′a+r′, 0 ≤r,r′<a.Ifr<r′, then from
(q−q′)a=r′−r,we obtain firstq>q′and thenr′−r≥a, which is a contradiction. Ifr′<r,we
obtain a contradiction similarly. Hencer=r′, which impliesq=q′.
We consider next existence. LetSbe the set of all integersy≥0 which can be
represented in the formy=b−xafor somex∈Z.ThesetSis not empty, since it
containsb−0ifb≥0andb−baifb<0. HenceScontains a least elementr.Then
b=qa+r,whereq,r∈Zandr≥0. Sincer−a=b−(q+ 1 )aandris the least
element inS,wemustalsohaver<a.
The concept of number will now be further extended to include ‘fractions’ or
‘rational numbers’. For measuring lengths the integers do not suffice, since the length
of a given segment may not be an exact multiple of the chosen unit of length. Similarly
for measuring weights, if we find that threeidentical coins balance five of the chosen
unit weights, then we ascribe to each coin the weight 5/3. In the realm of integers the
equationax=bfrequently has no solution; in the extended realm of rational numbers
it will always have a solution ifa=0.
Intuitively, a rational number is the ratio or ‘quotient’a/bof two integersa,b,
whereb=0, with addition and multiplication defined by
a/b+c/d=(ad+cb)/bd,
a/b·c/d=ac/bd.However, two other integersa′,b′may have the same ratio asa,b,andanywaywhat
doesa/bmean? To make things precise, we proceed in much the same way as before.
PutZ×=Z{ 0 }and consider the setZ×Z×of all ordered pairs (a,b) witha∈Z
andb∈Z×. For any two such ordered pairs, (a,b)and(a′,b′), we write
(a,b)∼(a′,b′) ifab′=a′b.To show that this is an equivalence relation it is again enough to verify that(a,b)∼
(a′,b′)and(a′,b′) ∼(a′′,b′′)imply(a,b)∼ (a′′,b′′). The same calculation as
before, with addition replaced by multiplication, shows that(ab′′)b′=(a′′b)b′.Since
b′=0, it follows thatab′′=a′′b.
The equivalence class containing( 0 , 1 )evidently consists of all pairs( 0 ,b)with
b=0, and the equivalence class containing( 1 , 1 )consists of all pairs(b,b)with
b=0.
We d e fi n e arational numberto be an equivalence class of elements ofZ×Z×and,
as is now customary, we denote the set of all rational numbers byQ.