14 I The Expanding Universe of Numbers
We may write(P2)and(P3)symbolically in the form
P+P⊆P, P·P⊆P.
We now show that there are nodivisors of zeroinZ:Proposition 12If a= 0 and b= 0 ,thenab= 0.
Proof By(P1), eitheraor−ais positive, and eitherbor−bis positive. Ifa∈Pand
b∈Pthenab∈P,by(P3), and henceab=0, by(P1).Ifa∈Pand−b∈P,then
a(−b)∈ P. Henceab=−(a(−b)) ∈−Pandab =0. Similarly if−a ∈ P
andb ∈ P. Finally, if−a ∈ Pand−b ∈ P,thenab= (−a)(−b)∈ Pand
againab=0.
The proof of Proposition 12 also shows that any nonzero square is positive:Proposition 13If a= 0 ,thena^2 :=aa∈P.
It follows that 1∈P,since1=0and1^2 =1.
The setPof positive integers induces an order relation inZ. Write
a<b ifb−a∈P,so thata∈Pif and only if 0<a. From this definition and the properties ofPit
follows that the order properties(O1)–(O3)holdalsoinZ,andthat(O4)holds in the
modified form:
(O4)′if 0 <c,then ac<bc if and only if a<b.
We now show that we can represent anya∈Zin the forma=b−c,where
b,c∈P.Infact,ifa=0, we can takeb=1andc=1; ifa∈P, we can take
b=a+1andc=1; and if−a∈P, we can takeb=1andc= 1 −a.
An elementaofZis said to be alower boundfor a subsetXofZifa≤xfor every
x∈X. Proposition 3 immediately implies that if a subset ofZhas a lower bound, then
it has a least element.
For anyn∈N,letn′be the integer represented by(n+ 1 , 1 ).Thenn′∈P.We
are going to study the mapn→n′ofNintoP. The map is injective, sincen′=m′
impliesn=m. It is also surjective, since ifa∈Pis represented by (m,n), where
n<m, then it is also represented by(p+ 1 , 1 ),wherep∈Nsatisfiesn+p=m.It
is easily verified that the map preserves sums and products:
(m+n)′=m′+n′,(mn)′=m′n′.Since 1′=1, it follows thatS(n)′=n′+1. Furthermore, we have
m′<n′ if and only ifm<n.
Thus the mapn → n′ establishes an ‘isomorphism’ ofNwithP.Inother
words,P is a copy ofNsituated withinZ. By identifyingnwithn′,wemay
regardNitself as a subset ofZ(and stop talking aboutP). Then ‘natural num-
ber’ is the same as ‘positive integer’ and any integer is the difference of two natural
numbers.
Number theory, in its most basic form, is the study of the properties of the setZof
integers. It will be considered in some detail in later chapters of this book, but to relieve
the abstraction of the preceding discussion we consider here thedivision algorithm: