Number Theory: An Introduction to Mathematics

12 I The Expanding Universe of Numbers

It follows at once from the corresponding properties of natural numbers that, also in
Z, addition satisfies the commutative law(A2)and the associative law(A3). Moreover,
the equivalence class 0 (zero) containing (1,1) is anidentity elementfor addition:

(A4)a+ 0 =a for every a.

Furthermore, the equivalence class containing (n,m)isanadditive inversefor the
equivalence containing (m,n):

(A5)for each a, there exists−a such that a+(−a)=0.

From these properties we can now obtain

Proposition 8Fo r a l l a,b∈Z, the equation a+x=b has a unique solution x∈Z.

Proof It is clear thatx=(−a)+bis a solution. Moreover, this solution is unique,
since ifa+x=a+x′then, by adding−ato both sides, we obtainx=x′.

Proposition 8 shows that the cancellation law(A1)is a consequence of(A2)–(A5).
It also immediately implies

Corollary 9Fo r e a c h a∈Z, 0 is the only element such that a+ 0 =a,−a is uniquely
determined by a, and a=−(−a).

As usual, we will henceforth writeb−ainstead ofb+(−a).

Multiplication of integers is defined by

(m,n)·(p,q)=(mp+nq,mq+np).

To justify this definition we must show that(m,n)∼(m′,n′)and(p,q)∼(p′,q′)
imply

(mp+nq,mq+np)∼(m′p′+n′q′,m′q′+n′p′).

Fromm+n′=m′+n, by multiplying bypandqwe obtain

mp+n′p=m′p+np,

m′q+nq=mq+n′q,

and fromp+q′=p′+q, by multiplying bym′andn′we obtain

m′p+m′q′=m′p′+m′q,

n′p′+n′q=n′p+n′q′.

Adding these four equations and cancelling the terms common to both sides, we get

(mp+nq)+(m′q′+n′p′)=(m′p′+n′q′)+(mq+np),

as required.
It is easily verified that, also inZ, multiplication satisfies the commutative law
(M2)and the associative law(M3). Moreover, the distributive law(AM1)holds and,
if 1 is the equivalence class containing( 1 + 1 , 1 ),then(M4)also holds. (In prac-
tice it does not cause confusion to denote identity elements ofNandZby the same
symbol.)