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1. The integers and the rational numbers. 1.1. The integers. Let
 

Summary: 1. The integers and the rational numbers.
1.1. The integers. Let
Z = {((m, n), (p, q)) (N N) (N N) : m + q = p + n}.
It is a simple matter to verify that Z is an equivalence relation on N N. Let
Z = (N N)/Z
and call its members integers. Let
m - n = (m, n)/Z for (m, n) N N.
One easily verifies that
N n (n - 0)/Z
is univalent. In what follows we will not distinguish between a member of N and
its image under this mapping.
One verifies that there is a unique unary operation - on Z such that
-(m - n) = n - m
and that there are unique binary operation + and on Z such that
(m-n)+(p-q) = (m+p)-(n+q) and (m-n)(p-q) = (mp+nq)-(mq+np)
for (m, n), (p, q) N N; on the right hand side of these equations + and are the
operations on N.
One easily verifies that (Z, +, 0, -) is an Abelian group and that (Z, +, 0, ) is
an integral domain in which 1 is the neutral element with respect to .
1.2. The rational numbers. Let

  

Source: Allard, William K. - Department of Mathematics, Duke University

 

Collections: Mathematics