CHAP. 11] PROPERTIES OF THE INTEGERS 265
AppendixBshows that other mathematical structures have the above properties. One fundamental property
which distinguishes the integersZfrom other structures is the Principle of Mathematical Induction (Section 1.8)
which we rediscuss here. We also state and prove (Problem 11.30) the following theorem.
Fundamental Theorem of Arithmetic: Every positive integern>1 can be written uniquely as a product
of prime numbers.
This theorem already appeared in Euclid’sElements. Here we also develop the concepts and methods which
are used to prove this important theorem.
11.2Order and Inequalities, Absolute Value
This section discusses the elementary properties of order and absolute value.Order
Observe that we define order inZin terms of the positive integersN. All the usual properties of this order
relation are a consequence of the following two properties ofN:
[P 1 ]Ifaandbbelong toN, thena+bandabbelong toN.
[P 2 ] For any integera, eithera∈N,a=0, or−a∈N.The following notation is also used:
a>bmeansb<a; read:ais greater thanb.
a≤bmeansa<bora=b; read:ais less than or equal tob.
a≥bmeansb≥a; read:ais greater than or equal tob.
The relations<, >,≤and≥are calledinequalitiesin order to distinguish them from the relation=of
equality. The reader is certainly familiar with the representation of the integers as points on a straight line, called
thenumber lineR, as shown in Fig. 11-1.Fig. 11-1We note thata<bif and only ifalies to the left ofbon the number lineRin Fig. 11-1. For example,
2 < 5 ;− 6 <− 3 ; 4 ≤ 4 ; 5 >− 8 ; 6 ≥ 0 ;− 7 ≤ 0We also note thatais positive iffa>0, andais negative iffa<0. (Recall “iff” means “if and only if.”)
Basic properties of the inequality relations follow.
Proposition 11.1: The relation≤inZhas the following properties:
(i) a≤a, for any integera.
(ii) Ifa≤bandb≤a, thena=b.
(iii) Ifa≤bandb≤c, thena≤c.
Proposition 11.2 (Law of Trichotomy): For any integersaandb, exactly one of the following holds:
a<b, a=b, or a>b
Proposition 11.3: Supposea≤b, and letcbe any integer. Then:
(i) a+c≤b+c.
(ii) ac≤bcwhenc> 0 ;butac≤bcwhenc<0.
(Problem 11.5 proves Proposition 11.3.)