Next: Ordering Up: The integers Previous: An equivalence relation on   Contents

## The set of integers

Definition 6.2.7   The quotient set is denoted by and its elements are called integers.

As usual, we denote by the equivalence class of the pair , i.e.

 in     in

In this set, we can define an addition and a multiplication, as follows:

Definition 6.2.8   For any two elements and in :
1. ;
2. .

In other words, the sum of two classes is the class of the sum'' and the product of two classes is the class of the product''. By Proposition 2.6, these operations are well defined, i.e. the result does not depend on the choice of the representants of the equivalence classes. The description we gave until now of the integres does not fit the intuitive notion that everybody knows. Let us define a mapping as follows:

Proposition 6.2.9
1. The mapping is an injection.
2. For any two natural numbers and , .
3. For any two natural numbers and , .

Proof.
1. Let and be two natural numbers such that . We have:

2. For any two natural numbers and , we have:

3. For any two natural numbers and , we have:

As a consequence, we can identify the set of the natural numbers with a subset of the set of integers. Now we can determine a useful partition of . Take .
• If , then , i.e. .
• If , then . In other words, for any , .
• If , then , i.e. .

Definition 6.2.10   Let be a non zero natural number. An integer of the form is called a positive integer; we will denote it by or simply by . An integer of the form is called a negative integer;we will denote it by . The set of non negative integers is denoted by and the set of non positive integers is denoted .

Thus we have:

With the identification determined by the mapping , we can write . The sign rules are given by the following proposition:

Proposition 6.2.11   For any two natural numbers and we have:

Proof.

An additional property is described in the following proposition:

Proposition 6.2.12   Let be any integer. Then there exists a unique integer such that .

The integer will be called the opposite of and will be denoted by .

Proof. Take for some pair and . The w have:

The other equation is obtained now by the commutative law of the addition in . Assume now that an integer has two opposites and . Then we have:

 by Prop 1.15

Next: Ordering Up: The integers Previous: An equivalence relation on   Contents
root 2002-06-10