1.1: Algebraic Operations With Integers

The set of all integers, which this book is all about, consists of all positive and negative integers as well as 0. Thus is the set given by While the set of all positive integers, denoted by , is defined by

On , there are two basic binary operations, namely addition (denoted by ) and multiplication (denoted by ), that satisfy some basic properties from which every other property for emerges.

1. The Commutativity property for addition and multiplication
2. Associativity property for addition and multiplication
3. The distributivity property of multiplication over addition

In the set there are "identity elements" for the two operations and , and these are the elements and respectively, that satisfy the basic properties for every .
The set allows additive inverses for its elements, in the sense that for every there exists another integer in , denoted by , such that While for multiplication, only the integer 1 has a multiplicative inverse in the sense that 1 is the only integer such that there exists another integer, denoted by or by , (namely 1 itself in this case) such that

From the operations of addition and multiplication one can define two other operations on , namely subtraction (denoted by ) and division (denoted by ). Subtraction is a binary operation on , i.e. defined for any two integers in , while division is not a binary operation and thus is defined only for some specific couple of integers in . Subtraction and division are defined as follows:

1. is defined by , i.e. for every
2. is defined by the integer if and only if .

Contributors and Attributions

• Dr. Wissam Raji, Ph.D., of the American University in Beirut. His work was selected by the Saylor Foundation’s Open Textbook Challenge for public release under a Creative Commons Attribution (CC BY) license.