1.1: Algebraic Operations With Integers
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The set Z of all integers, which this book is all about, consists of all positive and negative integers as well as 0. Thus Z is the set given by Z={...,−4,−3,−2,−1,0,1,2,3,4,...}. While the set of all positive integers, denoted by N, is defined by N={1,2,3,4,...}.
On Z, 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 Z emerges.
- The Commutativity property for addition and multiplication a+b=b+aa⋅b=b⋅a
- Associativity property for addition and multiplication (a+b)+c=a+(b+c)(a⋅b)⋅c=a⋅(b⋅c)
- The distributivity property of multiplication over addition a⋅(b+c)=a⋅b+a⋅c.
In the set Z there are "identity elements" for the two operations + and ⋅, and these are the elements 0 and 1 respectively, that satisfy the basic properties a+0=0+a=aa⋅1=1⋅a=a for every a∈Z.
The set Z allows additive inverses for its elements, in the sense that for every a∈Z there exists another integer in Z, denoted by −a, such that a+(−a)=0. While for multiplication, only the integer 1 has a multiplicative inverse in the sense that 1 is the only integer a such that there exists another integer, denoted by a−1 or by 1/a, (namely 1 itself in this case) such that a⋅a−1=1.
From the operations of addition and multiplication one can define two other operations on Z, namely subtraction (denoted by −) and division (denoted by /). Subtraction is a binary operation on Z, i.e. defined for any two integers in Z, while division is not a binary operation and thus is defined only for some specific couple of integers in Z. Subtraction and division are defined as follows:
- a−b is defined by a+(−b), i.e. a−b=a+(−b) for every a,b∈Z
- a/b is defined by the integer c if and only if a=b⋅c.
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.