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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.

  1. The Commutativity property for addition and multiplication a+b=b+aab=ba
  2. Associativity property for addition and multiplication (a+b)+c=a+(b+c)(ab)c=a(bc)
  3. The distributivity property of multiplication over addition a(b+c)=ab+ac.

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=aa1=1a=a for every aZ.
The set Z allows additive inverses for its elements, in the sense that for every aZ 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 a1 or by 1/a, (namely 1 itself in this case) such that aa1=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:

  1. ab is defined by a+(b), i.e. ab=a+(b) for every a,bZ
  2. a/b is defined by the integer c if and only if a=bc.

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.


This page titled 1.1: Algebraic Operations With Integers is shared under a CC BY license and was authored, remixed, and/or curated by Wissam Raji.

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