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1.1: Algebraic Operations With Integers

• • Contributed by Wissam Raji
• Associate Professor and the Chairman (Mathematics) at American University of Beirut

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The set $$\mathbb{Z}$$ of all integers, which this book is all about, consists of all positive and negative integers as well as 0. Thus $$\mathbb{Z}$$ is the set given by $\mathbb{Z}=\{...,-4,-3,-2,-1,0,1,2,3,4,...\}.$ While the set of all positive integers, denoted by $$\mathbb{N}$$, is defined by $\mathbb{N}=\{1,2,3,4,...\}.$

On $$\mathbb{Z}$$, there are two basic binary operations, namely addition (denoted by $$+$$) and multiplication (denoted by $$\cdot$$), that satisfy some basic properties from which every other property for $$\mathbb{Z}$$ emerges.

1. The Commutativity property for addition and multiplication \begin{aligned} a+b=b+a\\ a\cdot b=b\cdot a\end{aligned}
2. Associativity property for addition and multiplication \begin{aligned} (a+b)+c&=&a+(b+c)\\ (a\cdot b)\cdot c&=& a\cdot (b\cdot c)\end{aligned}
3. The distributivity property of multiplication over addition \begin{aligned} a\cdot (b+c)&=&a\cdot b+a\cdot c.\end{aligned}

In the set $$\mathbb{Z}$$ there are "identity elements" for the two operations $$+$$ and $$\cdot$$, and these are the elements $$0$$ and $$1$$ respectively, that satisfy the basic properties \begin{aligned} a + 0 =0+a=a\\ a\cdot 1 = 1\cdot a=a\end{aligned} for every $$a\in\mathbb{Z}$$.
The set $$\mathbb{Z}$$ allows additive inverses for its elements, in the sense that for every $$a\in\mathbb{Z}$$ there exists another integer in $$\mathbb{Z}$$, 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 $a\cdot a^{-1}=1.$

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

1. $$a-b$$ is defined by $$a+(-b)$$, i.e. $$a-b=a+(-b)$$ for every $$a,b\in\mathbb{Z}$$

2. $$a/b$$ is defined by the integer $$c$$ if and only if $$a=b\cdot c$$.

Contributors

• 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.