1.1: Algebraic Operations With Integers

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 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.