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.
- The Commutativity property for addition and multiplication \[\begin{aligned} a+b=b+a\\ a\cdot b=b\cdot a\end{aligned}\]
- Associativity property for addition and multiplication \[\begin{aligned} (a+b)+c&=&a+(b+c)\\ (a\cdot b)\cdot c&=& a\cdot (b\cdot c)\end{aligned}\]
- 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:
- \(a-b\) is defined by \(a+(-b)\), i.e. \(a-b=a+(-b)\) for every \(a,b\in\mathbb{Z}\)
- \(a/b\) is defined by the integer \(c\) if and only if \(a=b\cdot c\).
Contributors and Attributions
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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.