1.1: Algebraic Operations With Integers
- Page ID
- 8817
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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
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.