1.20: Zm and Complete Residue Systems
- Page ID
- 82302
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Throughout this section we assume a fixed modulus \(m>0\).
Definition \(\PageIndex{1}\)
We define \[\mathbb{Z}_m=\{[a]\mid a\in\mathbb{Z}\},\nonumber \] that is, \(\mathbb{Z}_m\) is the set of all residue classes modulo \(m\). We call \(\mathbb{Z}_m\) the ring of integers modulo \(m\). In the next chapter we shall show how to add and multiply residue classes. This makes \(\mathbb{Z}_m\) into a ring. See Appendix A for the definition of ring. Often we drop the ring and just call \(\mathbb{Z}_m\) the integers modulo \(m\). From Theorem 19.4 \[\mathbb{Z}_m=\{[0],[1],\dotsc,[m-1]\}\nonumber \] and since no two of the residue classes \([0],[1],\dotsc,[m-1]\) are equal we see that \(\mathbb{Z}_m\) has exactly \(m\) elements. By Exercise 19.4 if we choose \[a_0\in[0],a_1\in[1],\dotsc,a_{m-1}\in[m-1]\nonumber \] then \[[a_0]=[0],[a_1]=[1],\dotsc,[a_{m-1}]=[m-1].\nonumber \] So we also have \[\mathbb{Z}_m=\{[a_0],[a_1],\dotsc,[a_{m-1}]\}.\nonumber \]
Example \(\PageIndex{1}\)
If \(m=4\) we have, for example, \[8\in [0], 5 \in [1], -6 \in [2], 11\in [3].\nonumber \] And hence: \[\mathbb{Z}_4 = \{[8],[5],[-6],[11] \}.\nonumber \]
Definition \(\PageIndex{2}\)
A set of \(m\) integers \[\{a_0,a_1,\dotsc,a_{m-1}\}\nonumber \] is called a complete residue system modulo \(m\) if \[\mathbb{Z}_m=\{[a_0],[a_1],\dotsc,[a_{m-1}]\}.\nonumber \]
Remark \(\PageIndex{1}\)
A complete residue system modulo \(m\) is sometimes called a complete set of representatives for \(\mathbb{Z}_m\).
Example \(\PageIndex{2}\)
By Theorem 19.4, for \(m>0\) \[\{0,1,2,\dotsc,m-1\}\nonumber\] is a complete residue system modulo \(m\).
Example \(\PageIndex{3}\)
From the above discussion it is clear that for each \(m > 0\) there are infinitely many distinct complete residue systems modulo \(m\). For example, here are some examples of complete residue systems modulo 5:
- \(\{0,1,2,3,4\}\)
- \(\{0,1,2,-2,-1\}\)
- \(\{10,-9,12,8,14\}\)
- \(\{0+5n_1,1+5n_2,2+5n_3,3+5n_4,4+5n_4 \}\) where \(n_1,n_2,n_3,n_4,n_5\) may be any integers.
Definition \(\PageIndex{3}\)
The set \(\{0,1,\dotsc,m-1\}\) is called the set of least nonnegative residues modulo \(m\).
Let \(m>0\) be given.
- If \(m=2k\), then \[\{0,1,2,\dotsc,k-1,k,-(k-1),\dotsc,-2,-1\}\nonumber\] is a complete residue system modulo \(m\).
- If \(m=2k+1\), then \[\{0,1,2,\dotsc,k,-k,\dotsc,-2,-1\}\nonumber\] is a complete residue system modulo \(m\).
- Proof
-
Proof of (1). Since if \(m=2k\) \[\mathbb{Z}_m=\{[0],[1],\dotsc,[k],[k+1],\dotsc,[k+i],[k+k-1]\},\nonumber \] it suffices to note that by Exercise 19.3 we have \[[k+i]=[k+i-2k]=[-k+i]=[-(k-i)].\nonumber \] So \[[k+1]=[-(k-1)],[k+2]=[-(k-2)],\dotsc,[k+k-1]=[-1],\nonumber \] as desired.
Proof of (2). In this case \[[k+i]=[-(2k+1)+k+i]=[-k+i+1]=[-(k-i+1)]\nonumber \] so \[[k+1]=[-k],[k+2]=[-(k-1)],\dotsc,[2k]=[-1],\nonumber \] as desired.
Definition \(\PageIndex{4}\)
The complete residue system modulo \(m\) given in Theorem \(\PageIndex{1}\) is called the least absolute residue system modulo \(m\).
Remark \(\PageIndex{2}\)
If one chooses in each residue class \([a]\) the smallest nonnegative integer one obtains the least nonnegative residue system. If one chooses in each residue class \([a]\) an element of smallest possible absolute value one obtains the least absolute residue system.
Exercise \(\PageIndex{1}\)
Find both the least nonnegative residue system and the least absolute residues for each of the moduli given below. Also, in each case find a third complete residue system different from these two. \[m=3,\quad m=4,\quad m=5,\quad m=6,\quad m=7,\quad m=8.\nonumber\]