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1  %$Header: /home/dashley/cvsrep/e3ft_gpl01/e3ft_gpl01/dtaipubs/esrgubka/c_glo0/c_glo0.tex,v 1.9 2003/03/13 06:28:06 dtashley Exp $  %$Header$
2    
3  \chapter*{Glossary Of Terms}  \chapter*{Glossary Of Terms}
4  \markboth{GLOSSARY OF TERMS}{GLOSSARY OF TERMS}  \markboth{GLOSSARY OF TERMS}{GLOSSARY OF TERMS}
5    
6  \label{cglo0}  \label{cglo0}
7    
8  \begin{vworktermglossaryenum}  \begin{vworktermglossaryenum}
9    
10  \item \textbf{axiom}\index{axiom}  \item \textbf{axiom}\index{axiom}
11    
12        A statement used in the premises of arguments and assumed to be true        A statement used in the premises of arguments and assumed to be true
13            without proof.  In some cases axioms are held to be self-evident, as in            without proof.  In some cases axioms are held to be self-evident, as in
14            Euclidian geometry, while in others they are assumptions put forward for            Euclidian geometry, while in others they are assumptions put forward for
15            the sake of argument.            the sake of argument.
16        (Taken verbatim from \cite{bibref:b:penguindictionaryofmathematics:2ded}.)        (Taken verbatim from \cite{bibref:b:penguindictionaryofmathematics:2ded}.)
17    
18  \item \textbf{cardinality}\index{cardinality}  \item \textbf{cardinality}\index{cardinality}
19    
20        The cardinality of a set is the        The cardinality of a set is the
21        number of elements in the set.  In this work, the cardinality        number of elements in the set.  In this work, the cardinality
22        of a set is denoted $n()$.  For example,        of a set is denoted $n()$.  For example,
23        $n(\{12,29,327\}) = 3$.        $n(\{12,29,327\}) = 3$.
24    
25  \item \textbf{coprime}\index{coprime}  \item \textbf{coprime}\index{coprime}
26    
27        Two integers that share no prime factors are \emph{coprime}.        Two integers that share no prime factors are \emph{coprime}.
28        \emph{Example:}        \emph{Example:}
29        6 and 7 are coprime, whereas 6 and 8 are not.        6 and 7 are coprime, whereas 6 and 8 are not.
30    
31  \item \textbf{GMP}\index{GMP}  \item \textbf{GMP}\index{GMP}
32    
33        The \emph{G}NU \emph{M}ultiple \emph{P}recision library.        The \emph{G}NU \emph{M}ultiple \emph{P}recision library.
34        The GMP is an arbitrary-precision integer, rational number,        The GMP is an arbitrary-precision integer, rational number,
35        and floating-point library that places no restrictions on        and floating-point library that places no restrictions on
36        size of integers or number of significant digits in floating-point        size of integers or number of significant digits in floating-point
37        numbers.  This        numbers.  This
38        library is famous because it is the fastest of its        library is famous because it is the fastest of its
39        kind, and generally uses asymptotically superior algorithms.        kind, and generally uses asymptotically superior algorithms.
40    
41  \item \textbf{greatest common divisor (g.c.d.)}  \item \textbf{greatest common divisor (g.c.d.)}
42    
43        The greatest common divisor of two integers is the largest        The greatest common divisor of two integers is the largest
44        integer which divides both integers without a remainder.        integer which divides both integers without a remainder.
45        \emph{Example:} the g.c.d. of 30 and 42 is 6.        \emph{Example:} the g.c.d. of 30 and 42 is 6.
46    
47  \item \textbf{integer}\index{integer}\index{sets of integers}\index{Z@$\vworkintset$}%  \item \textbf{integer}\index{integer}\index{sets of integers}\index{Z@$\vworkintset$}%
48        \index{integer!Z@$\vworkintset$}\index{integer!sets of}        \index{integer!Z@$\vworkintset$}\index{integer!sets of}
49    
50        (Nearly verbatim from \cite{bibref:w:wwwwhatiscom}) An \emph{integer}        (Nearly verbatim from \cite{bibref:w:wwwwhatiscom}) An \emph{integer}
51        (pronounced \emph{IN-tuh-jer}) is a whole number        (pronounced \emph{IN-tuh-jer}) is a whole number
52        (not a fractional number) that can be positive, negative, or zero.        (not a fractional number) that can be positive, negative, or zero.
53    
54        Examples of integers are: -5, 1, 5, 8, 97, and 3,043.        Examples of integers are: -5, 1, 5, 8, 97, and 3,043.
55    
56        Examples of numbers that are not integers are: -1.43, 1 3/4, 3.14,        Examples of numbers that are not integers are: -1.43, 1 3/4, 3.14,
57        0.09, and 5,643.1.        0.09, and 5,643.1.
58    
59        The set of integers, denoted $\vworkintset{}$, is formally defined as:        The set of integers, denoted $\vworkintset{}$, is formally defined as:
60    
61        \begin{equation}        \begin{equation}
62        \vworkintset{} = \{\ldots{}, -3, -2, -1, 0, 1, 2, 3, \ldots{} \}        \vworkintset{} = \{\ldots{}, -3, -2, -1, 0, 1, 2, 3, \ldots{} \}
63        \end{equation}        \end{equation}
64    
65        In mathematical equations, unknown or unspecified integers are        In mathematical equations, unknown or unspecified integers are
66        represented by lowercase, italicized letters from the        represented by lowercase, italicized letters from the
67        ``late middle'' of the alphabet.  The most common        ``late middle'' of the alphabet.  The most common
68        are $p$, $q$, $r$, and $s$.        are $p$, $q$, $r$, and $s$.
69    
70  \item \textbf{irreducible}  \item \textbf{irreducible}
71    
72        A rational number $p/q$ where $p$ and $q$ are coprime        A rational number $p/q$ where $p$ and $q$ are coprime
73        is said to be \emph{irreducible}.        is said to be \emph{irreducible}.
74        Equivalently, it may be stated that $p$ and $q$ share no prime factors        Equivalently, it may be stated that $p$ and $q$ share no prime factors
75        or that the greatest common divisor of        or that the greatest common divisor of
76        $p$ and $q$ is 1.        $p$ and $q$ is 1.
77    
78  \item \textbf{KPH}  \item \textbf{KPH}
79    
80        Kilometers per hour.        Kilometers per hour.
81    
82  \item \textbf{limb}\index{limb}  \item \textbf{limb}\index{limb}
83    
84        An integer of a size which a machine can manipulate natively        An integer of a size which a machine can manipulate natively
85        that is arranged in an array to create a larger        that is arranged in an array to create a larger
86        integer which the machine cannot manipulate natively and must be        integer which the machine cannot manipulate natively and must be
87        manipulated through arithmetic subroutines.        manipulated through arithmetic subroutines.
88    
89  \item \textbf{limbsize}\index{limbsize}  \item \textbf{limbsize}\index{limbsize}
90    
91        The size, in bits, of a limb.  The limbsize usually represents        The size, in bits, of a limb.  The limbsize usually represents
92        the size of integer that a machine can manipulate directly        the size of integer that a machine can manipulate directly
93        through machine instructions.  For an inexpensive microcontroller,        through machine instructions.  For an inexpensive microcontroller,
94        8 or 16 is a typical limbsize.  For a personal computer or        8 or 16 is a typical limbsize.  For a personal computer or
95        workstation, 32 or 64 is a typical limbsize.        workstation, 32 or 64 is a typical limbsize.
96    
97  \item \textbf{MPH}  \item \textbf{MPH}
98    
99        Miles per hour.        Miles per hour.
100    
101  \item \textbf{mediant}\index{mediant}  \item \textbf{mediant}\index{mediant}
102    
103        The mediant of two fractions $m/n$ and $m'/n'$ is the fraction        The mediant of two fractions $m/n$ and $m'/n'$ is the fraction
104            $\frac{m+m'}{n+n'}$ (see Definition            $\frac{m+m'}{n+n'}$ (see Definition
105            \cfryzeroxrefhyphen{}\ref{def:cfry0:spfs:02}).  Note that the            \cfryzeroxrefhyphen{}\ref{def:cfry0:spfs:02}).  Note that the
106            mediant of two fractions with non-negative integer components            mediant of two fractions with non-negative integer components
107            is always between them, but not usually exactly at the            is always between them, but not usually exactly at the
108            midpoint (see Lemma \cfryzeroxrefhyphen{}\ref{lem:cfry0:spfs:02c}).            midpoint (see Lemma \cfryzeroxrefhyphen{}\ref{lem:cfry0:spfs:02c}).
109    
110  \item \textbf{natural number}\index{natural number}\index{integer!natural number}%  \item \textbf{natural number}\index{natural number}\index{integer!natural number}%
111        \index{sets of integers}\index{N@$\vworkintsetpos$}%        \index{sets of integers}\index{N@$\vworkintsetpos$}%
112        \index{integer!N@$\vworkintsetpos$}\index{integer!sets of}        \index{integer!N@$\vworkintsetpos$}\index{integer!sets of}
113                    
114        (Nearly verbatim from \cite{bibref:w:wwwwhatiscom})        (Nearly verbatim from \cite{bibref:w:wwwwhatiscom})
115        A \emph{natural number}        A \emph{natural number}
116        is a number that occurs commonly and obviously in nature.          is a number that occurs commonly and obviously in nature.  
117        As such, it is a whole, non-negative number.          As such, it is a whole, non-negative number.  
118        The set of natural numbers, denoted $\vworkintsetpos{}$,        The set of natural numbers, denoted $\vworkintsetpos{}$,
119        can be defined in either of two ways:        can be defined in either of two ways:
120    
121        \begin{equation}        \begin{equation}
122        \label{cglo0:eq0001}        \label{cglo0:eq0001}
123        \vworkintsetpos{} = \{ 0, 1, 2, 3, \ldots{} \}        \vworkintsetpos{} = \{ 0, 1, 2, 3, \ldots{} \}
124        \end{equation}        \end{equation}
125    
126        \begin{equation}        \begin{equation}
127        \label{cglo0:eq0002}        \label{cglo0:eq0002}
128        \vworkintsetpos{} = \{ 1, 2, 3, 4, \ldots{} \}        \vworkintsetpos{} = \{ 1, 2, 3, 4, \ldots{} \}
129        \end{equation}        \end{equation}
130                
131        In mathematical equations, unknown or unspecified natural numbers        In mathematical equations, unknown or unspecified natural numbers
132        are represented by lowercase, italicized letters from the        are represented by lowercase, italicized letters from the
133        middle of the alphabet.  The most common is $n$, followed by        middle of the alphabet.  The most common is $n$, followed by
134        $m$, $p$, and $q$.          $m$, $p$, and $q$.  
135        In subscripts, the lowercase $i$ is sometimes used to represent        In subscripts, the lowercase $i$ is sometimes used to represent
136        a non-specific natural number when denoting the elements in a        a non-specific natural number when denoting the elements in a
137        sequence or series.  However, $i$ is more often used to represent        sequence or series.  However, $i$ is more often used to represent
138        the positive square root of -1, the unit imaginary number.        the positive square root of -1, the unit imaginary number.
139    
140        \textbf{Important Note:}  The definition above is reproduced nearly        \textbf{Important Note:}  The definition above is reproduced nearly
141        verbatim from \cite{bibref:w:wwwwhatiscom}, and (\ref{cglo0:eq0001})        verbatim from \cite{bibref:w:wwwwhatiscom}, and (\ref{cglo0:eq0001})
142        is supplied only for perspective.  In this work, a natural        is supplied only for perspective.  In this work, a natural
143        number is defined by (\ref{cglo0:eq0002}) rather than (\ref{cglo0:eq0001}).        number is defined by (\ref{cglo0:eq0002}) rather than (\ref{cglo0:eq0001}).
144        In this work, the set of non-negative integers is denoted by        In this work, the set of non-negative integers is denoted by
145        $\vworkintsetnonneg{}$ rather than $\vworkintsetpos{}$.\index{Z+@$\vworkintsetnonneg$}%        $\vworkintsetnonneg{}$ rather than $\vworkintsetpos{}$.\index{Z+@$\vworkintsetnonneg$}%
146        \index{integer!Z+@$\vworkintsetnonneg$}\index{integer!non-negative}        \index{integer!Z+@$\vworkintsetnonneg$}\index{integer!non-negative}
147    
148  \item \textbf{postulate}\index{postulate!definition}  \item \textbf{postulate}\index{postulate!definition}
149    
150        An axiom (see \emph{axiom} earlier in this glossary).  The term is usually        An axiom (see \emph{axiom} earlier in this glossary).  The term is usually
151            used in certain contexts, e.g. Euclid's postulates or Peano's postulates.            used in certain contexts, e.g. Euclid's postulates or Peano's postulates.
152            (Taken verbatim from \cite{bibref:b:penguindictionaryofmathematics:2ded}.)            (Taken verbatim from \cite{bibref:b:penguindictionaryofmathematics:2ded}.)
153    
154  \item \textbf{prime number}\index{prime number!definition}  \item \textbf{prime number}\index{prime number!definition}
155    
156        (Nearly verbatim from \cite{bibref:w:wwwwhatiscom}) A \emph{prime number}        (Nearly verbatim from \cite{bibref:w:wwwwhatiscom}) A \emph{prime number}
157        is a whole number greater than 1, whose only two whole-number        is a whole number greater than 1, whose only two whole-number
158        factors are 1 and itself.  The first few prime numbers are        factors are 1 and itself.  The first few prime numbers are
159        2, 3, 5, 7, 11, 13, 17, 19, 23, and 29.  As we proceed in the set of        2, 3, 5, 7, 11, 13, 17, 19, 23, and 29.  As we proceed in the set of
160        natural numbers $\vworkintsetpos{} = \{ 1, 2, 3, \ldots{} \} $, the        natural numbers $\vworkintsetpos{} = \{ 1, 2, 3, \ldots{} \} $, the
161        primes become less and less frequent in general.          primes become less and less frequent in general.  
162        However, there is no largest prime number.          However, there is no largest prime number.  
163        For every prime number $p$, there exists a prime number $p'$ such that        For every prime number $p$, there exists a prime number $p'$ such that
164        $p'$ is greater than $p$.  This was demonstrated in ancient times by the        $p'$ is greater than $p$.  This was demonstrated in ancient times by the
165        Greek mathematician \index{Euclid}Euclid.\index{prime number!no largest prime number}%        Greek mathematician \index{Euclid}Euclid.\index{prime number!no largest prime number}%
166        \index{Euclid!Second Theorem}        \index{Euclid!Second Theorem}
167    
168        Suppose $n$ is a whole number, and we want to test it to see if it is prime.          Suppose $n$ is a whole number, and we want to test it to see if it is prime.  
169        First, we take the square root (or the 1/2 power) of $n$; then we round this        First, we take the square root (or the 1/2 power) of $n$; then we round this
170        number up to the next highest whole number.  Call the result $m$.          number up to the next highest whole number.  Call the result $m$.  
171        We must find all of the following quotients:        We must find all of the following quotients:
172    
173        \begin{equation}        \begin{equation}
174        \begin{array}{rcl}        \begin{array}{rcl}
175           q_m     & =        & n / m              \\           q_m     & =        & n / m              \\
176           q_{m-1} & =        & n / (m-1)          \\           q_{m-1} & =        & n / (m-1)          \\
177           q_{m-2} & =        & n / (m-2)          \\           q_{m-2} & =        & n / (m-2)          \\
178           q_{m-3} & =        & n / (m-3)          \\           q_{m-3} & =        & n / (m-3)          \\
179                   & \ldots{} &                    \\                   & \ldots{} &                    \\
180           q_3     & =        & n / 3              \\           q_3     & =        & n / 3              \\
181           q_2     & =        & n / 2              \\           q_2     & =        & n / 2              \\
182        \end{array}        \end{array}
183        \end{equation}        \end{equation}
184    
185        The number $n$ is prime if and only if none of the $q$'s, as        The number $n$ is prime if and only if none of the $q$'s, as
186        derived above, are whole numbers.        derived above, are whole numbers.
187    
188        A computer can be used to test extremely large numbers to see if they are prime.          A computer can be used to test extremely large numbers to see if they are prime.  
189        But, because there is no limit to how large a natural number can be,        But, because there is no limit to how large a natural number can be,
190        there is always a point where testing in this manner becomes too great        there is always a point where testing in this manner becomes too great
191        a task even for the most powerful supercomputers.          a task even for the most powerful supercomputers.  
192        Various algorithms have been formulated in an attempt to generate        Various algorithms have been formulated in an attempt to generate
193        ever-larger prime numbers.  These schemes all have limitations.        ever-larger prime numbers.  These schemes all have limitations.
194    
195  \end{vworktermglossaryenum}  \end{vworktermglossaryenum}
196    
197  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
198    
199  \noindent\begin{figure}[!b]  \noindent\begin{figure}[!b]
200  \noindent\rule[-0.25in]{\textwidth}{1pt}  \noindent\rule[-0.25in]{\textwidth}{1pt}
201  \begin{tiny}  \begin{tiny}
202  \begin{verbatim}  \begin{verbatim}
203  $RCSfile: c_glo0.tex,v $  $RCSfile: c_glo0.tex,v $
204  $Source: /home/dashley/cvsrep/e3ft_gpl01/e3ft_gpl01/dtaipubs/esrgubka/c_glo0/c_glo0.tex,v $  $Source: /home/dashley/cvsrep/e3ft_gpl01/e3ft_gpl01/dtaipubs/esrgubka/c_glo0/c_glo0.tex,v $
205  $Revision: 1.9 $  $Revision: 1.9 $
206  $Author: dtashley $  $Author: dtashley $
207  $Date: 2003/03/13 06:28:06 $  $Date: 2003/03/13 06:28:06 $
208  \end{verbatim}  \end{verbatim}
209  \end{tiny}  \end{tiny}
210  \noindent\rule[0.25in]{\textwidth}{1pt}  \noindent\rule[0.25in]{\textwidth}{1pt}
211  \end{figure}  \end{figure}
212    
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214  % $Log: c_glo0.tex,v $  % $Log: c_glo0.tex,v $
215  % Revision 1.9  2003/03/13 06:28:06  dtashley  % Revision 1.9  2003/03/13 06:28:06  dtashley
216  % Cardinality definition and notation added.  % Cardinality definition and notation added.
217  %  %
218  % Revision 1.8  2002/08/26 17:57:03  dtashley  % Revision 1.8  2002/08/26 17:57:03  dtashley
219  % Additional solutions chapter added.  Precautionary checkin to be sure  % Additional solutions chapter added.  Precautionary checkin to be sure
220  % that I've captured all changes.  % that I've captured all changes.
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222  % Revision 1.7  2001/08/25 22:51:25  dtashley  % Revision 1.7  2001/08/25 22:51:25  dtashley
223  % Complex re-organization of book.  % Complex re-organization of book.
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225  % Revision 1.6  2001/08/16 19:53:27  dtashley  % Revision 1.6  2001/08/16 19:53:27  dtashley
226  % Beginning to prepare for v1.05 release.  % Beginning to prepare for v1.05 release.
227  %  %
228  % Revision 1.5  2001/07/11 18:42:05  dtashley  % Revision 1.5  2001/07/11 18:42:05  dtashley
229  % Safety check-in.  Beginning work now on using GNU GMP in the tool set  % Safety check-in.  Beginning work now on using GNU GMP in the tool set
230  % and must cease work on book temporarily.  % and must cease work on book temporarily.
231  %  %
232  % Revision 1.4  2001/07/01 19:06:17  dtashley  % Revision 1.4  2001/07/01 19:06:17  dtashley
233  % Version control keywords changed.  % Version control keywords changed.
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235  % Revision 1.3  2001/07/01 19:05:20  dtashley  % Revision 1.3  2001/07/01 19:05:20  dtashley
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239  % $History: c_glo0.tex $  % $History: c_glo0.tex $
240  %  %
241  % *****************  Version 3  *****************  % *****************  Version 3  *****************
242  % User: Dashley1     Date: 1/31/01    Time: 4:20p  % User: Dashley1     Date: 1/31/01    Time: 4:20p
243  % Updated in $/uC Software Multi-Volume Book (A)/Chapter, GLO0, Glossary Of Terms  % Updated in $/uC Software Multi-Volume Book (A)/Chapter, GLO0, Glossary Of Terms
244  % Edits.  % Edits.
245  %  %
246  % *****************  Version 2  *****************  % *****************  Version 2  *****************
247  % User: David T. Ashley Date: 7/30/00    Time: 8:21p  % User: David T. Ashley Date: 7/30/00    Time: 8:21p
248  % Updated in $/uC Software Multi-Volume Book (A)/Chapter, GLO0, Glossary Of Terms  % Updated in $/uC Software Multi-Volume Book (A)/Chapter, GLO0, Glossary Of Terms
249  % Edits.  % Edits.
250  %  %
251  % *****************  Version 1  *****************  % *****************  Version 1  *****************
252  % User: David T. Ashley Date: 7/30/00    Time: 6:47p  % User: David T. Ashley Date: 7/30/00    Time: 6:47p
253  % Created in $/uC Software Multi-Volume Book (A)/Chapter, GLO0, Glossary Of Terms  % Created in $/uC Software Multi-Volume Book (A)/Chapter, GLO0, Glossary Of Terms
254  % Initial check-in.  % Initial check-in.
255  %  %
256  %End of file C_GLO0.TEX  %End of file C_GLO0.TEX

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