trunk/c_glo0/c_glo0.tex

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\chapter*{Glossary Of Terms}
\markboth{GLOSSARY OF TERMS}{GLOSSARY OF TERMS}

\label{cglo0}

\begin{vworktermglossaryenum}

\item \textbf{axiom}\index{axiom}

      A statement used in the premises of arguments and assumed to be true
          without proof.  In some cases axioms are held to be self-evident, as in 
          Euclidian geometry, while in others they are assumptions put forward for
          the sake of argument.
      (Taken verbatim from \cite{bibref:b:penguindictionaryofmathematics:2ded}.)

\item \textbf{cardinality}\index{cardinality}

      The cardinality of a set is the
      number of elements in the set.  In this work, the cardinality
      of a set is denoted $n()$.  For example, 
      $n(\{12,29,327\}) = 3$.

\item \textbf{coprime}\index{coprime}

      Two integers that share no prime factors are \emph{coprime}.
      \emph{Example:}
      6 and 7 are coprime, whereas 6 and 8 are not.

\item \textbf{GMP}\index{GMP}

      The \emph{G}NU \emph{M}ultiple \emph{P}recision library.
      The GMP is an arbitrary-precision integer, rational number,
      and floating-point library that places no restrictions on
      size of integers or number of significant digits in floating-point
      numbers.  This 
      library is famous because it is the fastest of its
      kind, and generally uses asymptotically superior algorithms.

\item \textbf{greatest common divisor (g.c.d.)}

      The greatest common divisor of two integers is the largest
      integer which divides both integers without a remainder.
      \emph{Example:} the g.c.d. of 30 and 42 is 6.

\item \textbf{integer}\index{integer}\index{sets of integers}\index{Z@$\vworkintset$}%
      \index{integer!Z@$\vworkintset$}\index{integer!sets of}

      (Nearly verbatim from \cite{bibref:w:wwwwhatiscom}) An \emph{integer}
      (pronounced \emph{IN-tuh-jer}) is a whole number
      (not a fractional number) that can be positive, negative, or zero. 

      Examples of integers are: -5, 1, 5, 8, 97, and 3,043. 

      Examples of numbers that are not integers are: -1.43, 1 3/4, 3.14, 
      0.09, and 5,643.1. 

      The set of integers, denoted $\vworkintset{}$, is formally defined as:

      \begin{equation}
      \vworkintset{} = \{\ldots{}, -3, -2, -1, 0, 1, 2, 3, \ldots{} \}
      \end{equation}

      In mathematical equations, unknown or unspecified integers are 
      represented by lowercase, italicized letters from the 
      ``late middle'' of the alphabet.  The most common 
      are $p$, $q$, $r$, and $s$.

\item \textbf{irreducible}

      A rational number $p/q$ where $p$ and $q$ are coprime
      is said to be \emph{irreducible}.
      Equivalently, it may be stated that $p$ and $q$ share no prime factors
      or that the greatest common divisor of
      $p$ and $q$ is 1.

\item \textbf{KPH}

      Kilometers per hour.

\item \textbf{limb}\index{limb}

      An integer of a size which a machine can manipulate natively
      that is arranged in an array to create a larger
      integer which the machine cannot manipulate natively and must be
      manipulated through arithmetic subroutines.

\item \textbf{limbsize}\index{limbsize}

      The size, in bits, of a limb.  The limbsize usually represents
      the size of integer that a machine can manipulate directly
      through machine instructions.  For an inexpensive microcontroller,
      8 or 16 is a typical limbsize.  For a personal computer or 
      workstation, 32 or 64 is a typical limbsize.

\item \textbf{MPH}

      Miles per hour.

\item \textbf{mediant}\index{mediant}

      The mediant of two fractions $m/n$ and $m'/n'$ is the fraction 
          $\frac{m+m'}{n+n'}$ (see Definition 
          \cfryzeroxrefhyphen{}\ref{def:cfry0:spfs:02}).  Note that the
          mediant of two fractions with non-negative integer components
          is always between them, but not usually exactly at the 
          midpoint (see Lemma \cfryzeroxrefhyphen{}\ref{lem:cfry0:spfs:02c}).

\item \textbf{natural number}\index{natural number}\index{integer!natural number}%
      \index{sets of integers}\index{N@$\vworkintsetpos$}%
      \index{integer!N@$\vworkintsetpos$}\index{integer!sets of}
         
      (Nearly verbatim from \cite{bibref:w:wwwwhatiscom})
      A \emph{natural number}
      is a number that occurs commonly and obviously in nature.  
      As such, it is a whole, non-negative number.  
      The set of natural numbers, denoted $\vworkintsetpos{}$, 
      can be defined in either of two ways:

      \begin{equation}
      \label{cglo0:eq0001}
      \vworkintsetpos{} = \{ 0, 1, 2, 3, \ldots{} \}
      \end{equation}

      \begin{equation}
      \label{cglo0:eq0002}
      \vworkintsetpos{} = \{ 1, 2, 3, 4, \ldots{} \}
      \end{equation}
      
      In mathematical equations, unknown or unspecified natural numbers 
      are represented by lowercase, italicized letters from the 
      middle of the alphabet.  The most common is $n$, followed by 
      $m$, $p$, and $q$.  
      In subscripts, the lowercase $i$ is sometimes used to represent 
      a non-specific natural number when denoting the elements in a 
      sequence or series.  However, $i$ is more often used to represent 
      the positive square root of -1, the unit imaginary number.

      \textbf{Important Note:}  The definition above is reproduced nearly
      verbatim from \cite{bibref:w:wwwwhatiscom}, and (\ref{cglo0:eq0001})
      is supplied only for perspective.  In this work, a natural
      number is defined by (\ref{cglo0:eq0002}) rather than (\ref{cglo0:eq0001}).
      In this work, the set of non-negative integers is denoted by
      $\vworkintsetnonneg{}$ rather than $\vworkintsetpos{}$.\index{Z+@$\vworkintsetnonneg$}%
      \index{integer!Z+@$\vworkintsetnonneg$}\index{integer!non-negative}

\item \textbf{postulate}\index{postulate!definition}

      An axiom (see \emph{axiom} earlier in this glossary).  The term is usually
          used in certain contexts, e.g. Euclid's postulates or Peano's postulates.
          (Taken verbatim from \cite{bibref:b:penguindictionaryofmathematics:2ded}.)

\item \textbf{prime number}\index{prime number!definition}

      (Nearly verbatim from \cite{bibref:w:wwwwhatiscom}) A \emph{prime number}
      is a whole number greater than 1, whose only two whole-number 
      factors are 1 and itself.  The first few prime numbers are 
      2, 3, 5, 7, 11, 13, 17, 19, 23, and 29.  As we proceed in the set of 
      natural numbers $\vworkintsetpos{} = \{ 1, 2, 3, \ldots{} \} $, the 
      primes become less and less frequent in general.  
      However, there is no largest prime number.  
      For every prime number $p$, there exists a prime number $p'$ such that 
      $p'$ is greater than $p$.  This was demonstrated in ancient times by the 
      Greek mathematician \index{Euclid}Euclid.\index{prime number!no largest prime number}%
      \index{Euclid!Second Theorem}

      Suppose $n$ is a whole number, and we want to test it to see if it is prime.   
      First, we take the square root (or the 1/2 power) of $n$; then we round this 
      number up to the next highest whole number.  Call the result $m$.  
      We must find all of the following quotients:

      \begin{equation}
      \begin{array}{rcl}
         q_m     & =        & n / m              \\
         q_{m-1} & =        & n / (m-1)          \\
         q_{m-2} & =        & n / (m-2)          \\
         q_{m-3} & =        & n / (m-3)          \\
                 & \ldots{} &                    \\
         q_3     & =        & n / 3              \\
         q_2     & =        & n / 2              \\
      \end{array}
      \end{equation}

      The number $n$ is prime if and only if none of the $q$'s, as 
      derived above, are whole numbers.

      A computer can be used to test extremely large numbers to see if they are prime.  
      But, because there is no limit to how large a natural number can be, 
      there is always a point where testing in this manner becomes too great 
      a task even for the most powerful supercomputers.  
      Various algorithms have been formulated in an attempt to generate 
      ever-larger prime numbers.  These schemes all have limitations.

\end{vworktermglossaryenum}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\noindent\begin{figure}[!b]
\noindent\rule[-0.25in]{\textwidth}{1pt}
\begin{tiny}
\begin{verbatim}
$RCSfile: c_glo0.tex,v $
$Source: /home/dashley/cvsrep/e3ft_gpl01/e3ft_gpl01/dtaipubs/esrgubka/c_glo0/c_glo0.tex,v $
$Revision: 1.9 $
$Author: dtashley $
$Date: 2003/03/13 06:28:06 $
\end{verbatim}
\end{tiny}
\noindent\rule[0.25in]{\textwidth}{1pt}
\end{figure}

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% Revision 1.9  2003/03/13 06:28:06  dtashley
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% Revision 1.8  2002/08/26 17:57:03  dtashley
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% that I've captured all changes.
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% Revision 1.7  2001/08/25 22:51:25  dtashley
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% Revision 1.6  2001/08/16 19:53:27  dtashley
% Beginning to prepare for v1.05 release.
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% Revision 1.5  2001/07/11 18:42:05  dtashley
% Safety check-in.  Beginning work now on using GNU GMP in the tool set
% and must cease work on book temporarily.
%
% Revision 1.4  2001/07/01 19:06:17  dtashley
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%
% Revision 1.3  2001/07/01 19:05:20  dtashley
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% $History: c_glo0.tex $
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% *****************  Version 3  *****************
% User: Dashley1     Date: 1/31/01    Time: 4:20p
% Updated in $/uC Software Multi-Volume Book (A)/Chapter, GLO0, Glossary Of Terms
% Edits.
% 
% *****************  Version 2  *****************
% User: David T. Ashley Date: 7/30/00    Time: 8:21p
% Updated in $/uC Software Multi-Volume Book (A)/Chapter, GLO0, Glossary Of Terms
% Edits.
%
% *****************  Version 1  *****************
% User: David T. Ashley Date: 7/30/00    Time: 6:47p
% Created in $/uC Software Multi-Volume Book (A)/Chapter, GLO0, Glossary Of Terms
% Initial check-in.
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%End of file C_GLO0.TEX
1	dashley	5	%$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 $
2
3			\chapter*{Glossary Of Terms}
4			\markboth{GLOSSARY OF TERMS}{GLOSSARY OF TERMS}
5
6			\label{cglo0}
7
8			\begin{vworktermglossaryenum}
9
10			\item \textbf{axiom}\index{axiom}
11
12			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
14			Euclidian geometry, while in others they are assumptions put forward for
15			the sake of argument.
16			(Taken verbatim from \cite{bibref:b:penguindictionaryofmathematics:2ded}.)
17
18			\item \textbf{cardinality}\index{cardinality}
19
20			The cardinality of a set is the
21			number of elements in the set. In this work, the cardinality
22			of a set is denoted $n()$. For example,
23			$n(\{12,29,327\}) = 3$.
24
25			\item \textbf{coprime}\index{coprime}
26
27			Two integers that share no prime factors are \emph{coprime}.
28			\emph{Example:}
29			6 and 7 are coprime, whereas 6 and 8 are not.
30
31			\item \textbf{GMP}\index{GMP}
32
33			The \emph{G}NU \emph{M}ultiple \emph{P}recision library.
34			The GMP is an arbitrary-precision integer, rational number,
35			and floating-point library that places no restrictions on
36			size of integers or number of significant digits in floating-point
37			numbers. This
38			library is famous because it is the fastest of its
39			kind, and generally uses asymptotically superior algorithms.
40
41			\item \textbf{greatest common divisor (g.c.d.)}
42
43			The greatest common divisor of two integers is the largest
44			integer which divides both integers without a remainder.
45			\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@$\vworkintset$}%
48			\index{integer!Z@$\vworkintset$}\index{integer!sets of}
49
50			(Nearly verbatim from \cite{bibref:w:wwwwhatiscom}) An \emph{integer}
51			(pronounced \emph{IN-tuh-jer}) is a whole number
52			(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.
55
56			Examples of numbers that are not integers are: -1.43, 1 3/4, 3.14,
57			0.09, and 5,643.1.
58
59			The set of integers, denoted $\vworkintset{}$, is formally defined as:
60
61			\begin{equation}
62			\vworkintset{} = \{\ldots{}, -3, -2, -1, 0, 1, 2, 3, \ldots{} \}
63			\end{equation}
64
65			In mathematical equations, unknown or unspecified integers are
66			represented by lowercase, italicized letters from the
67			``late middle'' of the alphabet. The most common
68			are $p$, $q$, $r$, and $s$.
69
70			\item \textbf{irreducible}
71
72			A rational number $p/q$ where $p$ and $q$ are coprime
73			is said to be \emph{irreducible}.
74			Equivalently, it may be stated that $p$ and $q$ share no prime factors
75			or that the greatest common divisor of
76			$p$ and $q$ is 1.
77
78			\item \textbf{KPH}
79
80			Kilometers per hour.
81
82			\item \textbf{limb}\index{limb}
83
84			An integer of a size which a machine can manipulate natively
85			that is arranged in an array to create a larger
86			integer which the machine cannot manipulate natively and must be
87			manipulated through arithmetic subroutines.
88
89			\item \textbf{limbsize}\index{limbsize}
90
91			The size, in bits, of a limb. The limbsize usually represents
92			the size of integer that a machine can manipulate directly
93			through machine instructions. For an inexpensive microcontroller,
94			8 or 16 is a typical limbsize. For a personal computer or
95			workstation, 32 or 64 is a typical limbsize.
96
97			\item \textbf{MPH}
98
99			Miles per hour.
100
101			\item \textbf{mediant}\index{mediant}
102
103			The mediant of two fractions $m/n$ and $m'/n'$ is the fraction
104			$\frac{m+m'}{n+n'}$ (see Definition
105			\cfryzeroxrefhyphen{}\ref{def:cfry0:spfs:02}). Note that the
106			mediant of two fractions with non-negative integer components
107			is always between them, but not usually exactly at the
108			midpoint (see Lemma \cfryzeroxrefhyphen{}\ref{lem:cfry0:spfs:02c}).
109
110			\item \textbf{natural number}\index{natural number}\index{integer!natural number}%
111			\index{sets of integers}\index{N@$\vworkintsetpos$}%
112			\index{integer!N@$\vworkintsetpos$}\index{integer!sets of}
113
114			(Nearly verbatim from \cite{bibref:w:wwwwhatiscom})
115			A \emph{natural number}
116			is a number that occurs commonly and obviously in nature.
117			As such, it is a whole, non-negative number.
118			The set of natural numbers, denoted $\vworkintsetpos{}$,
119			can be defined in either of two ways:
120
121			\begin{equation}
122			\label{cglo0:eq0001}
123			\vworkintsetpos{} = \{ 0, 1, 2, 3, \ldots{} \}
124			\end{equation}
125
126			\begin{equation}
127			\label{cglo0:eq0002}
128			\vworkintsetpos{} = \{ 1, 2, 3, 4, \ldots{} \}
129			\end{equation}
130
131			In mathematical equations, unknown or unspecified natural numbers
132			are represented by lowercase, italicized letters from the
133			middle of the alphabet. The most common is $n$, followed by
134			$m$, $p$, and $q$.
135			In subscripts, the lowercase $i$ is sometimes used to represent
136			a non-specific natural number when denoting the elements in a
137			sequence or series. However, $i$ is more often used to represent
138			the positive square root of -1, the unit imaginary number.
139
140			\textbf{Important Note:} The definition above is reproduced nearly
141			verbatim from \cite{bibref:w:wwwwhatiscom}, and (\ref{cglo0:eq0001})
142			is supplied only for perspective. In this work, a natural
143			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
145			$\vworkintsetnonneg{}$ rather than $\vworkintsetpos{}$.\index{Z+@$\vworkintsetnonneg$}%
146			\index{integer!Z+@$\vworkintsetnonneg$}\index{integer!non-negative}
147
148			\item \textbf{postulate}\index{postulate!definition}
149
150			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.
152			(Taken verbatim from \cite{bibref:b:penguindictionaryofmathematics:2ded}.)
153
154			\item \textbf{prime number}\index{prime number!definition}
155
156			(Nearly verbatim from \cite{bibref:w:wwwwhatiscom}) A \emph{prime number}
157			is a whole number greater than 1, whose only two whole-number
158			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
160			natural numbers $\vworkintsetpos{} = \{ 1, 2, 3, \ldots{} \} $, the
161			primes become less and less frequent in general.
162			However, there is no largest prime number.
163			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
165			Greek mathematician \index{Euclid}Euclid.\index{prime number!no largest prime number}%
166			\index{Euclid!Second Theorem}
167
168			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
170			number up to the next highest whole number. Call the result $m$.
171			We must find all of the following quotients:
172
173			\begin{equation}
174			\begin{array}{rcl}
175			q_m & = & n / m \\
176			q_{m-1} & = & n / (m-1) \\
177			q_{m-2} & = & n / (m-2) \\
178			q_{m-3} & = & n / (m-3) \\
179			& \ldots{} & \\
180			q_3 & = & n / 3 \\
181			q_2 & = & n / 2 \\
182			\end{array}
183			\end{equation}
184
185			The number $n$ is prime if and only if none of the $q$'s, as
186			derived above, are whole numbers.
187
188			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,
190			there is always a point where testing in this manner becomes too great
191			a task even for the most powerful supercomputers.
192			Various algorithms have been formulated in an attempt to generate
193			ever-larger prime numbers. These schemes all have limitations.
194
195			\end{vworktermglossaryenum}
196
197			%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
198
199			\noindent\begin{figure}[!b]
200			\noindent\rule[-0.25in]{\textwidth}{1pt}
201			\begin{tiny}
202			\begin{verbatim}
203			$RCSfile: c_glo0.tex,v $
204			$Source: /home/dashley/cvsrep/e3ft_gpl01/e3ft_gpl01/dtaipubs/esrgubka/c_glo0/c_glo0.tex,v $
205			$Revision: 1.9 $
206			$Author: dtashley $
207			$Date: 2003/03/13 06:28:06 $
208			\end{verbatim}
209			\end{tiny}
210			\noindent\rule[0.25in]{\textwidth}{1pt}
211			\end{figure}
212
213			%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
214			% $Log: c_glo0.tex,v $
215			% Revision 1.9 2003/03/13 06:28:06 dtashley
216			% Cardinality definition and notation added.
217			%
218			% Revision 1.8 2002/08/26 17:57:03 dtashley
219			% Additional solutions chapter added. Precautionary checkin to be sure
220			% that I've captured all changes.
221			%
222			% Revision 1.7 2001/08/25 22:51:25 dtashley
223			% Complex re-organization of book.
224			%
225			% Revision 1.6 2001/08/16 19:53:27 dtashley
226			% Beginning to prepare for v1.05 release.
227			%
228			% 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
230			% and must cease work on book temporarily.
231			%
232			% Revision 1.4 2001/07/01 19:06:17 dtashley
233			% Version control keywords changed.
234			%
235			% Revision 1.3 2001/07/01 19:05:20 dtashley
236			% Move out of binary mode (second attempt) for use with CVS.
237			%
238			%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
239			% $History: c_glo0.tex $
240			%
241			% *************** Version 3 ***************
242			% User: Dashley1 Date: 1/31/01 Time: 4:20p
243			% Updated in $/uC Software Multi-Volume Book (A)/Chapter, GLO0, Glossary Of Terms
244			% Edits.
245			%
246			% *************** Version 2 ***************
247			% 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
249			% Edits.
250			%
251			% *************** Version 1 ***************
252			% 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
254			% Initial check-in.
255			%
256			%End of file C_GLO0.TEX