trunk/c_glo0/c_glo0.tex

%$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 $

\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}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% $Log: c_glo0.tex,v $
% Revision 1.9  2003/03/13 06:28:06  dtashley
% Cardinality definition and notation added.
%
% Revision 1.8  2002/08/26 17:57:03  dtashley
% Additional solutions chapter added.  Precautionary checkin to be sure
% that I've captured all changes.
%
% Revision 1.7  2001/08/25 22:51:25  dtashley
% Complex re-organization of book.
%
% Revision 1.6  2001/08/16 19:53:27  dtashley
% Beginning to prepare for v1.05 release.
%
% 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
% Version control keywords changed.
%
% Revision 1.3  2001/07/01 19:05:20  dtashley
% Move out of binary mode (second attempt) for use with CVS.
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% $History: c_glo0.tex $
% 
% *****************  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 $/uC Software Multi-Volume Book (A)/Chapter, GLO0, Glossary Of Terms
% Initial check-in.
%
%End of file C_GLO0.TEX
1	%$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 $
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