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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
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234     %
235     % Revision 1.3 2001/07/01 19:05:20 dtashley
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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

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