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 1 %$Header$ 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 62 \vworkintset{} = \{\ldots{}, -3, -2, -1, 0, 1, 2, 3, \ldots{} \} 63 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: 120 121 122 \label{cglo0:eq0001} 123 \vworkintsetpos{} = \{ 0, 1, 2, 3, \ldots{} \} 124 125 126 127 \label{cglo0:eq0002} 128 \vworkintsetpos{} = \{ 1, 2, 3, 4, \ldots{} \} 129 130 131 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: 172 173 174 \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} 183 184 185 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

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