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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 $ |

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 |

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