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1 | dashley | 140 | %$Header$ |

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 | \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 | dashley | 278 | $HeadURL$ |

204 | $Revision$ | ||

205 | $Date$ | ||

206 | $Author$ | ||

207 | dashley | 140 | \end{verbatim} |

208 | \end{tiny} | ||

209 | \noindent\rule[0.25in]{\textwidth}{1pt} | ||

210 | \end{figure} | ||

211 | |||

212 | dashley | 278 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |

213 | dashley | 140 | % |

214 | %End of file C_GLO0.TEX |

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