%$Header$ \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: $$\vworkintset{} = \{\ldots{}, -3, -2, -1, 0, 1, 2, 3, \ldots{} \}$$ 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: $$\label{cglo0:eq0001} \vworkintsetpos{} = \{ 0, 1, 2, 3, \ldots{} \}$$ $$\label{cglo0:eq0002} \vworkintsetpos{} = \{ 1, 2, 3, 4, \ldots{} \}$$ 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: $$\begin{array}{rcl} q_m & = & n / m \\ q_{m-1} & = & n / (m-1) \\ q_{m-2} & = & n / (m-2) \\ q_{m-3} & = & n / (m-3) \\ & \ldots{} & \\ q_3 & = & n / 3 \\ q_2 & = & n / 2 \\ \end{array}$$ 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