\chapter{Numbers} %version définitive (!) 2000-09-10   
   
\Citation{I have often admired the mystical way of Pythagoras\footnotemark, and the
  secret magic of numbers.\\  
  \auteur{Sir Thomas}{Browne}{1605}{1682}{English writer and physician}{Religio Medici}{1643}} 
   \footnotetext{Pythagoras \pron{p£p£h£f£ag£or£es}} %prononciation ajoutée 2003-06-15
\section{Reminder}   
   
\subsection{Writing and reading numbers}   
   
To write numbers one uses ten \emph{figures} \pron{£hf£ig£e£Z} or \emph{digits} \pron{£hd£i£d£it}:%
\index{figure}\index{digit} 
   
\begin{longtable}[c]{l l l l l l}   
  $0$ & zero \emph{or} nought & \pron{£hz£Qr£w -- n£Ot}& $1$ & one & \pron{w£vn} \\ 
  $2$ & two & \pron{t£U} & $3$ & three& \pron{£fr£I} \\ 
  $4$ & four & \pron{f£O£Z}& $5$ & five & \pron{f£pv} \\ 
  $6$ & six & \pron{s£iks}& $7$ & seven & \pron{£hsevn}\\ 
  $8$ & eight & \pron{£qt}& $9$ & nine & \pron{n£pn} \\ 
\end{longtable}   
   
`Naught' is a variant spelling (especially in the USA) % correction 2003-06-15
of `nought'.\index{naught}\index{nought} 

\enlargethispage{\baselineskip}
   
$100$ is `one hundred' \pron{£hh£vndr£ed}; $256$ is `two hundred \textbf{and} fifty six'; $7,547$ is `seven thousand 
\pron{£h£f£Wznd} five hundred and forty seven'. 
   
$3{\cdot}142$ is `three point one four two'. The \emph{decimal point}\index{decimal point}\index{point!decimal} used to 
be centered in British English, e.g.\,\footnote{\parex\ (\emph{abbr.}) `for example', Latin `exempli gratia'. It is read 
  \pron{£b£I£hd£j£I} or \pron{f£er£ig£hz£Ampl}.}  $3\ept14$, but not in American English, \parex\ $3.14$.%
\index{e.g.\cfnote} 
   
$\cdot472$ is $0{\cdot}472$ i.e.\,\footnote{\idest\ (\emph{abbr.})  `that is to say', Latin `id est' \pron{£pd £Ist}. It 
  is read \pron{£b£p£h£I} or \pron{£b£z£at£h£iz}} `point four seven two' or `nought point four seven two'.  $\cdot01$ is 
`nought point nought one' or `point nought one'.\index{i.e.\cfnote} 
   
\subsection{Scientific Notation --- Standard Form}   
   
Scientific \pron{£bs£p£en£ht£if£ik} notation is a way of writing numbers that makes it easier to work with the very 
large and small numbers often found in scientific measurements \pron{£hme£j£em£ent}. 
   
When a number is written in scientific notation, the number is written as the base number $m$ (or mantissa 
\pron{m£an£ht£is£e}) times ten raised to a power $p$ (or exponent \pron{£ik£hsp£wn£ent}), expressed as $m\times 10 
^{p}$. The mantissa is not less than $1$ and less than $10$ ($1 \leq m < 10$). The power $p$ is an 
integer\pron{£h£int£i£d£e£Z}. 
   
The speed of light expressed in standard form is $2{\cdot}9979 \times 10^{8}\text{ m.s}^{-1}$. 
   
\subsection{Large Number Names}   
   
\begin{center}   
\noindent   
\begin{tabular}{|l|l|l|}\hline   
Number & US Name & British Name  \\ \hline \hline   
$10^{6}$  & million     & million  \\ \hline   
$10^{9}$  & billion     & milliard  \\ \hline   
$10^{12}$ & trillion    & billion  \\ \hline   
$10^{15}$ & quadrillion & 1,000 billion  \\ \hline   
$10^{18}$ & quintillion & trillion  \\ \hline   
$10^{21}$ & sextillion  & 1,000 trillion \\ \hline   
$10^{24}$ & septillion  & quadrillion \\ \hline   
$10^{27}$ & octillion   & 1,000 quadrillion \\ \hline   
$10^{30}$ & nonillion   & quintillion \\ \hline   
$10^{33}$ & decillion   & 1,000 quintillion \\ \hline   
\end{tabular}   
\end{center}   
   
{\small 
  In the \oldstylenums{1996} edition of the Oxford English Reference Dictionary it can be read:  \\ 
  `\textsf{milliard}: Now largely superseded by \emph{billion}.' 
   
  Around \oldstylenums{1484} Nicolas \textsc{Chuquet}\footnote{French mathematician \Detantatant{1445}{1500}} coined 
  the words \emph{billion, trillion, \dots, nonillion}, which also appeared in print in a %correction 2002-08-12
  \oldstylenums{1520} book by Émile de la \textsc{Roche}. These mathematicians used `illion' after the prefixes b-,
  tr-, \dots, non- to denote the 2nd, 3rd, \dots, 9th powers of a million. But around the middle of the 17th century,
  some other French mathematicians used them instead for the 3rd, 4th, \dots, 10th powers of a thousand.
   
  Although condemned by the greatest lexicographers as `erroneous' (\textsc{Littr\'e}) and `an entire perversion of the
  original nomenclature \pron{n£w£hmenkl£et£s£e£Z} of Chuquet and de la Roche' (\textsc{Murray}), the newer usage is now
  standard in the US, although the older one survives in Britain and is still standard in the continental countries (but
  the French spelling is nowadays `llon' rather than `llion').  }
   
\subsection{Operations}   
The basic \emph{operations} \pron{£b£op£e£hr£q£sn} are \emph{addition} \pron{£e£hd£i£sn}, \emph{subtraction} 
\pron{s£eb\-£htr£ak\-£sn}, \emph{multiplication} \pron{£bm£vlt£ipl£i£hk£q£sn} and \emph{division} \pron{d£i£hv£i£jn}. 
   
\subsubsection{Addition and subtraction}   
Addition is the operation of combining two numbers to form a \emph{sum} \pron{s£vm}.\index{addition} % correction 2003-06-15
The symbol for addition is $+$ `plus \pron{pl£vs}', from the Latin, meaning more; it is placed between two numbers to be
added together. Thus, $8+7$ means `eight plus seven' or `seven added to eight'.\index{plus}\index{$+$}
   
The symbol $=$ means `is equal \pron{£h£Ikw(£e)l} to' or `equals', so $8+7 = 15$ `eight plus seven equals fifteen'.%
\index{equal}\index{is equal to}\index{$=$} 
   
Let us consider $a+b=c$. $c$ is the sum, $a$ and $b$ are the summands \pron{£hs£v£bm£and}, addends \pron{£e£hdend} or 
\emph{terms} \pron{t£Em}.\index{term}\index{summand}\index{addend}\index{sum} 
   
The symbol for subtraction is $-$ `minus \pron{£hm£pn£es}', from the Latin, meaning less; it is placed between two 
numbers, when the second is to be taken away from the first. For example, $12-9$ means `twelve minus nine' or `nine 
taken away from 12'.\index{minus}\index{$-$} 
   
Let us consider $a-b=c$. $c$ is the \emph{difference} \pron{£hd£ifr£ens}, % correction 2003-06-15
$a$ is the minuend \pron{£hm£inj£u£bend} and $b$ the subtrahend \pron{£hs£vbtr£e£bhend}.\index{difference}
   
Let $a$ be a number. $-a$ is the \emph{additive inverse} \pron{£h£ad£it£iv £b£in£hv£Es} of $a$. We have $-a + a = 0$.%
\index{inverse}\index{inverse!additive} 
   
\subsubsection{Multiplication and division}   
The symbol for multiplication is $\times$ `times' \pron{t£pmz}. $12\times5=60$ is read `twelve times five equals sixty' 
or `twelve multiplied \pron{£hm£vlt£ipl£pd} by five equals sixty'. 
   
In $a\times b=c$, $a$ and $b$ are the \emph{factors} \pron{£hf£akt£e£Z} and $c$ is the \emph{product} 
\pron{£hpr£od£vkt}.\index{times}\index{multiplied by}\index{factor}\index{multiplication} 
   
The sign for division is $\div$ `divided \pron{d£i£hv£pd£id} by'.  Generally we write $a/b$ or \Dfrac{a}{b} `$a$ over 
$b$' instead of $a \div b$.\index{division}\index{divided by}\index{over} 
   
In $\dfrac{a}{b} = c$, $a$ is the \emph{dividend} \pron{£hd£iv£i£bdend}, 
$b$ is the \emph{divisor} \pron{d£i£hv£pz£e£Z} and $c$ is the \emph{quotient} \pron{£hkw£w£snt}.%   
\index{dividend}\index{divisor}\index{quotient} 
   
$67$ when divided by $7$ gives a quotient of $9$ and a \emph{remainder} \pron{r£i£hm£qnd£e£Z} of $4$ for we have $67 = 7 
\times 9 + 4$.\index{remainder} 
   
If $a$ is any number but $0$, $a$ has a \emph{multiplicative} \pron{£bm£vlt£i£hpl£ik£et£iv} \emph{inverse} or 
\nuvo{reciprocal} \pron{r£i£hs£ipr£ekl} which is denoted by \Dfrac{1}{a} or by $a^{-1}$ `a to the power minus 1'. 
We have $ a\times \dfrac{1}{a} = 1$.\index{inverse!multiplicative}    
   
\subsubsection{Factors}   
   
Consider $12 = 3\times 4$; $3$ and $4$ are said to be \emph{factors} of $12$. $12$ is said to be a \emph{multiple} 
\pron{£hm£vlt£ipl} \index{multiple} of $3$. One can say also that $12$ is \emph{exactly divided} or \emph{divisible} 
\pron{d£i£hv£iz£ebl} \index{divisible} by $3$.\index{exactly divided} 
   
An integer is said to be \emph{even} \pron{£h£Ivn}\index{even} if and only if it is divisible by $2$. If an integer is 
not even then it is \emph{odd} \pron{£od}.\index{odd} 
   
A \emph{common factor}\index{factor!common} of two (or more) numbers is a factor which occurs in both of them (or all of 
them). The \emph{highest common factor}\index{factor!common!highest} (H.C.F.) of a group of numbers is the largest 
number which will divide into all of them.\index{factor!common!highest@H.C.F.} 
   
The \emph{lowest common multiple}\index{multiple!common!lowest} (L.C.M.) of two or more numbers is the smallest number 
into which they will divide exactly.\index{multiple!common!lowest@L.C.M.} 
   
\section{The different kinds of numbers}   
   
\subsection{Natural numbers}   
   
The \emph{natural numbers} \pron{£hn£at£sr£el £hn£vmb£e£Z} or \emph{whole} \pron{h£wl} \emph{numbers} are $0$, $1$, $2$, 
$3$, \dots The set of all natural numbers is denoted by \ent. To denote that the number $n$ is a natural number one can 
write $n \in \ent$ `$n$ is in \ent' \emph{or} `$n$ belongs to \ent' \emph{or} `$n$ is an element \pron{£hel£im£ent} of 
\ent'.\index{number!natural}\index{$\in$}\index{belongs to}\index{$\ set natural$@$\ent$}\index{number!whole} 
   
\subsubsection{Prime numbers}   
   
Many natural numbers have no factors other than themselves and \nuvo{unity} \pron{£hj£Un£et£i} (\idest\ one), \parex\ $7$. 
These numbers are called \nuvo{prime} \pron{pr£pm} \emph{numbers} or \emph{primes}.\index{number!prime} 
   
A number greater than $1$ which is not prime is said to be \nuvo{composite} \pron{£hk£omp£ez£it}. % correction
                                                                                                  % 2001-09-09 !
   
\begin{prop}   
  Every natural number greater than $1$ is a product of prime numbers. 
\end{prop}   
   
For example: $630 = 2\times3\times3\times5\times7$. 
   
\subsection{Integers}   
   
The \emph{integers} \pron{£h£int£i£d£e£Z} are also called \emph{directed} \pron{d£i£hrekt£id} \emph{numbers} or
\emph{signed} \pron{s£pnd} \emph{numbers}.  The integers can be obtained as sum or difference of two natural numbers.
The set of all the integers is denoted by \zent. We have $\zent = \accol{\dots, -2, -1, 0, 1, 2,
  \dots}$.\index{integer}\index{$\ set zinteger$@$\zent$}
   
The natural numbers are the positive \pron{£hp£oz£et£iv} integers and $0$.  So all the elements of \ent{} are also 
elements of \zent.  One may write $\ent \subset \zent$: `the set of the natural numbers is a subset of the set of the 
integers' or 
`the set of the natural numbers is included in the set of the integers'.\index{$\subset$}\index{subset}%   
\index{included} 
   
\subsection{Rational numbers}   
   
A \nuvo{fraction} \pron{£hfr£ak£sn} is the \nuvo{ratio} \pron{£hr£q£s£i£w} or \nuvo{quotient} of two numbers.  In the 
fraction \Dfrac{a}{b} the number $a$ is said to be the \nuvo{numerator} \pron{£hnj£Um£er£qt£e£Z} and the number $b$ 
which is not equal to $0$ is said to be the \nuvo{denominator} \pron{d£i£hn£om£in£qt£e£Z}. 
   
A fraction like \Frac{6}{7} (written in words `six sevenths'), in which the numerator is smaller than the denominator, 
is called a \emph{proper fraction}, whereas one like \Frac{17}{12}, in which the numerator is greater than the 
denominator, is called an \emph{improper fraction}.\index{fraction!proper}\index{fraction!improper} The latter can always 
be reduced to a whole number ---~or integer~--- and a proper fraction, known as a \emph{mixed 
  number}.\index{number!whole}\index{number!mixed} 
   
Here we have $\dfrac{17}{12} = \dfrac{12+5}{12} = 1+\dfrac{5}{12}$ which is written $1\dfrac{5}{12}$ `one and five 
twelfths'. 
   
$\frac{1}{2}$ is `one half', $\frac{1}{3}$ \quad `one third', $\frac{2}{3}$ \quad `two thirds', $\frac{1}{4}$ \quad `one 
quarter', and $\frac{3}{4}$ \quad `three quarters'. 
   
We can \nuvo{cancel out} common factors in numerator and denominator, e.g. 
$$ 
\dfrac{24}{28}=\dfrac{4\times 6}{4 \times 7}=\dfrac{6}{7} 
$$ 
on dividing out top and bottom by 4. 
   
When the numerator and denominator have no common factor the fraction is said to be `in its lowest terms'.  To reduce a 
fraction to its lowest terms, divide the numerator and the denominator by their highest common factor.\index{lowest 
  terms}\index{reduce} 
   
To find the sum of two unlike fractions ---~fractions having different denominators~--- change them to like fractions 
---~fractions having the same denominator~--- and add the numerators. 
   
To find the quotient of two fractions, multiply the dividend by the inverted \pron{£in£hv£Et£id} divisor, \parex\ 
$$ 
\dfrac{5}{6}\div\dfrac{1}{2}= \dfrac{5}{6}\times\dfrac{2}{1} = \dfrac{5}{3}=1\dfrac{2}{3}. 
$$\index{inverted} 
   
The \emph{rational numbers} or \emph{rationals} \pron{£hr£a£s£enl}\index{rational}\index{number!rational} are the 
numbers that can be expressed as a ratio or fraction \Dfrac{a}{b} of two integers, $a$ and $b$, of which the latter may 
not be zero. 
   
The set of the rationals is denoted by \ratio.\index{$\ set ratio$@$\ratio$} 
   
\begin{prop}   
  For every rational $q$, there exists one and only one fraction in its lowest terms which is equal to $q$. 
\end{prop}   
   
A \nuvo{decimal fraction} or \nuvo{decimal} \pron{£hdes£im£el} is a rational which can be written with a denominator equal to a power of 
$10$. For example, $0.125$, \Dfrac{3}{4}, and $-\dfrac{245}{70}$ are decimal for we have: 
$$ 
0.125 = \dfrac{125}{1,000}, \quad \Dfrac{3}{4}= \Dfrac{3\times25}{4\times25}=\Dfrac{75}{100}=0.75, \quad\text{ and } 
-\dfrac{245}{70}=-\dfrac{35}{10}=3.5 
$$ 
   
It is always possible to write an integer as a fraction. So all integers are rationals. We can write $\zent \subset 
\ratio$. 
   
\subsection{Real numbers}   
   
Some numbers are not rational. For instance, §p \pron{p£p} is not a rational. It has been proven\,\footnote{The proof 
  was given in \oldstylenums{1761} by Johann Heinrich \textsc{Lambert} (\detantatant{1728}{1777}), mathematician born in 
  Mülhausen, Germany.}  that there exists no fraction equal to §p. The number §p is said to be \emph{irrational} 
\pron{£i£hr£a£s£enl}.\index{irrational}\index{number!real} It is also the case for such numbers as $\sqrt{2}$ `square 
root \pron{skw£P r£Ut} of 2',\index{real} $\sqrt{3}$ `square root of 3', \dots The rational and irrational numbers 
together form the set of the \emph{real} \pron{r£i£el} \emph{numbers} or \emph{reals}.  The set of the real numbers is 
denoted by \reel.\index{$\ set real$@$\reel$} 
   
All the rationals are reals and so $\ratio \subset \reel$. 
   
\enlargethispage{-\baselineskip} 
   
Let \Li be a line \pron{l£pn}. Let $O$ and $U$ be two points \pron{p£rnt} on that line such that $OU = 1$. % correction 2003
\couple{O}{U} % correction 2002-03-11
is said to be a \emph{coordinate} \pron{k£w£h£Odn£et} \emph{system} of line~\Li.\index{coordinate!system}\index{system!coordinate} 
   
To every point $M$ on \Li one can associate \pron{£e£hs£w£s£i£qt} a real number $x_M$ ($x$ \nuvo{sub} $M$), its 
\nuvo{abscissa} \pron{£a£hbs£is£e}\,\footnote{plurals: `abscissas' \pron{£a£hbs£is£ez} or `abscissae' 
  \pron{£a£hbs£is£I}} which is defined by: 
\begin{itemize}   
\item $x_M = OM$ if $M$ is on the \nuvo{ray} \pron{r£q} ---~or half-line~--- $[OU)$; 
\item $x_M = - OM$ if $M$ is not on the ray $[OU)$. 
\end{itemize}   
Conversely \pron{£bk£on£hv£Esl£i}, to every real number $x$ one can associate on \Li a point $M$ the abscissa of which 
is $x$.  So \reel{} is the set of all the absissae of the points of \Li. 
   
In such a case \Li together with \couple{O}{U} is said to be the \emph{real axis} 
\pron{£h£aks£is}.\index{real!axis}\index{axis!real} 
   
\begin{fgre}
  \IG{droit_grad}
\end{fgre}
   
The \nuvo{positive} \pron{£hp£oz£et£iv} real numbers are the abscissae of the points of $[OU)$ but $O$.  $0$ is the 
abscissa of $O$. The \nuvo{negative} \pron{£hneg£et£iv} real numbers are the abscissae of the points which are not on 
$[OU)$. 
   
$\reel^{+}$ is the set of all the positive real numbers together with $0$, in other terms, $\reel^{+}$ is the set of all 
the non-negative real numbers. $\reel^{-}$ is the set of all the non-positive real numbers. 
$\reel^{\ast}$ is the set of all non-null \pron{n£vl} real numbers.%
\index{$\ set real$@$\reel^{+}$}\index{$\ set real$@$\reel^{-}$}\index{$\ set real$@$\reel^{\ast}$} 
\endinput 
% correction (à finir) le 17/11/2000 & 17/01/2001   
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% correction pour v1.1 2001-08-02
% correction pour v1.2 2002-03-11 2002-08-12
% correction pour v1.3 2003-06-15
% correction pour v1.4 2003-07-04
