numberNumber2008-05-25 11:24:512008-06-03 04:24:34Featurecardinal number% this is the default PlanetMath preamble. as your knowledge
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{\em Number} is an abstract concept which is \textbf{not defined generally} in mathematics.\, The numbers can be used for counting \PMlinkescapetext{objects, comparing quantities and expressing order}.
In mathematics one \textbf{can define} different \textbf{kinds} of numbers; some of the most common are \PMlinkescapetext{the integers, the fractional numbers, the real numbers and the complex numbers}.\, There are also many special kinds of e.g. \PMlinkescapetext{integers}: odd numbers, prime numbers, triangular numbers etc.\, The \PMlinkname{algebraic integers}{AlgebraicNumberTheory} are a special kind of \PMlinkescapetext{complex numbers, having similar} divisibility properties as the \PMlinkescapetext{rational integers} but much richer.
Usually the \PMlinkescapetext{word ``number'' refers only to the complex} numbers, which can be thought to be formed by gradually expanding the available system of numbers:
\begin{itemize}
\item natural numbers 1, 2, 3, \ldots
\item integers (added \PMlinkname{0}{Null} and negative integers)
\item rational numbers (added fractional numbers)
\item real numbers (added irrational numbers)
\item complex numbers (added imaginary numbers)
\end{itemize}
A usual applier of the mathematics, e.g. an engineer, probably believes that there are no other numbers than the complex numbers.\, Some school book may tell that the complex numbers form the widest possible field of numbers.\, However, the mathematicians know that there exist infinitely many extension fields of the field $\mathbb{C}$ of the complex numbers, e.g. the rational function field $\mathbb{C}(X)$ or the formal Laurent series field $\mathbb{C}((X))$.\, That's a different matter if one wants to call {\em numbers} the elements of the last fields.
The field $\mathbb{Q}$ of the rational numbers can be extended also in another direction than the real and complex numbers:\, the field of \PMlinkname{$p$-adic numbers}{PAdicNumbers} makes a completion of $\mathbb{Q}$ which resembles $\mathbb{R}$ but which is not contained neither in $\mathbb{R}$ \PMlinkescapetext{nor} in $\mathbb{C}$.