number

Number is an abstract concept which isnot defined generally in mathematics.  The numberscan be used for counting .

In mathematics one can define differentkinds of numbers; some of the most common are.  There are also manyspecial kinds of e.g. : odd numbers,prime numbers, triangular numbers, Fibonacci numbers, etc. The algebraic integers (http://planetmath.org/AlgebraicNumberTheory) area special kind of complex numbers, havingsimilar (http://planetmath.org/Number) divisibility properties as the but much richer.

Usually the numbers, which can be thought to be formed by gradually expanding the available system of numbers:

• natural numbers 1, 2, 3, …

• integers (added 0 (http://planetmath.org/Null) and negative integers)

• rational numbers (added fractional numbers)

• real numbers (added irrational numbers)

• complex numbers (added imaginary numbers)

A usual applier of the mathematics, e.g. an engineer, probablybelieves that there are no other numbers than the complexnumbers.  Some school book may tell that the complex numbersform the widest possible field (http://planetmath.org/Field) of numbers.  However, the mathematicians know that there exist infinitely many extension fields of the field $ℂ$ of the complex numbers, e.g. the rational function field $ℂ(X)$ or the formal Laurent series field $ℂ((X))$.  That’s a different matter if one wants to call numbers the elements of the last fields.

The field $\mathbb{Q}$ of the rational numbers can be extendedalso in another direction than the real and complex numbers: the field of $p$-adic numbers (http://planetmath.org/PAdicIntegers) makes a completion of $\mathbb{Q}$ which resembles $ℝ$ but which is not contained neither in $ℝ$ in $ℂ$.