number

Number is an abstract concept which is not defined generally in mathematics.  The numbers can be used for counting .

In mathematics one can define different kinds of numbers; some of the most common are .  There are also many special kinds of e.g. : odd numbers, prime numbers, triangular numbers, Fibonacci numbers, etc.  The algebraic integers (http://planetmath.org/AlgebraicNumberTheory) are a special kind of complex numbers, having similar (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, 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 (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 extended also 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 $ℂ$.

Title	number
Canonical name	Number
Date of creation	2015-11-17 12:09:30
Last modified on	2015-11-17 12:09:30
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	10
Author	pahio (2872)
Entry type	Feature
Classification	msc 03E10
Related topic	ClassificationOfComplexNumbers
Related topic	Fraction
Related topic	CardinalNumber
Related topic	OrdinalNumber
Related topic	RuleOfProduct