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Number
Number is an abstract concept which is not defined generally in mathematics. The numbers can be used for counting objects, comparing quantities and expressing order.
In mathematics one can define different kinds of numbers; some of the most common are the integers, the fractional numbers, the real numbers and the complex numbers. There are also many special kinds of e.g. integers: odd numbers, prime numbers, triangular numbers etc. The algebraic integers are a special kind of complex numbers, having similar divisibility properties as the rational integers but much richer.
Usually the word “number” refers only to the complex numbers, which can be thought to be formed by gradually expanding the available system of numbers:
- natural numbers 1, 2, 3, ...
- integers (added 0 and negative integers)
- rational numbers (added fractional numbers)
- real numbers (added irrational numbers)
- complex numbers (added imaginary numbers)
of the complex numbers, e.g. the rational function field
or the formal Laurent series
field
. That's a different matter if one wants to call numbers the elements of the last fields.
The field
of the rational numbers can be extended also in another direction than the real and complex numbers: the field of 
-adic numbers
makes a completion
of
which resembles
but which is not contained
neither in
nor in
.