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In mathematics, a 'number system' is a [[Set (mathematics)|set]] of [[number]]s, (in the broadest sense of the word), together with one or more operations, such as [[addition]] or [[multiplication]].  

Examples of number systems include: [[natural number]]s, [[integer]]s, [[rational number]]s, [[algebraic number]]s, [[real number]]s, [[complex number]]s, [[p-adic number]]s, [[surreal number]]s, and [[hyperreal number]]s.

For a history of number systems, see [[number]].  For a history of the symbols used to represent numbers, see [[numeral system]].

==Logical construction of number systems==
===Natural numbers===
{{main|Set-theoretic definition of natural numbers}}
Simply put, the natural numbers consist of the set of all whole numbers greater than zero. The set is denoted with a bold face capital '''N''' or with the special symbol <math>\mathbb{N}</math> .  (In some books, the natural numbers begin with 0.  There is no general agreement on this subject.)<ref>Billstein, Libeskind, and Lott, ''Mathematics for Elementary School Teachers'' 8th edition, Pearson, 2004, ISBN 0-321-15680-3</ref><ref>Usage varies among mathematicians as to whether zero is to be included in the natural numbers.  The Peano axioms include zero, but substituting "1" for "0" in rule one, rule three and the induction rule accurately describes the natural numbers without a zero.</ref>
 
[[Giuseppe Peano]] developed [[Peano axioms|axiom]]s for the [[natural number]]s, and is considered the founder of axiomatic number theory.

==More advanced number systems==
The word ''number'' has no generally agreed upon mathematical meaning, nor does the word ''number system''.  Instead, we have many examples.  Thus there is no rule to say what is a number and what is not.  Some of the more interesting examples of abstractions that can be considered numbers include the [[quaternion]]s, the [[octonion]]s, [[ordinal number]]s, and the [[transfinite number]]s.

==See also==
*[[Abstract algebra]]
*[[Affinely extended real number system]]
*[[Approximate number system]]
*[[Large numbers]]

==Notes==
<references />

==References==
* [[Richard Dedekind]], 1888. ''Was sind und was sollen die Zahlen?'' ("What are and what should the numbers be?"). Braunschweig.
* [[Edmund Landau]], 2001, ISBN 082182693X, ''Foundations of Analysis'', [[American Mathematical Society]]. 
* [[Giuseppe Peano]], 1889. ''Arithmetices principia, nova methodo exposita'' (The principles of arithmetic, presented by a new method). Bocca, Torino. [[Jean van Heijenoort]], trans., 1967. ''A Source Book of Mathematical Logic: 1879-1931''. Harvard Univ. Press: 83-97.
* {{cite book | author=B. A. Sethuraman | title=Rings, Fields, Vector Spaces, and Group Theory: An Introduction to Abstract Algebra via Geometric Constructibility | publisher=Springer | year=1996 | isbn=0-387-94848-1}}
* {{cite book | author=[[Solomon Feferman]] | title=The Number Systems : Foundations of Algebra and Analysis | publisher=Addison-Wesley | year = 1964 }}
* Stoll, Robert R., 1979 (1963). ''Set Theory and Logic''. Dover.



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