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

Examples of number systems include: natural numbers, integers, rational numbers, algebraic numbers, real numbers, complex numbers, p-adic numbers, surreal numbers, and hyperreal numbers.

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

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 $\mathbb {N}$ . (In some books, the natural numbers begin with 0. There is no general agreement on this subject.)^[1]^[2]

Giuseppe Peano developed axioms for the natural numbers, 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 quaternions, the octonions, ordinal numbers, and the transfinite numbers.

Notes

^ Billstein, Libeskind, and Lott, Mathematics for Elementary School Teachers 8th edition, Pearson, 2004, ISBN 0-321-15680-3
^ 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.

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.
B. A. Sethuraman (1996). Rings, Fields, Vector Spaces, and Group Theory: An Introduction to Abstract Algebra via Geometric Constructibility. Springer. ISBN 0-387-94848-1.
Solomon Feferman (1964). The Number Systems : Foundations of Algebra and Analysis. Addison–Wesley.
Stoll, Robert R., 1979 (1963). Set Theory and Logic. Dover.

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[1] Billstein, Libeskind, and Lott, Mathematics for Elementary School Teachers 8th edition, Pearson, 2004, ISBN 0-321-15680-3

[2] 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.

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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]].
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 ==References==
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 * [[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}}
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 * Stoll, Robert R., 1979 (1963). ''Set Theory and Logic''. Dover.
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v t e Number systems
Sets of definable numbers	Natural numbers ( $\mathbb {N}$ ) Integers ( $\mathbb {Z}$ ) Rational numbers ( $\mathbb {Q}$ ) Constructible numbers Algebraic numbers ( $\mathbb {A}$ ) Closed-form numbers Periods Computable numbers Arithmetical numbers Set-theoretically definable numbers Gaussian integers
Composition algebras	Division algebras: Real numbers ( $\mathbb {R}$ ) Complex numbers ( $\mathbb {C}$ ) Quaternions ( $\mathbb {H}$ ) Octonions ( $\mathbb {O}$ )
Split types	Over $\mathbb {R}$ : Split-complex numbers Split-quaternions Split-octonions Over $\mathbb {C}$ : Bicomplex numbers Biquaternions Bioctonions
Other hypercomplex	Dual numbers Dual quaternions Dual-complex numbers Hyperbolic quaternions Sedenions ( $\mathbb {S}$ ) Split-biquaternions Multicomplex numbers Geometric algebra/Clifford algebra Algebra of physical space Spacetime algebra Plane-based geometric algebra
Infinities and infinitesimals	Cardinal numbers Extended natural numbers Extended real numbers Projective Hyperreal numbers Levi-Civita field Ordinal numbers Supernatural numbers Surreal numbers Superreal numbers
Other types	Irrational numbers Fuzzy numbers Transcendental numbers p-adic numbers (p-adic solenoids) Profinite integers Normal numbers
Classification List