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Difference between revisions of "Natural number"

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====Comments====
 
====Comments====
A definition more elegant than the definition given above (the one of Frege–Russell) as cardinal numbers is von Neumann's definition, identifying a number with the set of its predecessors: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066090/n06609013.png" />,  "n+ 1"  <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066090/n06609014.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066090/n06609015.png" /> denotes  "successorsuccessor" . In this definition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066090/n06609016.png" /> is taken to belong to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066090/n06609017.png" /> (this is often done). In this case, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066090/n06609018.png" /> is the neutral element for addition.
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A definition more elegant than the definition given above (the one of Frege–Russell) as cardinal numbers is von Neumann's definition, identifying a number with the set of its predecessors: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066090/n06609013.png" />,  "n+ 1"  <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066090/n06609014.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066090/n06609015.png" /> denotes  "successor" . In this definition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066090/n06609016.png" /> is taken to belong to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066090/n06609017.png" /> (this is often done). In this case, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066090/n06609018.png" /> is the neutral element for addition.
  
 
Cf. also [[Natural sequence|Natural sequence]].
 
Cf. also [[Natural sequence|Natural sequence]].

Revision as of 20:30, 12 January 2016

One of the fundamental concepts in mathematics. Natural numbers may be interpreted as the cardinal numbers (cf. Cardinal number) of non-empty finite sets. The set of all natural numbers, together with the operations of addition and multiplication , forms the natural number system . In this system, both binary operations are associative and commutative and satisfy the distributivity law; 1 is the neutral element for multiplication, i.e. for any natural number ; there is no neutral element for addition, and, moreover, for any natural numbers . Finally, the following condition, known as the axiom of induction, is satisfied. Any subset of that contains 1 and, together with any element also contains the sum , is necessarily the whole of . See Natural sequence; Arithmetic, formal.

References

[1] , The history of mathematics from Antiquity to the beginning of the XIX-th century , 1 , Moscow (1970) (In Russian)
[2] V.I. Nechaev, "Number systems" , Moscow (1975) (In Russian)
[3] H. Davenport, "The higher arithmetic" , Hutchinson (1952)


Comments

A definition more elegant than the definition given above (the one of Frege–Russell) as cardinal numbers is von Neumann's definition, identifying a number with the set of its predecessors: , "n+ 1" . Here denotes "successor" . In this definition is taken to belong to (this is often done). In this case, is the neutral element for addition.

Cf. also Natural sequence.

References

[a1] C.J. Scriba, "The concept of number, a chapter in the history of mathematics, with applications of interest to teachers" , B.I. Wissenschaftsverlag Mannheim (1968)
How to Cite This Entry:
Natural number. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Natural_number&oldid=17093
This article was adapted from an original article by A.A. BukhshtabV.I. Nechaev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article