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One of the fundamental concepts in mathematics. Natural numbers may be interpreted as the cardinal numbers (cf. [[Cardinal number|Cardinal number]]) of non-empty finite sets. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066090/n0660901.png" /> of all natural numbers, together with the operations of addition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066090/n0660902.png" /> and multiplication <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066090/n0660903.png" />, forms the natural number system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066090/n0660904.png" />. In this system, both binary operations are associative and commutative and satisfy the distributivity law; 1 is the neutral element for multiplication, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066090/n0660905.png" /> for any natural number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066090/n0660906.png" />; there is no neutral element for addition, and, moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066090/n0660907.png" /> for any natural numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066090/n0660908.png" />. Finally, the following condition, known as the axiom of induction, is satisfied. Any subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066090/n0660909.png" /> that contains 1 and, together with any element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066090/n06609010.png" /> also contains the sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066090/n06609011.png" />, is necessarily the whole of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066090/n06609012.png" />. See [[Natural sequence|Natural sequence]]; [[Arithmetic, formal|Arithmetic, formal]].
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One of the fundamental concepts in mathematics. Natural numbers may be interpreted as the [[cardinal number]]s of non-empty finite sets. The set $\mathbf{N} = \{1,2,3,\ldots\}$ of all natural numbers, together with the operations of [[addition]] $({+})$ and [[multiplication]] $({\times})$, forms the natural number system $(\mathbf{N},{+},{\times},1)$. In this system, both [[binary operation]]s are [[Associativity|associative]] and [[Commutativity|commutative]] and satisfy the [[distributive law]]; 1 is the [[neutral element]] for multiplication, i.e. $a \times 1 = a = 1 \times a$ for any natural number $a$; there is no neutral element for addition, and, moreover, $a + b \neq a$ for any natural numbers $a,b$. Finally, the following condition, known as the principle of [[mathematical induction]], is satisfied. Any subset of $\mathbf{N}$ that contains 1 and, together with any element $a$ also contains the sum $a+1$, is necessarily the whole of $\mathbf{N}$. See [[Natural sequence]]; [[Arithmetic, formal]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> , ''The history of mathematics from Antiquity to the beginning of the XIX-th century'' , '''1''' , Moscow  (1970)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.I. Nechaev,  "Number systems" , Moscow  (1975)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  H. Davenport,  "The higher arithmetic" , Hutchinson  (1952)</TD></TR></table>
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top"> , ''The history of mathematics from Antiquity to the beginning of the XIX-th century'' , '''1''' , Moscow  (1970)  (In Russian)</TD></TR>
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<TR><TD valign="top">[2]</TD> <TD valign="top">  V.I. Nechaev,  "Number systems" , Moscow  (1975)  (In Russian)</TD></TR>
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<TR><TD valign="top">[3]</TD> <TD valign="top">  H. Davenport,  "The higher arithmetic" , Hutchinson  (1952)</TD></TR>
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</table>
  
  
  
 
====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  "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.
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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: $0 = \emptyset$, $n+1 = Sn = \{0,\ldots,n\}$. Here $S$ denotes  "successor" . In this definition $0$ is taken to belong to $\mathbf{N}$ (this is often done). In this case, $0$ is the neutral element for addition and the [[zero element]] for multiplication.
  
Cf. also [[Natural sequence|Natural sequence]].
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====References====
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  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)</TD></TR>
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</table>
  
====References====
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{{TEX|done}}
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  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)</TD></TR></table>
 

Latest revision as of 19:08, 5 February 2016

One of the fundamental concepts in mathematics. Natural numbers may be interpreted as the cardinal numbers of non-empty finite sets. The set $\mathbf{N} = \{1,2,3,\ldots\}$ of all natural numbers, together with the operations of addition $({+})$ and multiplication $({\times})$, forms the natural number system $(\mathbf{N},{+},{\times},1)$. In this system, both binary operations are associative and commutative and satisfy the distributive law; 1 is the neutral element for multiplication, i.e. $a \times 1 = a = 1 \times a$ for any natural number $a$; there is no neutral element for addition, and, moreover, $a + b \neq a$ for any natural numbers $a,b$. Finally, the following condition, known as the principle of mathematical induction, is satisfied. Any subset of $\mathbf{N}$ that contains 1 and, together with any element $a$ also contains the sum $a+1$, is necessarily the whole of $\mathbf{N}$. 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: $0 = \emptyset$, $n+1 = Sn = \{0,\ldots,n\}$. Here $S$ denotes "successor" . In this definition $0$ is taken to belong to $\mathbf{N}$ (this is often done). In this case, $0$ is the neutral element for addition and the zero element for multiplication.

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=37506
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