View source for Natural number

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====
<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>



====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====
<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>

{{TEX|done}}