Natural number

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 axiom of 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)