Natural number
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) |
Natural number. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Natural_number&oldid=17093