# 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)