 # Natural number

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In mathematics, the natural numbers are those numbers used for counting (as in "there are six coins on the table") and ordering (as in "this is the third largest city in the country"). Numbers used for counting are called cardinal numbers, and numbers used for ordering are called ordinal numbers. Natural numbers are sometimes used as labels, known as nominal numbers, having none of the properties of numbers in a mathematical sense (e.g. sports jersey numbers).

The natural numbers form a set. Many other number sets are built by successively extending the set of natural numbers: the integers, by including an additive identity 0 (if not yet in) and an additive inverse n for each nonzero natural number n; the rational numbers, by including a multiplicative inverse $1/n$ for each nonzero integer n (and also the product of these inverses by integers); the real numbers by including the limits of (converging) Cauchy sequences of rationals; the complex numbers, by adjoining to the real numbers a square root of −1 (and also the sums and products thereof); and so on.[lower-alpha 3][lower-alpha 4] This chain of extensions canonically embeds the natural numbers in the other number systems.