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This article is about “positive integers” and “non-negative integers”. For all the numbers ..., −2, −1, 0, 1, 2, ..., see Integer.

In mathematics, the natural numbers (sometimes called the whole numbers) are those 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"). In common language, words used for counting are "cardinal numbers" and words used for ordering are "ordinal numbers".

Another use of natural numbers is for what linguists call nominal numbers, such as the model number of a product, where the "natural number" is used only for naming (as distinct from a serial number where the order properties of the natural numbers distinguish later uses from earlier uses) and generally lacks any meaning of number as used in mathematics.

The natural numbers are the basis from which many other number sets may be built by extension: the integers, by including an unresolved negation operation; the rational numbers, by including with the integers an unresolved division operation; the real numbers by including with the rationals the termination of Cauchy sequences; the complex numbers, by including with the real numbers the unresolved square root of minus one; the hyperreal numbers, by including with real numbers the infinitesimal value epsilon; vectors, by including a vector structure with reals; matrices, by having vectors of vectors; the nonstandard integers; and so on. Therefore, the natural numbers are canonically embedded (identified) in the other number systems.

Properties of the natural numbers, such as divisibility and the distribution of prime numbers, are studied in number theory. Problems concerning counting and ordering, such as partitioning and enumerations, are studied in combinatorics.

There is no universal agreement about whether to include zero in the set of natural numbers. Some authors begin the natural numbers with 0, corresponding to the non-negative integers 0, 1, 2, 3, ..., whereas others start with 1, corresponding to the positive integers 1, 2, 3, .... This distinction is of no fundamental concern for the natural numbers (even when viewed via additional axioms as semigroup with respect to addition and monoid for multiplication). Including the number 0 just supplies an identity element for the former (binary) operation to achieve a monoid structure for both, and a (trivial) zero divisor for the multiplication.

In common language, for example in primary school, natural numbers may be called counting numbers to distinguish them from the real numbers which are used for measurement.


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