**natural numbers**are the ordinary whole numbers used for counting ("there are 6 coins on the table") and ordering ("this is the 3rd largest city in the country"). These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively (see English numerals). A more recent notion is that of a nominal number, which is used only for naming.

Properties of the natural numbers related to divisibility, such as the distribution of prime numbers, are studied in number theory. Problems concerning counting and ordering, such as partition enumeration, are studied in combinatorics.

There is no universal agreement about whether to include zero in the set of natural numbers: some define the natural numbers to be the

**positive integers**{1, 2, 3, ...}, while for others the term designates the

**non-negative integers**{0, 1, 2, 3, ...}. The former definition is the traditional one, with the latter definition first appearing in the 19th century. Some authors use the term

*natural number*to exclude zero and

*whole number*to include it; others use

*whole number*in a way that excludes zero, or in a way that includes both zero and the negative integers.

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