nmathematics, thenatural numbersare those used forcounting("there are six coins on the table") andordering("this is the third largest city in the country"). These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively (seeEnglish numerals). A later notion is that of anominal number, which is used only for naming.

Properties of the natural numbers related todivisibility, such as the distribution ofprime numbers, are studied innumber theory. Problems concerning counting and ordering, such aspartitionenumeration, are studied incombinatorics.

There is no universal agreement about whether to includezeroin the set of natural numbers: some define the natural numbers to be thepositiveintegers{1,2,3, ...}, while for others the term designates thenon-negativeintegers{0, 1, 2, 3, ...}. The former definition is the traditional one, with the latter definition having first appeared in the 19thcentury.