## Natural numbers

In mathematics, the natural numbers are those used for counting ("there are six coins on the table") and ordering ("this is the third largest city in the country"). These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively (see English numerals). A later notion is that of a nominal number, which is used only for naming.

Natural numbers are used to count objects and order them, in this image 3 coloured circles are shown.  Each colour circle is labelled with a numeral. image: R.Conan-Davies

Properties

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.

Is Zero a natural number?

There is no universal agreement about whether to include zero in the set of natural numbers. In 1763 W. Emerson's Method of Increments contains, on page 113, the phrase "To find the product of all natural numbers from 1 to 100 ... ."

But the Peano axioms (1889) begin the natural numbers with zero. Today some textbooks, especially college textbooks, define the natural numbers to be the positive integers {1, 2, 3, ...}, while others, especially primary and secondary textbooks, define the term as the non-negative integers {0, 1, 2, 3, …}.