Addition and multiplication are compatible, which is expressed in the distribution law: .These properties of addition and multiplication make the natural numbers an instance of a commutative semiring.Peano arithmetic is equiconsistent with several weak systems of set theory.One such system is ZFC with the axiom of infinity replaced by its negation.

The most primitive method of representing a natural number is to put down a mark for each object.The use of a numeral 0 in modern times originated with the Indian mathematician Brahmagupta in 628.However, 0 had been used as a number in the medieval computus (the calculation of the date of Easter), beginning with Dionysius Exiguus in 525, without being denoted by a numeral (standard Roman numerals do not have a symbol for 0); instead nulla (or the genitive form nullae) from nullus, the Latin word for "none", was employed to denote a 0 value.Problems concerning counting and ordering, such as partitioning and enumerations, are studied in combinatorics.In common language, for example in primary school, natural numbers may be called counting numbers both to intuitively exclude the negative integers and zero, and also to contrast the discreteness of counting to the continuity of measurement, established by the real numbers.