In mathematics, the natural numbers are those used for counting (as in "there are six coins on the table") and total order (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".
Some definitions, including the standard ISO 800002, begin the natural numbers with , corresponding to the nonnegative integers , whereas others start with 1, corresponding to the positive integers .
says: "ℕ is the set of natural numbers (positive integers)" (p. 3) include zero in the natural numbers: 'Intuitively, the set of all natural numbers may be described as follows: contains an "initial" number 0; ...'. They follow that with their version of the Peano Postulates. (p. 15) Texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, but in other writings, that term is used instead for the integers (including negative integers).
The natural numbers are the basis from which many other number sets may be built by extension: the , by including (if not yet in) the neutral element 0 and an additive inverse (− n) for each nonzero natural number n; the , by including a multiplicative inverse (1/ n) for each nonzero integer n (and also the product of these inverses by integers); the by including with the rationals the limits of (converging) Cauchy sequences of rationals; the , by including with the real numbers the unresolved Imaginary unit (and also the sums and products thereof); and so on. says: "The whole fantastic hierarchy of number systems is built up by purely settheoretic means from a few simple assumptions about natural numbers." (Preface, p. x): "Numbers make up the foundation of mathematics." (p. 1) These chains of extensions make the natural numbers canonically Embedding (identified) in the other number systems.
Properties of the natural numbers, such as divisibility and the distribution of , are studied in number theory. 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 .
The natural numbers can, at times, appear as a convenient set of names (labels), that is, as what linguists call , foregoing many or all of the properties of being a number in a mathematical sense.
The most primitive method of representing a natural number is to put down a mark for each object. Later, a set of objects could be tested for equality, excess or shortage, by striking out a mark and removing an object from the set.
The first major advance in abstraction was the use of numeral system to represent numbers. This allowed systems to be developed for recording large numbers. The ancient Egyptians developed a powerful system of numerals with distinct hieroglyphs for 1, 10, and all the powers of 10 up to over 1 million. A stone carving from Karnak, dating from around 1500 BC and now at the Louvre in Paris, depicts 276 as 2 hundreds, 7 tens, and 6 ones; and similarly for the number 4,622. The had a placevalue system based essentially on the numerals for 1 and 10, using base sixty, so that the symbol for sixty was the same as the symbol for one, its value being determined from context.Georges Ifrah, The Universal History of Numbers, Wiley, 2000,
A much later advance was the development of the idea that can be considered as a number, with its own numeral. The use of a 0 numerical digit in placevalue notation (within other numbers) dates back as early as 700 BC by the Babylonians, but they omitted such a digit when it would have been the last symbol in the number. The Olmec and Maya civilizations used 0 as a separate number as early as the , but this usage did not spread beyond Mesoamerica... The use of a numeral 0 in modern times originated with the 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.
The first systematic study of numbers as is usually credited to the ancient Greece philosophers Pythagoras and Archimedes. Some Greek mathematicians treated the number 1 differently than larger numbers, sometimes even not as a number at all.This convention is used, for example, in Euclid's Elements, see Book VII, definitions 1 and 2.
Independent studies also occurred at around the same time in India, China, and Mesoamerica.Morris Kline, Mathematical Thought From Ancient to Modern Times, Oxford University Press, 1990 1972,
In opposition to the Naturalists, the constructivists saw a need to improve the logical rigor in the foundations of mathematics."Much of the mathematical work of the twentieth century has been devoted to examining the logical foundations and structure of the subject." In the 1860s, Hermann Grassmann suggested a recursive definition for natural numbers thus stating they were not really natural but a consequence of definitions. Later, two classes of such formal definitions were constructed; later, they were shown to be equivalent in most practical applications.
Settheoretical definitions of natural numbers were initiated by Frege and he initially defined a natural number as the class of all sets that are in onetoone correspondence with a particular set, but this definition turned out to lead to paradoxes including Russell's paradox. Therefore, this formalism was modified so that a natural number is defined as a particular set, and any set that can be put into onetoone correspondence with that set is said to have that number of elements.
The second class of definitions was introduced by Charles Sanders Peirce, refined by Richard Dedekind, and further explored by Giuseppe Peano; this approach is now called Peano arithmetic. It is based on an axiomatization of the properties of : each natural number has a successor and every nonzero natural number has a unique predecessor. 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. Theorems that can be proved in ZFC but cannot be proved using the Peano Axioms include Goodstein's theorem.L. Kirby; J. Paris, Accessible Independence Results for Peano Arithmetic, Bulletin of the London Mathematical Society 14 (4): 285. doi:10.1112/blms/14.4.285, 1982.
With all these definitions it is convenient to include 0 (corresponding to the empty set) as a natural number. Including 0 is now the common convention among set theory and .
To be unambiguous about whether 0 is included or not, sometimes an index (or superscript) "0" is added in the former case, and a superscript "" or subscript "" is added in the latter case:
Alternatively, natural numbers may be distinguished from positive integers with the index notation, but it must be understood by context that since both symbols are used, the natural numbers contain zero.
If 1 is defined as , then . That is, is simply the successor of .
If the natural numbers are taken as "excluding 0", and "starting at 1", the definitions of + and × are as above, except that they begin with and .
A total order on the natural numbers is defined by letting if and only if there exists another natural number where . This order is compatible with the arithmetical operations in the following sense: if , and are natural numbers and , then and .
An important property of the natural numbers is that they are : every nonempty set of natural numbers has a least element. The rank among wellordered sets is expressed by an ordinal number; for the natural numbers, this is denoted as (omega).
While it is in general not possible to divide one natural number by another and get a natural number as result, the procedure of division with remainder is available as a substitute: for any two natural numbers and with there are natural numbers and such that
The number is called the quotient and is called the remainder of the division of by . The numbers and are uniquely determined by and . This Euclidean division is key to several other properties (divisibility), algorithms (such as the Euclidean algorithm), and ideas in number theory.
Many wellordered sets with cardinal number have an ordinal number greater than (the latter is the lowest possible). The least ordinal of cardinality (i.e., the initial ordinal) is .
For finite set wellordered sets, there is a onetoone correspondence between ordinal and cardinal numbers; therefore they can both be expressed by the same natural number, the number of elements of the set. This number can also be used to describe the position of an element in a larger finite, or an infinite, sequence.
A countable nonstandard model of arithmetic satisfying the Peano Arithmetic (i.e., the firstorder Peano axioms) was developed by Skolem in 1933. The hypernatural numbers are an uncountable model that can be constructed from the ordinary natural numbers via the ultrapower construction.
Georges Reeb used to claim provocatively that The naïve integers don't fill up . Other generalizations are discussed in the article on .
uses the language of set theory instead of the language of arithmetic for his five axioms. He begins with "(I) (where, of course, " ( is the set of all natural numbers). (p. 46)
gives "a twopart axiom" in which the natural numbers begin with 1. (Section 10.1: ''An Axiomatization for the System of Positive Integers'')
These are not the original axioms published by Peano, but are named in his honor. Some forms of the Peano axioms have 1 in place of 0. In ordinary arithmetic, the successor of $x$ is $x\; +\; 1$. Replacing Axiom Five by an axiom schema one obtains a (weaker) firstorder theory called Peano Arithmetic.
With this definition, a natural number is a particular set with elements, and if and only if is a subset of . The standard definition, now called definition of von Neumann ordinals, is: "each ordinal is the wellordered set of all smaller ordinals."
Also, with this definition, different possible interpretations of notations like (tuples versus mappings of into ) coincide.
Even if one finitism and therefore cannot accept that the set of all natural numbers exists, it is still possible to define any one of these sets.

