In

ordinal number
In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets.
A finite set can be enumerated by successively labeling each element with the least n ...

, the sets considered below are sometimes called von Neumann ordinals.
The definition proceeds as follows:
* Call , the ordinal number
In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets.
A finite set can be enumerated by successively labeling each element with the least n ...

s, including the infinite ones: "each ordinal is the well-ordered set of all smaller ordinals."
If one does not accept the axiom of infinity, the natural numbers may not form a set. Nevertheless, the natural numbers can still be individually defined as above, and they still satisfy the Peano axioms.
There are other set theoretical constructions. In particular,

mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the natural numbers are those number
A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers can ...

s used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called '' cardinal numbers'', and numbers used for ordering are called ''ordinal numbers
In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets.
A finite set can be enumerated by successively labeling each element with the least n ...

''. Natural numbers are sometimes used as labels, known as '' nominal numbers'', having none of the properties of numbers in a mathematical sense (e.g. sports jersey numbers).
Some definitions, including the standard ISO 80000-2, begin the natural numbers with , corresponding to the non-negative integers , whereas others start with , corresponding to the positive integers Texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, while in other writings, that term is used instead for the integers (including negative integers).
The natural numbers form a set. Many other number sets are built by successively extending the set of natural numbers: the integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language of ...

s, by including an additive identity In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mode ...

0 (if not yet in) and an additive inverse
In mathematics, the additive inverse of a number
A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with n ...

for each nonzero natural number ; the rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ) ...

s, by including a multiplicative inverse
In mathematics, a multiplicative inverse or reciprocal for a number ''x'', denoted by 1/''x'' or ''x''−1, is a number which when Multiplication, multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a rat ...

$1/n$ for each nonzero integer (and also the product of these inverses by integers); the real number
In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...

s by including the limits
Limit or Limits may refer to:
Arts and media
* Limit (manga), ''Limit'' (manga), a manga by Keiko Suenobu
* Limit (film), ''Limit'' (film), a South Korean film
* Limit (music), a way to characterize harmony
* Limit (song), "Limit" (song), a 201 ...

of (converging) Cauchy sequences of rationals; the complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a ...

s, by adjoining to the real numbers a square root of (and also the sums and products thereof); and so on. This chain of extensions canonically embeds the natural numbers in the other number systems.
Properties of the natural numbers, such as divisibility
In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...

and the distribution of prime number
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only wa ...

s, are studied in number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative intege ...

. Problems concerning counting and ordering, such as partitioning and enumerations, are studied in combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite set, finite Mathematical structure, structures. It is closely related to many other ar ...

.
In common language, particularly in primary school education, natural numbers may be called counting numbers to intuitively exclude the negative integers and zero, and also to contrast the discreteness of counting
Counting is the process of determining the number of Element (mathematics), elements of a finite set of objects, i.e., determining the size (mathematics), size of a set. The traditional way of counting consists of continually increasing a (mental ...

to the continuity of measurement
Measurement is the quantification (science), quantification of variable and attribute (research), attributes of an object or event, which can be used to compare with other objects or events.
In other words, measurement is a process of determi ...

—a hallmark characteristic of real number
In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...

s.
History

Ancient roots

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 numerals to represent numbers. This allowed systems to be developed for recording large numbers. The ancientEgyptians
Egyptians ( arz, المَصرِيُون, translit=al-Maṣriyyūn, ; arz, المَصرِيِين, translit=al-Maṣriyyīn, ; cop, ⲣⲉⲙⲛ̀ⲭⲏⲙⲓ, remenkhēmi) are an ethnic group native to the Nile Valley in Egypt. Egyptian ident ...

developed a powerful system of numerals with distinct hieroglyphs
A hieroglyph (Ancient Greek, Greek for "sacred carvings") was a Character (symbol), character of the Egyptian hieroglyphs, ancient Egyptian writing system. logogram, Logographic scripts that are pictographic in form in a way reminiscent of ancien ...

for 1, 10, and all powers of 10 up to over 1 million. A stone carving from Karnak
The Karnak Temple Complex, commonly known as Karnak (, which was originally derived from ar, خورنق ''Khurnaq'' "fortified village"), comprises a vast mix of decayed Egyptian temple, temples, Pylon (architecture), pylons, chapels, and other ...

, dating back from around 1500 BCE and now at the Louvre
The Louvre ( ), or the Louvre Museum ( ), is the world's List of most-visited museums, most-visited museum, and an historic landmark in Paris, France. It is the home of some of the best-known works of art, including the ''Mona Lisa'' and the ' ...

in Paris, depicts 276 as 2 hundreds, 7 tens, and 6 ones; and similarly for the number 4,622. The Babylonia
Babylonia (; Akkadian: , ''māt Akkadī'') was an ancient Akkadian-speaking state and cultural area based in the city of Babylon in central-southern Mesopotamia (present-day Iraq and parts of Syria). It emerged as an Amorites, Amorite-ruled ...

ns had a place-value 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.
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 digit in place-value notation (within other numbers) dates back as early as 700 BCE by the Babylonians, who omitted such a digit when it would have been the last symbol in the number. The Olmec
The Olmecs () were the earliest known major Mesoamerican civilization. Following a progressive development in Soconusco, Veracruz, Soconusco, they occupied the tropical lowlands of the modern-day Mexican states of Veracruz and Tabasco. It has b ...

and Maya civilization
The Maya civilization () of the Mesoamerican people is known by its ancient temples and Glyph, glyphs. Its Maya script is the most sophisticated and highly developed writing system in the Pre-Columbian era, pre-Columbian Americas. It is also ...

s used 0 as a separate number as early as the , but this usage did not spread beyond Mesoamerica
Mesoamerica is a historical region and cultural area in southern North America and most of Central America. It extends from approximately central Mexico through Belize, Guatemala, El Salvador, Honduras, Nicaragua, and northern Costa Rica. Withi ...

. The use of a numeral 0 in modern times originated with the Indian mathematician Brahmagupta
Brahmagupta ( – ) was an Indian Indian mathematics, mathematician and Indian astronomy, astronomer. He is the author of two early works on mathematics and astronomy: the ''Brāhmasphuṭasiddhānta'' (BSS, "correctly established Siddhanta, do ...

in 628 CE. However, 0 had been used as a number in the medieval computus
As a moveable feast, the date of Easter is determined in each year through a calculation known as (). Easter is celebrated on the first Sunday after the Paschal full moon, which is the first full moon on or after 21 March (a fixed approxim ...

(the calculation of the date of Easter), beginning with Dionysius Exiguus
Dionysius Exiguus (Latin language, Latin for "Dionysius the Humble", Ancient Greek, Greek: Διονύσιος; – ) was a 6th-century Eastern Roman monk born in Scythia Minor (Roman province), Scythia Minor. He was a member of a community of S ...

in 525 CE, without being denoted by a numeral (standard Roman numerals
Roman numerals are a numeral system that originated in ancient Rome and remained the usual way of writing numbers throughout Europe well into the Late Middle Ages. Numbers are written with combinations of letters from the Latin alphabet, eac ...

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 abstraction
Abstraction in its main sense is a conceptual process wherein general rules and concept
Concepts are defined as abstract ideas. They are understood to be the fundamental building blocks of the concept behind principles, thoughts and beliefs.
T ...

s is usually credited to the Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece, a country in Southern Europe:
*Greeks, an ethnic group.
*Greek language, a branch of the Indo-European language family.
**Proto-Greek language, the assumed last common ancestor ...

philosophers Pythagoras
Pythagoras of Samos ( grc, Πυθαγόρας ὁ Σάμιος, Pythagóras ho Sámios, Pythagoras the Samos, Samian, or simply ; in Ionian Greek; ) was an ancient Ionians, Ionian Ancient Greek philosophy, Greek philosopher and the eponymou ...

and Archimedes
Archimedes of Syracuse (;; ) was a Greek mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Mathematicians are concerned with numbers, data, ...

. Some Greek mathematicians treated the number 1 differently than larger numbers, sometimes even not as a number at all. Euclid
Euclid (; grc-gre, Wikt:Εὐκλείδης, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the ''Euclid's Elements, Elements'' trea ...

, for example, defined a unit first and then a number as a multitude of units, thus by his definition, a unit is not a number and there are no unique numbers (e.g., any two units from indefinitely many units is a 2).
Independent studies on numbers also occurred at around the same time in India
India, officially the Republic of India (Hindi: ), is a country in South Asia. It is the List of countries and dependencies by area, seventh-largest country by area, the List of countries and dependencies by population, second-most populous ...

, China, and Mesoamerica
Mesoamerica is a historical region and cultural area in southern North America and most of Central America. It extends from approximately central Mexico through Belize, Guatemala, El Salvador, Honduras, Nicaragua, and northern Costa Rica. Withi ...

.
Modern definitions

In 19th century Europe, there was mathematical and philosophical discussion about the exact nature of the natural numbers. A school of Naturalism stated that the natural numbers were a direct consequence of the human psyche.Henri Poincaré
Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosophy of science, philosopher of science. He is often described as a polymath, a ...

was one of its advocates, as was Leopold Kronecker
Leopold Kronecker (; 7 December 1823 – 29 December 1891) was a Germans, German mathematician who worked on number theory, abstract algebra, algebra and mathematical logic, logic. He criticized Georg Cantor's work on set theory, and was quoted b ...

, who summarized his belief as "God made the integers, all else is the work of man".
In opposition to the Naturalists, the constructivists saw a need to improve upon the logical rigor in the foundations of mathematics
Foundations of mathematics is the study of the philosophy, philosophical and logical and/or algorithmic basis of mathematics, or, in a broader sense, the mathematical investigation of what underlies the philosophical theories concerning the natu ...

. In the 1860s, Hermann Grassmann
Hermann Günther Grassmann (german: link=no, Graßmann, ; 15 April 1809 – 26 September 1877) was a German polymath known in his day as a linguistics, linguist and now also as a mathematician. He was also a physicist, general scholar, and publi ...

suggested a recursive definition
In mathematics and computer science, a recursive definition, or inductive definition, is used to define the Element (mathematics), elements in a Set (mathematics), set in terms of other elements in the set (Peter Aczel, Aczel 1977:740ff). Some exa ...

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 still, they were shown to be equivalent in most practical applications.
Set-theoretical definitions of natural numbers were initiated by Frege. He initially defined a natural number as the class of all sets that are in one-to-one correspondence with a particular set. However, this definition turned out to lead to paradoxes, including Russell's paradox
In mathematical logic
Mathematical logic is the study of formal logic within mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are ...

. To avoid such paradoxes, the formalism was modified so that a natural number is defined as a particular set, and any set that can be put into one-to-one correspondence with that set is said to have that number of elements.
The second class of definitions was introduced by Charles Sanders Peirce
Charles Sanders Peirce ( ; September 10, 1839 – April 19, 1914) was an American philosopher, logician, mathematician and scientist who is sometimes known as "the father of pragmatism".
Educated as a chemist and employed as a scientist for t ...

, refined by Richard Dedekind
Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Mathemati ...

, and further explored by Giuseppe Peano
Giuseppe Peano (; ; 27 August 1858 – 20 April 1932) was an Italian mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Mathematicians are concer ...

; this approach is now called Peano arithmetic
In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th century Italian people, Italian mathematician Giuseppe Peano. These axioms have be ...

. It is based on an axiomatization of the properties of ordinal number
In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets.
A finite set can be enumerated by successively labeling each element with the least n ...

s: each natural number has a successor and every non-zero natural number has a unique predecessor. Peano arithmetic is equiconsistent with several weak systems of set theory. One such system is with the axiom of infinity
In axiomatic set theory and the branches of mathematics and philosophy that use it, the axiom of infinity is one of the axioms of Zermelo–Fraenkel set theory. It guarantees the existence of at least one infinite set, namely a set containing the ...

replaced by its negation. Theorems that can be proved in ZFC but cannot be proved using the Peano Axioms include Goodstein's theorem.
With all these definitions, it is convenient to include 0 (corresponding to the empty set
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...

) as a natural number. Including 0 is now the common convention among set theorists and logic
Logic is the study of correct reasoning. It includes both Mathematical logic, formal and informal logic. Formal logic is the science of Validity (logic), deductively valid inferences or of logical truths. It is a formal science investigating h ...

ians. Other mathematicians also include 0, and computer language
A computer language is a formal language used to communicate with a computer. Types of computer languages include:
* Software construction#Construction languages, Construction language – all forms of communication by which a human can Comput ...

s often start from zero when enumerating items like and string- or array-elements. On the other hand, many mathematicians have kept the older tradition to take 1 to be the first natural number.
Notation

The set of all natural numbers is standardly denoted or $\backslash mathbb\; N.$ Older texts have occasionally employed as the symbol for this set. Since natural numbers may contain or not, it may be important to know which version is referred to. This is often specified by the context, but may also be done by using a subscript or a superscript in the notation, such as: * Naturals without zero: $\backslash =\backslash mathbb^*=\; \backslash mathbb\; N^+=\backslash mathbb\_0\backslash smallsetminus\backslash \; =\; \backslash mathbb\_1$ * Naturals with zero: $\backslash ;\backslash =\backslash mathbb\_0=\backslash mathbb\; N^0=\backslash mathbb^*\backslash cup\backslash $ Alternatively, since the natural numbers naturally form asubset
In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...

of the integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language of ...

s (often they may be referred to as the positive, or the non-negative integers, respectively. To be unambiguous about whether 0 is included or not, sometimes a subscript (or superscript) "0" is added in the former case, and a superscript "" is added in the latter case:
:$\backslash \; =\; \backslash =\backslash mathbb\; Z^+=\; \backslash mathbb\_$
:$\backslash \; =\; \backslash =\backslash mathbb\; Z^\_=\backslash mathbb\_$
Properties

Addition

Given the set $\backslash mathbb$ of natural numbers and thesuccessor function
In mathematics, the successor function or successor operation sends a natural number to the next one. The successor function is denoted by ''S'', so ''S''(''n'') = ''n'' +1. For example, ''S''(1) = 2 and ''S''(2) = 3. The successor functio ...

$S\; \backslash colon\; \backslash mathbb\; \backslash to\; \backslash mathbb$ sending each natural number to the next one, one can define addition
Addition (usually signified by the Plus and minus signs#Plus sign, plus symbol ) is one of the four basic Operation (mathematics), operations of arithmetic, the other three being subtraction, multiplication and Division (mathematics), division. ...

of natural numbers recursively by setting and for all , . Then $(\backslash mathbb,\; +)$ is a commutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...

monoid
In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0.
Monoids ar ...

with identity element
In mathematics, an identity element, or neutral element, of a binary operation operating on a Set (mathematics), set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in alge ...

0. It is a free monoid In abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathematics), f ...

on one generator. This commutative monoid satisfies the cancellation property
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in m ...

, so it can be embedded in a group
A group is a number
A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, ...

. The smallest group containing the natural numbers is the integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language of ...

s.
If 1 is defined as , then . That is, is simply the successor of .
Multiplication

Analogously, given that addition has been defined, amultiplication
Multiplication (often denoted by the Multiplication sign, cross symbol , by the mid-line #Notation and terminology, dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four Elementary arithmetic, elementary Op ...

operator $\backslash times$ can be defined via and . This turns $(\backslash mathbb^*,\; \backslash times)$ into a free commutative monoid with identity element 1; a generator set for this monoid is the set of prime number
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only wa ...

s.
Relationship between addition and multiplication

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 acommutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...

semiring
In abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathematics), ...

. Semirings are an algebraic generalization of the natural numbers where multiplication is not necessarily commutative. The lack of additive inverses, which is equivalent to the fact that $\backslash mathbb$ is not closed under subtraction (that is, subtracting one natural from another does not always result in another natural), means that $\backslash mathbb$ is ''not'' a ring; instead it is a semiring
In abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathematics), ...

(also known as a ''rig'').
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 . Furthermore, $(\backslash mathbb,\; +)$ has no identity element.
Order

In this section, juxtaposed variables such as indicate the product , and the standardorder of operations
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...

is assumed.
A total order
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...

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
Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operation (mathematics), operations on numbers—addition, subtraction, multiplication, division (mathematics), division, exponent ...

in the following sense: if , and are natural numbers and , then and .
An important property of the natural numbers is that they are well-order
In mathematics, a well-order (or well-ordering or well-order relation) on a Set (mathematics), set ''S'' is a total order on ''S'' with the property that every non-empty subset of ''S'' has a least element in this ordering. The set ''S'' together ...

ed: every non-empty set of natural numbers has a least element. The rank among well-ordered sets is expressed by an ordinal number
In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets.
A finite set can be enumerated by successively labeling each element with the least n ...

; for the natural numbers, this is denoted as (omega).
Division

In this section, juxtaposed variables such as indicate the product , and the standardorder of operations
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...

is assumed.
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'' or Euclidean division
In arithmetic
Arithmetic () is an elementary part of mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantitie ...

is available as a substitute: for any two natural numbers and with there are natural numbers and such that
:$a\; =\; bq\; +\; r\; \backslash text\; r\; <\; b.$
The number is called the ''quotient
In arithmetic, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division (mathematics), division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to a ...

'' and is called the ''remainder
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in m ...

'' of the division of by . The numbers and are uniquely determined by and . This Euclidean division is key to the several other properties (divisibility
In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...

), algorithms (such as the Euclidean algorithm
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...

), and ideas in number theory.
Algebraic properties satisfied by the natural numbers

The addition (+) and multiplication (×) operations on natural numbers as defined above have several algebraic properties: * Closure under addition and multiplication: for all natural numbers and , both and are natural numbers. *Associativity
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...

: for all natural numbers , , and , and .
* Commutativity
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...

: for all natural numbers and , and .
* Existence of identity element
In mathematics, an identity element, or neutral element, of a binary operation operating on a Set (mathematics), set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in alge ...

s: for every natural number , if and .
** If the natural numbers are taken as "excluding 0", and "starting at 1", then for every natural number , . However, the "existence of additive identity element" property is not satisfied
* Distributivity
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...

of multiplication over addition for all natural numbers , , and , .
* No nonzero zero divisor
In abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathematics), ...

s: if and are natural numbers such that , then or (or both).
** If the natural numbers are taken as "excluding 0", and "starting at 1", the "no nonzero zero divisors" property is not satisfied.
Generalizations

Two important generalizations of natural numbers arise from the two uses of counting and ordering:cardinal number
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented ...

s and ordinal number
In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets.
A finite set can be enumerated by successively labeling each element with the least n ...

s.
* A natural number can be used to express the size of a finite set; more precisely, a cardinal number is a measure for the size of a set, which is even suitable for infinite sets. This concept of "size" relies on maps between sets, such that two sets have the same size, exactly if there exists a bijection
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function (mathematics), function between the elements of two set (mathematics), sets, where each element of one set is pair ...

between them. The set of natural numbers itself, and any bijective image of it, is said to be ''countably infinite
In mathematics, a Set (mathematics), set is countable if either it is finite set, finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function fro ...

'' and to have cardinality
In mathematics, the cardinality of a set (mathematics), set is a measure of the number of Element (mathematics), elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19 ...

aleph-null
In mathematics, particularly in set theory, the aleph numbers are a sequence of numbers used to represent the cardinality (or size) of infinite sets that can be well-ordered. They were introduced by the mathematician Georg Cantor and are named af ...

().
* Natural numbers are also used as linguistic ordinal numbers: "first", "second", "third", and so forth. This way they can be assigned to the elements of a totally ordered finite set, and also to the elements of any well-order
In mathematics, a well-order (or well-ordering or well-order relation) on a Set (mathematics), set ''S'' is a total order on ''S'' with the property that every non-empty subset of ''S'' has a least element in this ordering. The set ''S'' together ...

ed countably infinite set. This assignment can be generalized to general well-orderings with a cardinality beyond countability, to yield the ordinal numbers. An ordinal number may also be used to describe the notion of "size" for a well-ordered set, in a sense different from cardinality: if there is an order isomorphism
In the mathematics, mathematical field of order theory, an order isomorphism is a special kind of monotone function that constitutes a suitable notion of isomorphism for partially ordered sets (posets). Whenever two posets are order isomorphic, the ...

(more than a bijection!) between two well-ordered sets, they have the same ordinal number. The first ordinal number that is not a natural number is expressed as ; this is also the ordinal number of the set of natural numbers itself.
The least ordinal of cardinality (that is, the initial ordinal
In a written or published work, an initial capital, also referred to as a drop capital or simply an initial cap, initial, initcapital, initcap or init or a drop cap or drop, is a letter at the beginning of a word, a chapter, or a paragraph tha ...

of ) is but many well-ordered sets with cardinal number have an ordinal number greater than .
For finite
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation) Finite number may refer to:
* A countable number less than infinity, being the cardinality of a finite set – i.e., some natural number
In mathematics, the ...

well-ordered sets, there is a one-to-one 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
In mathematics, a sequence is an enumerated collection of mathematical object, objects in which repetitions are allowed and order theory, order matters. Like a Set (mathematics), set, it contains Element (mathematics), members (also called ''eleme ...

.
A countable non-standard model of arithmetic
Standardization or standardisation is the process of implementing and developing technical standard
A technical standard is an established norm or requirement for a repeatable technical task which is applied to a common and repeated use of rul ...

satisfying the Peano Arithmetic (that is, the first-order 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
Georges Henri Reeb (12 November 1920 – 6 November 1993) was a French mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Mathematicians are concer ...

used to claim provocatively that "The naïve integers don't fill up" $\backslash mathbb$. Other generalizations are discussed in the article on numbers.
Formal definitions

There are two standard methods for formally defining natural numbers. The first one, due toGiuseppe Peano
Giuseppe Peano (; ; 27 August 1858 – 20 April 1932) was an Italian mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Mathematicians are concer ...

, consists of an autonomous axiomatic theory called Peano arithmetic
In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th century Italian people, Italian mathematician Giuseppe Peano. These axioms have be ...

, based on few axioms called Peano axioms
In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th century Italian people, Italian mathematician Giuseppe Peano. These axioms have be ...

.
The second definition is based on set theory
Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, ...

. It defines the natural numbers as specific sets. More precisely, each natural number is defined as an explicitly defined set, whose elements allow counting the elements of other sets, in the sense that the sentence "a set has elements" means that there exists a one to one correspondence
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...

between the two sets and .
The sets used to define natural numbers satisfy Peano axioms. It follows that every theorem
In mathematics, a theorem is a statement (logic), statement that has been Mathematical proof, proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the th ...

that can be stated and proved in Peano arithmetic can also be proved in set theory. However, the two definitions are not equivalent, as there are theorems that can be stated in terms of Peano arithmetic and proved in set theory, which are not ''provable'' inside Peano arithmetic. A probable example is Fermat's Last Theorem
In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive number, positive integers , , and satisfy the equation for any integer value of greater than 2. The cases ...

.
The definition of the integers as sets satisfying Peano axioms provide a model
A model is an informative representation of an object, person or system. The term originally denoted the Plan_(drawing), plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin ''modulus'', a mea ...

of Peano arithmetic inside set theory. An important consequence is that, if set theory is consistent
In classical deductive logic, a consistent theory
A theory is a rational type of abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with such proc ...

(as it is usually guessed), then Peano arithmetic is consistent. In other words, if a contradiction could be proved in Peano arithmetic, then set theory would by contradictory, and every theorem of set theory would be both true and wrong.
Peano axioms

The five Peano axioms are the following: # 0 is a natural number. # Every natural number has a successor which is also a natural number. # 0 is not the successor of any natural number. # If the successor of $x$ equals the successor of $y$, then $x$ equals $y$. # The axiom of induction: If a statement is true of 0, and if the truth of that statement for a number implies its truth for the successor of that number, then the statement is true for every natural number. 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 $x\; +\; 1$.Set-theoretic definition

Intuitively, the natural number is the common property of all sets that have elements. So, its seems natural to define as anequivalence class
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...

under the relation "can be made in one to one correspondence
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...

". Unfortunately, this does not work in set theory
Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, ...

, as such an equivalence class would not be a set (because of Russell's paradox
In mathematical logic
Mathematical logic is the study of formal logic within mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are ...

). The standard solution is to define a particular set with elements that will be called the natural number .
The following definition was first published by John von Neumann
John von Neumann (; hu, Neumann János Lajos, ; December 28, 1903 – February 8, 1957) was a Hungarian-American mathematician, physicist, computer scientist, engineer and polymath. He was regarded as having perhaps the widest cover ...

, although Levy attributes the idea to unpublished work of Zermelo in 1916. As this definition extends to infinite set
In set theory
Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a bran ...

as a definition of empty set
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...

.
* Define the ''successor'' of any set by .
* By the axiom of infinity
In axiomatic set theory and the branches of mathematics and philosophy that use it, the axiom of infinity is one of the axioms of Zermelo–Fraenkel set theory. It guarantees the existence of at least one infinite set, namely a set containing the ...

, there exist sets which contain 0 and are closed under the successor function. Such sets are said to be ''inductive''. The intersection of all inductive sets is still an inductive set.
* This intersection is the set of the ''natural numbers''.
It follows that the natural numbers are defined iteratively as follows:
:*,
:*,
:*,
:*,
:*,
:* etc.
It can be checked that the natural numbers satisfies the Peano axioms
In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th century Italian people, Italian mathematician Giuseppe Peano. These axioms have be ...

.
With this definition, given a natural number , the sentence "a set has elements" can be formally defined as "there exists a bijection
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function (mathematics), function between the elements of two set (mathematics), sets, where each element of one set is pair ...

from to . This formalizes the operation of ''counting'' the elements of . Also, if and only if is a subset
In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...

of . In other words, the set inclusion
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...

defines the usual total order
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...

on the natural numbers. This order is a well-order
In mathematics, a well-order (or well-ordering or well-order relation) on a Set (mathematics), set ''S'' is a total order on ''S'' with the property that every non-empty subset of ''S'' has a least element in this ordering. The set ''S'' together ...

.
It follows from the definition that each natural number is equal to the set of all natural numbers less than it. This definition, can be extended to the von Neumann definition of ordinals for defining all Ernst Zermelo
Ernst Friedrich Ferdinand Zermelo (, ; 27 July 187121 May 1953) was a German logic
Logic is the study of correct reasoning. It includes both Mathematical logic, formal and informal logic. Formal logic is the science of Validity (logic), ded ...

provided a construction that is nowadays only of historical interest, and is sometimes referred to as . It consists in defining as the empty set, and .
With this definition each natural number is a singleton set
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...

. So, the property of the natural numbers to represent cardinalities is not directly accessible; only the ordinal property (being the th element of a sequence) is immediate. Unlike von Neumann's construction, the Zermelo ordinals do not extend to infinite ordinals.
See also

* * *Sequence
In mathematics, a sequence is an enumerated collection of mathematical object, objects in which repetitions are allowed and order theory, order matters. Like a Set (mathematics), set, it contains Element (mathematics), members (also called ''eleme ...

– Function of the natural numbers in another set
*
*
*
Notes

References

Bibliography

* * * * ** ** * * * * * * * * * * * * * * – English translation of .External links

* * {{Authority control Cardinal numbers Elementary mathematics Integers Number theory Sets of real numbers