TheInfoList

OR:

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 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 number ...
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 In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. The ...
'', 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 leas ...
''. Natural numbers are sometimes used as labels, known as ''
nominal number Nominal numbers are numerals used as labels to identify items uniquely. Importantly, the actual values of the numbers which these numerals represent are less relevant, as they do not indicate quantity, rank, or any other measurement. Labelling r ...
s'', having none of the properties of numbers in a mathematical sense (e.g. sports jersey numbers). Some definitions, including the standard
ISO 80000-2 ISO 80000 or IEC 80000 is an international standard introducing the International System of Quantities (ISQ). It was developed and promulgated jointly by the International Organization for Standardization (ISO) and the International Electrotech ...
, 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 Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
. 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 languag ...
s, by including an additive identity 0 (if not yet in) and an
additive inverse In mathematics, the additive inverse of a number is the number that, when added to , yields zero. This number is also known as the opposite (number), sign change, and negation. For a real number, it reverses its sign: the additive inverse (o ...
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 of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
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 multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a fraction ''a''/' ...
$1/n$ 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 measure a ''continuous'' one- dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s by including the limits 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 fo ...
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 integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mat ...
. Problems concerning counting and ordering, such as partitioning and
enumerations An enumeration is a complete, ordered listing of all the items in a collection. The term is commonly used in mathematics and computer science to refer to a listing of all of the elements of a set. The precise requirements for an enumeration (for ...
, 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 structures. It is closely related to many other areas of mathematics and has many a ...
. 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 elements of a finite set of objects, i.e., determining the size of a set. The traditional way of counting consists of continually increasing a (mental or spoken) counter by a unit for every el ...
to the continuity of
measurement Measurement is the quantification of 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 determining how large or small a physical quantity is as compared ...
—a hallmark characteristic of
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one- dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
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 ancient
Egyptians 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 identi ...
developed a powerful system of numerals with distinct
hieroglyphs A hieroglyph (Greek for "sacred carvings") was a character of the ancient Egyptian writing system. Logographic scripts that are pictographic in form in a way reminiscent of ancient Egyptian are also sometimes called "hieroglyphs". In Neoplatoni ...
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 temples, pylons, chapels, and other buildings near Luxor, Egypt. Constru ...
, dating back from around 1500 BCE and now at the
Louvre The Louvre ( ), or the Louvre Museum ( ), is the world's 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 ''Venus de Milo''. A central ...
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 Amorite-ruled state ...
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 Digit may refer to: Mathematics and science * Numerical digit, as used in mathematics or computer science ** Hindu-Arabic numerals, the most common modern representation of numerical digits * Digit (anatomy), the most distal part of a limb, suc ...
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, they occupied the tropical lowlands of the modern-day Mexican states of Veracruz and Tabasco. It has been speculated that ...
and
Maya civilization The Maya civilization () of the Mesoamerican people is known by its ancient temples and glyphs. Its Maya script is the most sophisticated and highly developed writing system in the pre-Columbian Americas. It is also noted for its art, arc ...
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 Ric ...
. The use of a numeral 0 in modern times originated with the Indian mathematician
Brahmagupta Brahmagupta ( – ) was an Indian mathematician and astronomer. He is the author of two early works on mathematics and astronomy: the '' Brāhmasphuṭasiddhānta'' (BSS, "correctly established doctrine of Brahma", dated 628), a theoretical tre ...
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 approx ...
(the calculation of the date of Easter), beginning with
Dionysius Exiguus Dionysius Exiguus (Latin for "Dionysius the Humble", Greek: Διονύσιος; – ) was a 6th-century Eastern Roman monk born in Scythia Minor. He was a member of a community of Scythian monks concentrated in Tomis (present day Constan ...
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, e ...
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 concepts are derived from the usage and classification of specific examples, literal ("real" or "concrete") signifiers, first principles, or other methods. "An abs ...
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 Samian, or simply ; in Ionian Greek; ) was an ancient Ionian Greek philosopher and the eponymous founder of Pythagoreanism. His poli ...
and
Archimedes Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientist ...
. Some Greek mathematicians treated the number 1 differently than larger numbers, sometimes even not as a number at all.
Euclid Euclid (; grc-gre, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of ...
, 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 seventh-largest country by area, the second-most populous country, and the most populous democracy in the world. Bounded by the Indian Ocean on th ...
, 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 Ric ...
.

## 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 philosopher of science. He is often described as a polymath, and in mathematics as "The ...
was one of its advocates, as was
Leopold Kronecker Leopold Kronecker (; 7 December 1823 – 29 December 1891) was a German mathematician who worked on number theory, algebra and logic. He criticized Georg Cantor's work on set theory, and was quoted by as having said, "'" ("God made the integers ...
, 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 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 nature of mathe ...
. 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 linguist and now also as a mathematician. He was also a physicist, general scholar, and publisher. His mat ...
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 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, Russell's paradox (also known as Russell's antinomy) is a set-theoretic paradox discovered by the British philosopher and mathematician Bertrand Russell in 1901. Russell's paradox shows that every set theory that contains a ...
. 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 ...
, refined by
Richard Dedekind Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to number theory, abstract algebra (particularly ring theory), and the axiomatic foundations of arithmetic. His ...
, and further explored by
Giuseppe Peano Giuseppe Peano (; ; 27 August 1858 – 20 April 1932) was an Italian mathematician and glottologist. The author of over 200 books and papers, he was a founder of mathematical logic and set theory, to which he contributed much notation. The ...
; 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 mathematician Giuseppe Peano. These axioms have been used nearly ...
. It is based on an
axiomatization In mathematics and logic, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. A theory is a consistent, relatively-self-contained body of knowledge which usually contains ...
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 leas ...
s: each natural number has a successor and every non-zero natural number has a unique predecessor. Peano arithmetic is
equiconsistent In mathematical logic, two theories are equiconsistent if the consistency of one theory implies the consistency of the other theory, and vice versa. In this case, they are, roughly speaking, "as consistent as each other". In general, it is not p ...
with several weak systems of set theory. One such system is ZFC 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 t ...
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, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in oth ...
) 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 formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from prem ...
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: * Construction language – all forms of communication by which a human can specify an executable problem solution to a comput ...
s often start from zero when enumerating items like loop counters 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 Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
of all natural numbers is standardly denoted or $\mathbb N.$ 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: $\=\mathbb^*= \mathbb N^+=\mathbb_0\smallsetminus\ = \mathbb_1$ * Naturals with zero: $\;\=\mathbb_0=\mathbb N^0=\mathbb^*\cup\$ Alternatively, since the natural numbers naturally form a
subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset o ...
of the
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s (often they may be referred to as the positive, or the non-negative integers, respectively. To be unambiguous about whether 0 is included or not, sometimes a subscript (or superscript) "0" is added in the former case, and a superscript "" is added in the latter case: :$\ = \=\mathbb Z^+= \mathbb_$ :$\ = \=\mathbb Z^_=\mathbb_$

# Properties

Given the set $\mathbb$ of natural numbers and the
successor 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 ...
$S \colon \mathbb \to \mathbb$ sending each natural number to the next one, one can define
addition Addition (usually signified by the plus symbol ) is one of the four basic operations of arithmetic, the other three being subtraction, multiplication and division. The addition of two whole numbers results in the total amount or '' sum'' ...
of natural numbers recursively by setting and for all , . Then $\left(\mathbb, +\right)$ 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 ...
with
identity element In mathematics, an identity element, or neutral element, of a binary operation operating on a 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 algebraic structures ...
0. It is a
free monoid In abstract algebra, the free monoid on a set is the monoid whose elements are all the finite sequences (or strings) of zero or more elements from that set, with string concatenation as the monoid operation and with the unique sequence of zero el ...
on one generator. This commutative monoid satisfies the
cancellation property In mathematics, the notion of cancellative is a generalization of the notion of invertible. An element ''a'' in a magma has the left cancellation property (or is left-cancellative) if for all ''b'' and ''c'' in ''M'', always implies that . A ...
, so it can be embedded in a
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
. 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 languag ...
s. If 1 is defined as , then . That is, is simply the successor of .

## Multiplication

Analogously, given that addition has been defined, a
multiplication Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary mathematical operations of arithmetic, with the other ones being ad ...
operator $\times$ can be defined via and . This turns $\left(\mathbb^*, \times\right)$ 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 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 ...
semiring In abstract algebra, a semiring is an algebraic structure similar to a ring, but without the requirement that each element must have an additive inverse. The term rig is also used occasionally—this originated as a joke, suggesting that rigs ...
. 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 $\mathbb$ is not closed under subtraction (that is, subtracting one natural from another does not always result in another natural), means that $\mathbb$ is ''not'' a
ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...
semiring In abstract algebra, a semiring is an algebraic structure similar to a ring, but without the requirement that each element must have an additive inverse. The term rig is also used occasionally—this originated as a joke, suggesting that rigs ...
(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, $\left(\mathbb, +\right)$ has no identity element.

## Order

In this section, juxtaposed variables such as indicate the product , and the standard
order of operations In mathematics and computer programming, the order of operations (or operator precedence) is a collection of rules that reflect conventions about which procedures to perform first in order to evaluate a given mathematical expression. For examp ...
is assumed. A
total order In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X: # a \leq a ( reflexi ...
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 operations on numbers—addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th ...
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 ''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 with the well- ...
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 leas ...
; for the natural numbers, this is denoted as (omega).

## Division

In this section, juxtaposed variables such as indicate the product , and the standard
order of operations In mathematics and computer programming, the order of operations (or operator precedence) is a collection of rules that reflect conventions about which procedures to perform first in order to evaluate a given mathematical expression. For examp ...
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, Euclidean division – or division with remainder – is the process of dividing one integer (the dividend) by another (the divisor), in a way that produces an integer quotient and a natural number remainder strictly smaller tha ...
is available as a substitute: for any two natural numbers and with there are natural numbers and such that :$a = bq + r \text r < b.$ The number is called the ''
quotient In arithmetic, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a ...
'' and is called the ''
remainder In mathematics, the remainder is the amount "left over" after performing some computation. In arithmetic, the remainder is the integer "left over" after dividing one integer by another to produce an integer quotient ( integer division). In alge ...
'' 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, the Euclidean algorithm,Some widely used textbooks, such as I. N. Herstein's ''Topics in Algebra'' and Serge Lang's ''Algebra'', use the term "Euclidean algorithm" to refer to Euclidean division or Euclid's algorithm, is an e ...
), 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, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacemen ...
: 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 is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures ...
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, the distributive property of binary operations generalizes the distributive law, which asserts that the equality x \cdot (y + z) = x \cdot y + x \cdot z is always true in elementary algebra. For example, in elementary arithmeti ...
of multiplication over addition for all natural numbers , , and , . * No nonzero
zero divisor In abstract algebra, an element of a ring is called a left zero divisor if there exists a nonzero in such that , or equivalently if the map from to that sends to is not injective. Similarly, an element of a ring is called a right ze ...
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, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. T ...
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 leas ...
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 between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
between them. The set of natural numbers itself, and any bijective image of it, is said to be ''
countably infinite In mathematics, a set is countable if either it is 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 from it into the natural number ...
'' and to have
cardinality In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...
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 name ...
(). * 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 ''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 with the well- ...
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 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, they can be con ...
(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 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 set, a set whose cardinality (number of elements) is some natural number * Finite verb Traditionally, a finite verb (from la, fīnītus, past partici ...
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 objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is cal ...
. A countable non-standard model of arithmetic 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 used to claim provocatively that "The naïve integers don't fill up" $\mathbb$. 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 to
Giuseppe Peano Giuseppe Peano (; ; 27 August 1858 – 20 April 1932) was an Italian mathematician and glottologist. The author of over 200 books and papers, he was a founder of mathematical logic and set theory, to which he contributed much notation. The ...
, 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 mathematician Giuseppe Peano. These axioms have been used nearly ...
, 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 mathematician Giuseppe Peano. These axioms have been used nearly ...
. The second definition is based on
set theory Set theory is the branch of mathematical logic that studies 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, is mostly concern ...
. It defines the natural numbers as specific
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
s. 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, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
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 that has been 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 theorem is a logical consequence of ...
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 integers , , and satisfy the equation for any integer value of greater than 2. The cases and have been ...
. 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 plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin ''modulus'', a measure. Models ...
of Peano arithmetic inside set theory. An important consequence is that, if set theory is
consistent In classical deductive logic, a consistent theory is one that does not lead to a logical contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent ...
(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
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
s that have elements. So, its seems natural to define as an
equivalence class In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements ...
under the relation "can be made in
one to one correspondence In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
". Unfortunately, this does not work in
set theory Set theory is the branch of mathematical logic that studies 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, is mostly concern ...
, as such an equivalence class would not be a set (because of
Russell's paradox In mathematical logic, Russell's paradox (also known as Russell's antinomy) is a set-theoretic paradox discovered by the British philosopher and mathematician Bertrand Russell in 1901. Russell's paradox shows that every set theory that contains a ...
). 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 cov ...
, although Levy attributes the idea to unpublished work of Zermelo in 1916. As this definition extends to
infinite set In set theory, an infinite set is a set that is not a finite set. Infinite sets may be countable or uncountable. Properties The set of natural numbers (whose existence is postulated by the axiom of infinity) is infinite. It is the only se ...
as a definition 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 leas ...
, the sets considered below are sometimes called von Neumann ordinals. The definition proceeds as follows: * Call , the
empty set In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in oth ...
. * 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 t ...
, 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 mathematician Giuseppe Peano. These axioms have been used nearly ...
. 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 between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
from to . This formalizes the operation of ''counting'' the elements of . Also, if and only if is a
subset In mathematics, set ''A'' is a subset of a set ''B'' if all 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 unequal, then ''A'' is a proper subset o ...
of . In other words, the set inclusion defines the usual
total order In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X: # a \leq a ( reflexi ...
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 ''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 with the well- ...
. 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
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 leas ...
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,
Ernst Zermelo Ernst Friedrich Ferdinand Zermelo (, ; 27 July 187121 May 1953) was a German logician and mathematician, whose work has major implications for the foundations of mathematics. He is known for his role in developing Zermelo–Fraenkel axiomatic se ...
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, a singleton, also known as a unit set or one-point set, is a set with exactly one element. For example, the set \ is a singleton whose single element is 0. Properties Within the framework of Zermelo–Fraenkel set theory, the ...
. So, the property of the natural numbers to represent
cardinalities In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...
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.

* * *
Sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is cal ...
– Function of the natural numbers in another set * * *

# Bibliography

* * * * ** ** * * * * * * * * * * * * * * – English translation of .