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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 ...
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''. 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, 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 set 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 are built by successively extending the set of natural numbers: the integers, by including an additive identity 0 (if not yet in) and an additive inverse for each nonzero natural number ; the rational numbers, by including a multiplicative inverse 1/n for each nonzero integer (and also the product of these inverses by integers); the real numbers 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 for ...
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 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 way ...
s, are studied in number theory. Problems concerning counting and ordering, such as partitioning and enumerations, are studied in combinatorics. 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 Discrete may refer to: *Discrete particle or quantum in physics, for example in quantum theory *Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit * Discrete group, a ...
of counting 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 numbers.


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 for 1, 10, and all powers of 10 up to over 1 million. A stone carving from Karnak, dating back from around 1500 BCE and now at the Louvre in Paris, depicts 276 as 2 hundreds, 7 tens, and 6 ones; and similarly for the number 4,622. The Babylonians 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, such ...
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 and Maya civilizations 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. W ...
. The use of a numeral 0 in modern times originated with the Indian mathematician Brahmagupta in 628 CE. However, 0 had been used as a number in the medieval computus (the calculation of the date of Easter), beginning with Dionysius Exiguus in 525 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, ...
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 abstractions is usually credited to the Greek philosophers Pythagoras and Archimedes. Some Greek mathematicians treated the number 1 differently than larger numbers, sometimes even not as a number at all. Euclid, 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 the ...
, 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. W ...
.


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, 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. In the 1860s, Hermann Grassmann suggested a recursive definition for natural numbers, thus stating they were not really natural—but a consequence of definitions. Later, two classes of such formal definitions were constructed; later still, they were shown to be equivalent in most practical applications.
Set-theoretical definitions of natural numbers In set theory, several ways have been proposed to construct the natural numbers. These include the representation via von Neumann ordinals, commonly employed in axiomatic set theory, and a system based on equinumerosity that was proposed by Got ...
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. 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; 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 u ...
. It is based on an
axiomatization In mathematics and logic, an axiomatic system is any Set (mathematics), set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. A Theory (mathematical logic), theory is a consistent, relatively-self-co ...
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 lea ...
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 ...
with several weak systems of set theory. One such system is ZFC with the axiom of infinity replaced by its negation. Theorems that can be proved in ZFC but cannot be proved using the Peano Axioms include Goodstein's theorem. 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 other ...
) as a natural number. Including 0 is now the common convention among set theorists and logicians. 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 compu ...
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 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 (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 integers (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


Addition

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 of natural numbers recursively by setting and for all , . Then (\mathbb, +) is a commutative monoid 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 s ...
 0. It is a free monoid on one generator. This commutative monoid satisfies the cancellation property, so it can be embedded in a group. The smallest group containing the natural numbers is the integers. 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 additi ...
operator \times can be defined via and . This turns (\mathbb^*, \times) into a
free commutative 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 eleme ...
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 way ...
s.


Relationship between addition and multiplication

Addition and multiplication are compatible, which is expressed in the
distribution law Distribution law or the Nernst's distribution law gives a generalisation which governs the distribution of a solute between two non miscible solvents. This law was first given by Nernst who studied the distribution of several solutes between dif ...
: . These properties of addition and multiplication make the natural numbers an instance of a commutative semiring. 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; instead it is a semiring (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, (\mathbb, +) 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 exampl ...
is assumed. A total order on the natural numbers is defined by letting if and only if there exists another natural number where . This order is compatible with the arithmetical operations in the following sense: if , and are natural numbers and , then and . An important property of the natural numbers is that they are well-ordered: 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 lea ...
; 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 exampl ...
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 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'' and is called the '' remainder'' of the division of by . The numbers and are uniquely determined by and . This Euclidean division is key to the several other properties ( divisibility), 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 effi ...
), 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: for all natural numbers , , and , and . * Commutativity: 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 ...
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 of multiplication over addition for all natural numbers , , and , . * No nonzero zero divisors: 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 lea ...
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 between them. The set of natural numbers itself, and any bijective image of it, is said to be '' countably infinite'' and to have cardinality aleph-null (). * 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-ordered 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 (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 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. A countable
non-standard model of arithmetic Standardization or standardisation is the process of implementing and developing technical standards based on the consensus of different parties that include firms, users, interest groups, standards organizations and governments. Standardization ...
satisfying the Peano Arithmetic (that is, the first-order Peano axioms) was developed by
Skolem Thoralf Albert Skolem (; 23 May 1887 – 23 March 1963) was a Norwegian mathematician who worked in mathematical logic and set theory. Life Although Skolem's father was a primary school teacher, most of his extended family were farmers. 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, 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 u ...
, based on few axioms called Peano axioms. The second definition is based on set theory. 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 between the two sets and . The sets used to define natural numbers satisfy Peano axioms. It follows that every theorem 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 bee ...
. The definition of the integers as sets satisfying Peano axioms provide a model of Peano arithmetic inside set theory. An important consequence is that, if set 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 sets 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 a ...
under the relation "can be made in one to one correspondence". Unfortunately, this does not work in set theory, as such an equivalence class would not be a set (because of Russell's paradox). 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, although Levy attributes the idea to unpublished work of Zermelo in 1916. As this definition extends to infinite set 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 lea ...
, 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 other ...
. * Define the ''successor'' of any set by . * By the axiom of infinity, 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. With this definition, given a natural number , the sentence "a set has elements" can be formally defined as "there exists a bijection 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 defines the usual total order on the natural numbers. This order is a well-order. 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 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 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 lea ...
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 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. 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 – 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