View source for Natural number

{{Short description|Number used for counting}}{{pp-vandalism|small=yes}}
{{Use dmy dates|date=May 2021}}
[[File:Three Baskets with Apples.svg|right|thumb|upright|Natural numbers can be used for counting: one apple; two apples are one apple added to another apple, three apples are one apple added to two apples, ...]]

In [[mathematics]], the '''natural numbers''' are the numbers [[0]], [[1]], [[2]], [[3]], and so on, possibly excluding 0.{{efn|It depends on authors and context whether 0 is considered a natural number.}}<ref name="Enderton">{{cite book |last1=Enderton |first1=Herbert B. |title=Elements of set theory |date=1977 |publisher=Academic Press |location=New York |isbn=0122384407 |page=66}}</ref> The terms '''positive integers''', '''non-negative integers''', '''whole numbers''', and '''counting numbers''' are also used.<ref name="Cooke2000">{{cite book |last1=Cooke |first1=Heather |url=https://www.google.com/books/edition/Primary_Mathematics/yNCL5iTJqLQC?hl=en&gbpv=1&dq=%22counting%20numbers%22%20%22positive%20integers%22&pg=PA14&printsec=frontcover |title=Primary Mathematics |date=26 October 2000 |publisher=SAGE |isbn=978-1-84787-949-3 |page=14 |language=en}}</ref><ref name="Zegarelli2014">{{cite book |last1=Zegarelli |first1=Mark |url=https://www.google.com/books/edition/Basic_Math_and_Pre_Algebra_For_Dummies/V8nDAgAAQBAJ?hl=en&gbpv=1&dq=%22counting%20numbers%22&pg=PA21&printsec=frontcover |title=Basic Math and Pre-Algebra For Dummies |date=28 January 2014 |publisher=John Wiley & Sons |isbn=978-1-118-79199-8 |page=21 |language=en |quote=Counting numbers (also called natural numbers): The set of numbers beginning 1, 2, 3, 4, ... and going on infinitely.}}</ref> The [[set (mathematics)|set]] of the natural numbers is commonly denoted by a bold {{math|'''N'''}} or a [[blackboard bold]] {{tmath|\N}}.

The natural numbers are used for counting,<!-- Please, do not link this word that is used in its common language meaning, and not in any technical meaning --> and for labeling the result of a count, like "there are ''seven'' days in a week", in which case they are called ''[[cardinal numeral|cardinal number]]s''. They are also used to label places in an ordered series,<!-- Please, do not link these words that are used in their common language meanings, and not in any technical meaning --> like "the ''third'' day of the month", in which case they are called ''[[ordinal numeral|ordinal number]]s''. Natural numbers may also be used to label, like the [[Number (sports)|jersey numbers]] of a sports team; in this case, they have no specific mathematical properties and are called ''[[nominal number]]s''.<ref>{{cite journal |last1=Woodin |first1=Greg |first2=Bodo |last2=Winter |title=Numbers in Context: Cardinals, Ordinals, and Nominals in American English |journal=Cognitive Science |volume=48|number=6 |year=2024 |article-number=e13471 |doi=10.1111/cogs.13471 |doi-access=free|pmid=38895756 |pmc=11475258 }}</ref>

Two natural [[operation (mathematics)|operation]]s are defined on natural numbers, [[addition]] and [[multiplication]]. [[Arithmetic]] is the study of the ways to perform these operations. [[Number theory]] is the study of the properties of these operations and their generalizations. Much of [[combinatorics]] involves counting mathematical objects, patterns and structures that are defined using natural numbers.

Many [[number system]]s are built from the natural numbers and contain them. For example, the [[integer]]s are made by including 0 and negative numbers. The [[rational number]]s add fractions, and the [[real number]]s add all infinite decimals. [[Complex number]]s add the [[Imaginary unit|square root of {{math|−1}}]].<ref>{{harvtxt|Mendelson|2008|page=x}} says: "The whole fantastic hierarchy of number systems is built up by purely set-theoretic means from a few simple assumptions about natural numbers."</ref>  This makes up natural numbers as foundational for all mathematics.<ref>{{harvtxt|Bluman|2010|page=1}}: "Numbers make up the foundation of mathematics."</ref>

== Terminology and notation ==

The term ''natural numbers'' has two common definitions: either {{math|1=0, 1, 2, ...}} or {{math|1, 2, 3, ...}}. Because there is no universal convention, the definition can be chosen to suit the context of use.<ref name="Enderton" /><ref name=":1">{{cite web |last=Weisstein |first=Eric W. |title=Natural Number |url=https://mathworld.wolfram.com/NaturalNumber.html |access-date=11 August 2020 |website=mathworld.wolfram.com |language=en}}</ref> To eliminate ambiguity, the sequences {{math|1, 2, 3, ...}} and {{math|1=0, 1, 2, ...}} are often called the '''positive integers''' and the '''non-negative integers''', respectively.

The phrase '''whole numbers''' is frequently used for the natural numbers that include 0, although it may also mean all integers, positive and negative.<ref>{{cite dictionary |year=2003 |title=integer |dictionary=Embedded Systems Dictionary |publisher=Taylor & Francis |url=https://books.google.com/books?id=zePGx82d_fwC |access-date=28 March 2017 |pages=138 (integer), 247 (signed integer), & 276 (unsigned integer) |isbn=978-1-57820-120-4 |url-status=live |archive-url=https://web.archive.org/web/20170329150719/https://books.google.com/books?id=zePGx82d_fwC |archive-date=29 March 2017 |last2=Barr |first2=Michael |name-list-style=amp |first1=Jack G. |last1=Ganssle |via=Google Books}}</ref><ref name="Cooke2000" /> In primary education, '''counting numbers''' usually refer to the natural numbers starting at 1,<ref name="Zegarelli2014" /> though this definition can vary.<ref>{{cite book |last1=Rice |first1=Harris |url=https://www.google.com/books/edition/The_Mathematics_Teacher/2kwH8pZKLLEC?hl=en&gbpv=1&dq=%22counting%20numbers%22&pg=PA393&printsec=frontcover |title=The Mathematics Teacher |date=1922 |publisher=National Council of Teachers of Mathematics |page=393 |language=en |chapter=Errors in computations and the rounded number |quote=A counting number is the number given in answer to the question "How many?" In this class of numbers belongs zero and positive integers/}}</ref><ref name="MathWorld_CountingNumber">{{MathWorld|title=Counting Number|id=CountingNumber}}</ref>

The [[Set (mathematics)|set]] of all natural numbers is typically denoted {{math|'''N'''}} or in [[blackboard bold]] as <math>\mathbb N.</math><ref name=":1" /><ref>{{cite web |title=Listing of the Mathematical Notations used in the Mathematical Functions Website: Numbers, variables, and functions |url=https://functions.wolfram.com/Notations/1/ |access-date=27 July 2020 |website=functions.wolfram.com}}</ref>{{Efn|Older texts have occasionally employed {{math|''J''}} as the symbol for this set.<ref>{{cite book |url=https://archive.org/details/1979RudinW |title=Principles of Mathematical Analysis |last=Rudin |first=W. |publisher=McGraw-Hill |year=1976 |isbn=978-0-07-054235-8 |location=New York |page=25}}</ref>}} Whether 0 is included is often determined by the context but may also be specified by using <math>\mathbb N</math> or <math>\mathbb Z</math> (the set of all integers) with a subscript or superscript. Examples include <math>\mathbb{N}_1</math>,<ref name="Peano19012">{{cite book |last1=Peano |first1=Giuseppe |url=https://archive.org/details/formulairedesmat00pean/page/38/mode/2up |title=Formulaire des mathematiques |date=1901 |publisher=Paris, Gauthier-Villars |page=39 |language=fr}}</ref> or <math>\mathbb Z^+</math><ref name="Grimaldi20042">{{cite book |last1=Grimaldi |first1=Ralph P. |title=Discrete and Combinatorial Mathematics: An applied introduction |publisher=Pearson Addison Wesley |year=2004 |isbn=978-0-201-72634-3 |edition=5th}}</ref> (for the set starting at 1) and <math>\mathbb{N}_0</math><ref>{{cite book |last1=Stewart |first1=Ian |url=https://books.google.com/books?id=xSN-BwAAQBAJ&dq=natural%20numbers%20%22N0%22&pg=PA160 |title=The Foundations of Mathematics |last2=Tall |first2=David |date=12 March 2015 |publisher=OUP Oxford |isbn=978-0-19-101648-6 |page=160 |language=en |access-date=30 July 2025}}</ref> or <math>\mathbb Z^{0+}</math><ref>{{cite book |last1=Fokas |first1=Athanassios |url=https://books.google.com/books?id=QwuhEAAAQBAJ&dq=natural%20numbers%20%22z0%2B%22&pg=PA4 |title=Modern Mathematical Methods For Scientists And Engineers: A Street-smart Introduction |last2=Kaxiras |first2=Efthimios |date=12 December 2022 |publisher=World Scientific |isbn=978-1-80061-182-5 |page=4 |language=en |access-date=30 July 2025}}</ref> (for the set including 0).

==Intuitive concept==
An intuitive and implicit understanding of natural numbers is developed ''naturally'' through using numbers for counting, ordering and basic arithmetic. Within this are two closely related aspects of what a natural number is: the ''size of a collection;'' and ''a'' ''position in a sequence''.

=== Size of a collection ===
Natural numbers can be used to answer questions like: "how many apples are on the table?".<ref>{{Cite book |last1=Frege |first1=Gottlob |title=The foundations of arithmetic: a logico-mathematical enquiry into the concept of number |last2=Frege |first2=Gottlob |orig-date=1953|date=1975 |publisher=Northwestern Univ. Press |isbn=978-0-8101-0605-5 |edition=2. revised |location=Evanston Ill |pages=5}}</ref> A natural number used in this way describes a characteristic of a ''collection of objects''. This characteristic, the ''size of a collection'' is called [[cardinality]] and a natural number used to describe or measure it is called a cardinal number.
[[File:Cardinality apples and oranges.svg|thumb|A group of apples and group of oranges with the same cardinality.]]
Two collections have the same size or cardinality if there is a [[one-to-one correspondence]] between the objects in each collection to the objects in the other. For example, in the image to the right every apple can be paired off with one orange and every orange can be paired off with one apple. From this, even without counting or using numbers it can be seen that the group of apples has the ''same'' cardinality as the group of oranges, meaning they are both assigned the same cardinal number. 

The natural number 3 is the thing used for the particular cardinal number described above and for the cardinal number of any other collection of objects that could be paired off in the same way to one of these groups.

=== Position in a sequence ===
The natural numbers have a fixed progression, which is the familiar sequence beginning with 1, 2, 3, and so on. A natural number can be used to denote a specific position in any other sequence, in which case it is called an [[ordinal number|ordinal number.]] To have a specific position in a sequence means to come either before or after every other position in the sequence in a defined way, which is the concept of [[Order theory|order]].

The natural number 3 then is the thing that ''comes after'' 2 and 1, and ''before'' 4, 5 and so on. The number 2 is the thing that ''comes after'' 1, and 1 is the first element in the sequence. Each number represents the relation that position bears to the rest of the infinite sequence.<ref>{{Cite journal |last=Benacerraf |first=Paul |date=January 1965 |title=What Numbers Could not Be |url=https://www.jstor.org/stable/2183530?origin=crossref |journal=The Philosophical Review |volume=74 |issue=1 |pages=47 |doi=10.2307/2183530 |url-access=subscription}}</ref>

=== Counting ===
The process of counting involves both the cardinal and ordinal use of the natural numbers and illustrates the way the two fit together. To count the number of objects in a collection, each object is paired off with a natural number, usually by mentally or verbally saying the name of the number and assigning it to a particular object. The numbers must be assigned in order starting with 1 (they are ordinal) but the order of the objects chosen is arbitrary as long as each object is assigned one and only one number. When all of the objects have been assigned a number, the ordinal number assigned to the final object gives the result of the count, which is the cardinal number of the whole collection.

==History==
<!--This section is linked from [[Numeral system]]-->

===Ancient roots===
{{further|Prehistoric counting}}

[[File:Ishango bone (cropped).jpg|thumb|The [[Ishango bone]] (on exhibition at the [[Royal Belgian Institute of Natural Sciences]])<ref name=RBINS_intro>{{cite web |title=Introduction |series=[[Ishango bone]] |publisher=[[Royal Belgian Institute of Natural Sciences]] |location=Brussels, Belgium |url=https://www.naturalsciences.be/expo/old_ishango/en/ishango/introduction.html |archive-url=https://web.archive.org/web/20160304051733/https://www.naturalsciences.be/expo/old_ishango/en/ishango/introduction.html |archive-date=4 March 2016}}</ref><ref name=RBINS_flash>{{cite web |title=Flash presentation |series=[[Ishango bone]] |publisher=[[Royal Belgian Institute of Natural Sciences]] |place=Brussels, Belgium |url=http://ishango.naturalsciences.be/Flash/flash_local/Ishango-02-EN.html |archive-url=https://web.archive.org/web/20160527164619/http://ishango.naturalsciences.be/Flash/flash_local/Ishango-02-EN.html |archive-date=27 May 2016}}</ref><ref name=UNESCO>{{cite web |title=The Ishango Bone, Democratic Republic of the Congo |website=[[UNESCO]]'s Portal to the Heritage of Astronomy |url=http://www2.astronomicalheritage.net/index.php/show-entity?identity=4&idsubentity=1 |archive-url=https://web.archive.org/web/20141110195426/http://www2.astronomicalheritage.net/index.php/show-entity?identity=4&idsubentity=1 |archive-date=10 November 2014}}, on permanent display at the [[Royal Belgian Institute of Natural Sciences]], Brussels, Belgium.</ref> is believed to have been used 20,000&nbsp;years ago for natural number arithmetic.]]

The most primitive method of representing a natural number is to use one's fingers, as in [[finger counting]]. Putting down a [[tally mark]] for each object is another primitive method. Later, a set of objects could be tested for equality, excess or shortage—by striking out a mark and removing an object from the set.

The first major advance in abstraction was the use of [[numeral system|numerals]] to represent numbers. This allowed systems to be developed for recording large numbers. The ancient [[History of Ancient Egypt|Egyptians]] developed a powerful system of numerals with distinct [[Egyptian hieroglyphs|hieroglyphs]] for&nbsp;1, 10, and all powers of&nbsp;10 up to over 1&nbsp;million. A stone carving from [[Karnak]], dating back from around 1500&nbsp;BCE and now at the [[Louvre]] in Paris, depicts&nbsp;276 as 2&nbsp;hundreds, 7&nbsp;tens, and 6&nbsp;ones; and similarly for the number&nbsp;4,622. The [[Babylonia]]ns had a [[Positional notation|place-value]] system based essentially on the numerals for&nbsp;1 and&nbsp;10, using base sixty, so that the symbol for sixty was the same as the symbol for one—its value being determined from context.<ref>{{cite book |first=Georges |last=Ifrah |year=2000 |title=The Universal History of Numbers |publisher=Wiley |isbn=0-471-37568-3}}</ref>

A much later advance was the development of the idea that&nbsp;{{num|0}} can be considered as a number, with its own numeral. The use of a&nbsp;0 [[numerical digit|digit]] in place-value notation (within other numbers) dates back as early as 700&nbsp;BCE by the Babylonians, who omitted such a digit when it would have been the last symbol in the number.{{efn| A tablet found at Kish ... thought to date from around 700&nbsp;BC, uses three hooks to denote an empty place in the positional notation. Other tablets dated from around the same time use a single hook for an empty place.<ref>{{cite web |title=A history of Zero |website=MacTutor History of Mathematics |url=http://www-history.mcs.st-and.ac.uk/history/HistTopics/Zero.html |url-status=live |access-date=23 January 2013 |archive-url=https://web.archive.org/web/20130119083234/http://www-history.mcs.st-and.ac.uk/history/HistTopics/Zero.html |archive-date=19 January 2013}}</ref>}} The [[Olmec]] and [[Maya civilization]]s used&nbsp;0 as a separate number as early as the {{nowrap|1st century BCE}}, but this usage did not spread beyond [[Mesoamerica]].<ref>{{cite book |first=Charles C. |last=Mann |year=2005 |title=1491: New Revelations of the Americas before Columbus |page=19 |publisher=Knopf |isbn=978-1-4000-4006-3 |url=https://books.google.com/books?id=Jw2TE_UNHJYC&pg=PA19 |url-status=live |via=Google Books |access-date=3 February 2015 |archive-url=https://web.archive.org/web/20150514105855/https://books.google.com/books?id=Jw2TE_UNHJYC&pg=PA19 |archive-date=14 May 2015}}</ref><ref>{{cite book |first=Brian |last=Evans |year=2014 |title=The Development of Mathematics Throughout the Centuries: A brief history in a cultural context |publisher=John Wiley & Sons |isbn=978-1-118-85397-9 |chapter=Chapter 10. Pre-Columbian Mathematics: The Olmec, Maya, and Inca Civilizations |via=Google Books |chapter-url=https://books.google.com/books?id=3CPwAgAAQBAJ&pg=PT73}}</ref> The use of a numeral&nbsp;0 in modern times originated with the Indian mathematician [[Brahmagupta]] in 628&nbsp;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&nbsp;CE, without being denoted by a numeral. Standard [[Roman numerals]] do not have a symbol for&nbsp;0; instead, ''nulla'' (or the genitive form ''nullae'') from {{Lang|la|nullus}}, the Latin word for "none", was employed to denote a&nbsp;0 value.<ref>{{cite web |first=Michael |last=Deckers |title=Cyclus Decemnovennalis Dionysii – Nineteen year cycle of Dionysius |url=http://hbar.phys.msu.ru/gorm/chrono/paschata.htm |publisher=Hbar.phys.msu.ru |date=25 August 2003 |access-date=13 February 2012 |archive-url=https://web.archive.org/web/20190115083618/http://hbar.phys.msu.ru/gorm/chrono/paschata.htm |archive-date=15 January 2019 |url-status=live }}</ref>

The first systematic study of numbers as [[abstraction]]s is usually credited to the [[ancient Greece|Greek]] philosophers [[Pythagoras]] and [[Archimedes]]. Some Greek mathematicians treated the number&nbsp;1 differently than larger numbers, sometimes even not as a number at all.{{efn|This convention is used, for example, in [[Euclid's Elements]], see D.&nbsp;Joyce's web edition of Book VII.<ref name=EuclidVIIJoyce>{{cite book |author=Euclid |author-link=Euclid |editor-first=D. |editor-last=Joyce|editor-link=David E. Joyce (mathematician) |chapter=Book VII, definitions 1 and 2 |title=[[Euclid's Elements|Elements]] |publisher=Clark University |chapter-url=http://aleph0.clarku.edu/~djoyce/java/elements/bookVII/defVII1.html }}</ref>}} [[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).<ref name="Mueller 2006 p. 58">{{cite book |last=Mueller |first=Ian |year=2006 |title=Philosophy of mathematics and deductive structure in [[Euclid's Elements]] |page=58 |publisher=Dover Publications |location=Mineola, New York |isbn=978-0-486-45300-2 |oclc=69792712}}</ref> However, in the definition of [[perfect number]] which comes shortly afterward, Euclid treats 1 as a number like any other.<ref>{{cite book |author=Euclid |author-link=Euclid |editor-first=D. |editor-last=Joyce |chapter=Book VII, definition 22 |title=[[Euclid's Elements|Elements]] |publisher=Clark University |chapter-url=http://aleph0.clarku.edu/~djoyce/java/elements/bookVII/defVII22.html |quote=A perfect number is that which is equal to the sum of its own parts. }} In definition VII.3 a "part" was defined as a number, but here 1 is considered to be a part, so that for example {{math|1=6 = 1 + 2 + 3}} is a perfect number.</ref>

Independent studies on numbers also occurred at around the same time in [[India]], China, and [[Mesoamerica]].<ref>{{cite book |first=Morris |last=Kline |year=1990 |orig-year=1972 |title=Mathematical Thought from Ancient to Modern Times |publisher=Oxford University Press |isbn=0-19-506135-7}}</ref>

===Emergence as a term===
[[Nicolas Chuquet]] used the term ''progression naturelle'' (natural progression) in 1484.<ref>{{cite book |last1=Chuquet |first1=Nicolas|author-link=Nicolas Chuquet |title=Le Triparty en la science des nombres|date=1881 |orig-date=1484 |url=https://gallica.bnf.fr/ark:/12148/bpt6k62599266/f75.image |language=fr}}</ref> The earliest known use of "natural number" as a complete English phrase is in 1763.<ref>{{cite book |last1=Emerson |first1=William |title=The method of increments|date=1763 |page=113 |url=https://archive.org/details/bim_eighteenth-century_the-method-of-increments_emerson-william_1763/page/112/mode/2up}}</ref><ref name="MacTutor"/> The 1771 Encyclopaedia Britannica defines natural numbers in the logarithm article.<ref name="MacTutor">{{cite web |title=Earliest Known Uses of Some of the Words of Mathematics (N) |url=https://mathshistory.st-andrews.ac.uk/Miller/mathword/n/ |website=Maths History |language=en}}</ref>

Starting at 0 or 1 has long been a matter of definition. In 1727, [[Bernard Le Bovier de Fontenelle]] wrote that his notions of distance and element led to defining the natural numbers as including or excluding 0.<ref>{{cite book |last1=Fontenelle |first1=Bernard de |title=Eléments de la géométrie de l'infini |date=1727 |page=3 |url=https://gallica.bnf.fr/ark:/12148/bpt6k64762n/f31.item |language=fr}}</ref> In 1889, [[Giuseppe Peano]] used N for the positive integers and started at 1,<ref>{{cite book |title=Arithmetices principia: nova methodo |date=1889 |publisher=Fratres Bocca |url=https://archive.org/details/arithmeticespri00peangoog/page/n12/mode/2up|page=12 |language=Latin}}</ref> but he later changed to using N<sub>0</sub> and N<sub>1</sub>.<ref name="Peano1901">{{cite book |last1=Peano |first1=Giuseppe |title=Formulaire des mathematiques |date=1901 |publisher=Paris, Gauthier-Villars |page=39 |url=https://archive.org/details/formulairedesmat00pean/page/38/mode/2up|language=fr}}</ref> Historically, most definitions have excluded 0,<ref name="MacTutor"/><ref>{{cite book |last1=Fine |first1=Henry Burchard |title=A College Algebra |date=1904 |publisher=Ginn |page=6 |url=https://books.google.com/books?id=RR4PAAAAIAAJ&dq=%22natural%20number%22&pg=PA6 |language=en}}</ref><ref>{{cite book |title=Advanced Algebra: A Study Guide to be Used with USAFI Course MC 166 Or CC166 |date=1958 |publisher=United States Armed Forces Institute |page=12 |url=https://books.google.com/books?id=184i06Py1ZYC&dq=%22natural%20number%22%201&pg=PA12 |language=en}}</ref> but many mathematicians such as [[George A. Wentworth]], [[Bertrand Russell]], [[Nicolas Bourbaki]], [[Paul Halmos]], [[Stephen Cole Kleene]], and [[John Horton Conway]] have preferred to include 0.<ref>{{cite web |title=Natural Number |url=https://archive.lib.msu.edu/crcmath/math/math/n/n035.htm |website=archive.lib.msu.edu}}</ref><ref name="MacTutor"/> This approach gained wider adoption in the 1960s<ref name="MacTutor"/> and was formalized in [[ISO 31-11]] (1978), which defines natural numbers to include zero, a convention retained in the current [[ISO/IEC 80000|ISO 80000-2]] standard.<ref name="ISO80000">{{cite book |url=https://www.iso.org/standard/64973.html |title=ISO 80000-2:2019 Quantities and units Part 2: Mathematics |date=24 June 2025 |publisher=[[International Organization for Standardization]] |chapter=Standard number sets and intervals |chapter-url=https://cdn.standards.iteh.ai/samples/64973/329519100abd447ea0d49747258d1094/ISO-80000-2-2019.pdf#page=10}}</ref>

===Formal construction===

In 19th century Europe, there was mathematical and philosophical discussion about the exact nature of the natural numbers. [[Henri Poincaré]] stated that axioms can only be demonstrated in their finite application, and concluded that it is "the power of the mind" which allows conceiving of the indefinite repetition of the same act.<ref>{{cite book |last1=Poincaré |first1=Henri|translator1-first=William John|translator1-last= Greenstreet |title=La Science et l'hypothèse|trans-title=Science and Hypothesis|orig-date=1902|date=1905|chapter=On the nature of mathematical reasoning|chapter-url=https://en.wikisource.org/wiki/Science_and_Hypothesis/Chapter_1|at=VI}}</ref> [[Leopold Kronecker]] summarized his belief as "God made the integers, all else is the work of man".{{efn|The English translation is from Gray. In a footnote, Gray attributes the German quote to: "Weber 1891–1892, 19, quoting from a lecture of Kronecker's of 1886."<ref>{{cite book
 |last=Gray |first=Jeremy |author-link=Jeremy Gray (mathematician)
 |year=2008
 |title=Plato's Ghost: The modernist transformation of mathematics
 |page=153
 |publisher=Princeton University Press
 |isbn=978-1-4008-2904-0
 |via=Google Books
 |url=https://books.google.com/books?id=ldzseiuZbsIC&q=%22God+made+the+integers%2C+all+else+is+the+work+of+man.%22
 |url-status=live
 |archive-url=https://web.archive.org/web/20170329150904/https://books.google.com/books?id=ldzseiuZbsIC&q=%22God+made+the+integers%2C+all+else+is+the+work+of+man.%22#v=snippet&q=%22God%20made%20the%20integers%2C%20all%20else%20is%20the%20work%20of%20man.%22&f=false
 |archive-date=29 March 2017
}}</ref><ref>{{cite book
 |last=Weber |first=Heinrich L.
 |year=1891–1892
 |chapter=Kronecker
 |chapter-url=http://www.digizeitschriften.de/dms/img/?PPN=PPN37721857X_0002&DMDID=dmdlog6 
 |archive-url=https://web.archive.org/web/20180809110042/http://www.digizeitschriften.de/dms/img/?PPN=PPN37721857X_0002&DMDID=dmdlog6
 |archive-date=9 August 2018
 |title=''Jahresbericht der Deutschen Mathematiker-Vereinigung''
 |trans-title=Annual report of the German Mathematicians Association
 |pages=2:5–23. (The quote is on p.&nbsp;19)
 |postscript=;
}} {{cite web
 |title=access to ''Jahresbericht der Deutschen Mathematiker-Vereinigung''
 |url=http://www.digizeitschriften.de/dms/toc/?PPN=PPN37721857X_0002
 |archive-url=https://web.archive.org/web/20170820201100/http://www.digizeitschriften.de/dms/toc/?PPN=PPN37721857X_0002
 |archive-date=20 August 2017 
}}</ref>}}

The [[Constructivism (mathematics)|constructivists]] saw a need to improve upon the logical rigor in the [[foundations of mathematics]].{{efn|"Much of the mathematical work of the twentieth century has been devoted to examining the logical foundations and structure of the subject." {{harv|Eves|1990|p=606}} }} 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 emerged, using set theory and Peano's axioms respectively. Later still, they were shown to be equivalent in most practical applications.

[[Set-theoretical definitions of natural numbers]] were initiated by [[Gottlob Frege|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.<ref>{{harvnb|Eves|1990|loc=Chapter 15}}</ref>

In 1881, [[Charles Sanders Peirce]] provided the first [[Axiomatic system#Axiomatization|axiomatization]] of natural-number arithmetic.<ref>{{cite journal|last= Peirce|first= C.&nbsp;S.|author-link= Charles Sanders Peirce|year= 1881|title= On the Logic of Number|url= https://archive.org/details/jstor-2369151|journal= American Journal of Mathematics|volume= 4|issue= 1|pages= 85–95|doi= 10.2307/2369151|mr= 1507856|jstor= 2369151}}</ref><ref>{{cite book|last= Shields|first= Paul|year= 1997|title= Studies in the Logic of Charles Sanders Peirce|url= https://archive.org/details/studiesinlogicof00nath|url-access= registration|chapter= 3. Peirce's Axiomatization of Arithmetic|chapter-url= https://books.google.com/books?id=pWjOg-zbtMAC&pg=PA43|editor1-last= Houser
|editor1-first= Nathan|editor2-last= Roberts|editor2-first= Don D.|editor3-last= Van Evra|editor3-first= James|publisher= Indiana University Press|isbn= 0-253-33020-3|pages= 43–52}}</ref> In 1888, [[Richard Dedekind]] proposed another axiomatization of natural-number arithmetic,<ref>{{cite book |title=Was sind und was sollen die Zahlen? |date=1893 |publisher=F. Vieweg |url=https://archive.org/details/wassindundwasso00dedegoog/page/n42/mode/2up |language=German|at=71–73}}</ref> and in&nbsp;1889, Peano published a simplified version of Dedekind's axioms in his book ''The principles of arithmetic presented by a new method'' ({{langx|la|[[Arithmetices principia, nova methodo exposita]]}}). This approach is now called [[Peano arithmetic]]. It is based on an [[axiomatization]] of the properties of [[ordinal number]]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 [[ZFC]] with the [[axiom of infinity]] replaced by its negation.<ref>{{cite journal
 | last1 = Baratella | first1 = Stefano
 | last2 = Ferro | first2 = Ruggero
 | doi = 10.1002/malq.19930390138
 | issue = 3
 | journal = Mathematical Logic Quarterly
 | mr = 1270381
 | pages = 338–352
 | title = A theory of sets with the negation of the axiom of infinity
 | volume = 39
 | year = 1993}}</ref> Theorems that can be proved in ZFC but cannot be proved using the Peano Axioms include [[Goodstein's theorem]].<ref>{{cite journal | last1=Kirby | first1=Laurie | last2=Paris | first2=Jeff | title=Accessible Independence Results for Peano Arithmetic | journal=Bulletin of the London Mathematical Society | publisher=Wiley | volume=14 | issue=4 | year=1982 | issn=0024-6093 | doi=10.1112/blms/14.4.285 | pages=285–293}}</ref>

==Formal definitions==

Formal definitions of the natural numbers take the existing, intuitive notion of natural numbers and the rules of arithmetic and define them both in the more fundamental terms of mathematical logic. The two standard methods for doing this are: the [[Peano axioms]]; and [[set theory]].

The Peano axioms (named for [[Giuseppe Peano]]) do not explicitly define what the natural numbers ''are'', but instead comprise a list of statements or [[Axiom|axioms]] that must be true of natural numbers, however they are defined. In contrast, set theory defines each natural number as a particular [[Set (mathematics)|set]], in which a set can be generally understood as a collection of distinct objects or [[Element of a set|elements]]. While the two approaches are different, they are consistent in that the natural number sets collectively ''satisfy'' the Peano axioms.

===Peano axioms===
{{Main|Peano axioms}}

The five Peano axioms are the following:<ref>{{cite encyclopedia
 |editor-first=G.E. |editor-last=Mints
 |title=Peano axioms
 |encyclopedia=Encyclopedia of Mathematics
 |publisher=[[Springer Science+Business Media|Springer]], in cooperation with the [[European Mathematical Society]]
 |url=http://www.encyclopediaofmath.org/index.php/Peano_axioms
 |url-status=live |access-date=8 October 2014
|archive-url=https://web.archive.org/web/20141013163028/http://www.encyclopediaofmath.org/index.php/Peano_axioms
 |archive-date=13 October 2014
}}</ref>{{efn|{{harvtxt|Hamilton|1988|pages=117&nbsp;ff}} calls them "Peano's Postulates" and begins with "1.{{spaces|2}}0 is a natural number."<br/>
{{harvtxt|Halmos|1974|page=46}} uses the language of set theory instead of the language of arithmetic for his five axioms. He begins with "(I){{spaces|2}}{{math|0 ∈ ω}} (where, of course, {{math|0 {{=}} ∅}}" ({{math|ω}} is the set of all natural numbers).<br/>
{{harvtxt|Morash|1991}} gives "a two-part axiom" in which the natural numbers begin with 1. (Section 10.1: ''An Axiomatization for the System of Positive Integers'')
}}

# 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 <math> x </math> equals the successor of <math> y </math>, then <math> x</math> equals <math> y</math>.
# 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 <math> x</math> is <math> x + 1</math>.

===Set-theoretic definition===
{{Main|Set-theoretic definition of natural numbers|von Neumann ordinal}}

In set theory each natural number {{mvar|n}} is defined as an explicitly defined set, whose elements allow counting the elements of other sets. A variety of different constructions have been proposed, however the standard solution (due to [[John von Neumann]])<ref name="vonNeumann1923pp199-208">{{Harvp|von&nbsp;Neumann|1923}}</ref> is to define each natural number {{mvar|n}} as a set containing {{mvar|n}} elements in the following way:
* Call {{math|0 {{=}} {{mset| }}}}, the [[empty set]].
* Define the ''successor'' {{math|''S''(''a'')}} of any set {{mvar|a}} by {{math|''S''(''a'') {{=}} ''a'' ∪ {{mset|''a''}}}}.
* By the [[axiom of infinity]], there exist sets which contain 0 and are [[closure (mathematics)|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''.

This produces an iterative definition of the natural numbers satisfying the Peano axioms, sometimes called [[von Neumann ordinals]]:

* {{math|0 {{=}} {{mset| }}}}
* {{math|1 {{=}} 0 ∪ {{mset|0}} {{=}} {{mset|0}} {{=}} {{mset|{{mset| }}}}}}
* {{math|2 {{=}} 1 ∪ {{mset|1}} {{=}} {{mset|0, 1}} {{=}} {{mset|{{mset| }}, {{mset|{{mset| }}}}}}}}
* {{math|3 {{=}} 2 ∪ {{mset|2}} {{=}} {{mset|0, 1, 2}}  {{=}} {{mset|{{mset| }}, {{mset|{{mset| }}}}, {{mset|{{mset| }}, {{mset|{{mset| }}}}}}}}}}
* {{math|''n'' {{=}} ''n''−1 ∪ {{mset|''n''−1}} {{=}} {{mset|0, 1, ..., ''n''−1}} {{=}} {{mset|{{mset| }}, {{mset|{{mset| }}}}, ..., {{mset|{{mset| }}, {{mset|{{mset| }}}}, ...}}}}}}

In this definition each natural number is equal to the set of all natural numbers less than it. Given a natural number {{math|''n''}}, the sentence "a set {{mvar|S}} has {{mvar|n}} elements" can be formally defined as "there exists a [[bijection]] from {{mvar|n}} to {{mvar|S}}." This formalizes the operation of ''counting'' the elements of {{mvar|S}}. Also, {{math|''n'' ≤ ''m''}} if and only if {{math|''n''}} is a [[subset]] of {{math|''m''}}. In other words, the [[set inclusion]] defines the usual [[total order]] on the natural numbers. This order is a [[well-order]].

Another construction sometimes called '''{{vanchor|Zermelo ordinals}}'''<ref name="Levy">{{harvp|Levy|1979|page=52}}</ref> defines {{math|0 {{=}} {{mset| }}}} and {{math|''S''(''a'') {{=}} {{mset|''a''}}}} and is now largely only of historical interest.

==Properties==
This section uses the convention <math>\mathbb{N}=\mathbb{N}_0=\mathbb{N}^*\cup\{0\}</math>.

===Addition===
Given the set <math>\mathbb{N}</math> of natural numbers and the [[successor function]] <math>S \colon \mathbb{N} \to \mathbb{N}</math> sending each natural number to the next one, one can define [[Addition in N|addition]] of natural numbers recursively by setting {{math|''a'' + 0 {{=}} ''a''}} and {{math|''a'' + ''S''(''b'') {{=}} ''S''(''a'' + ''b'')}} for all {{math|''a''}}, {{math|''b''}}. Thus, {{math|''a'' + 1 {{=}} ''a'' + S(0) {{=}} S(''a''+0) {{=}} S(''a'')}}, {{math|''a'' + 2 {{=}} ''a'' + S(1) {{=}} S(''a''+1) {{=}} S(S(''a''))}}, and so on. The [[algebraic structure]] <math>(\mathbb{N}, +)</math> is a [[commutative]] [[monoid]] with [[identity element]]&nbsp;0. It is a [[free object|free monoid]] on one generator. This commutative monoid satisfies the [[cancellation property]], so it can be embedded in a [[group (mathematics)|group]]. The smallest group containing the natural numbers is the [[integer]]s.

If 1 is defined as {{math|''S''(0)}}, then {{math|''b'' + 1 {{=}} ''b'' + ''S''(0) {{=}} ''S''(''b'' + 0) {{=}} ''S''(''b'')}}. That is, {{math|''b'' + 1}} is simply the successor of {{math|''b''}}.

===Multiplication===
Analogously, given that addition has been defined, a [[multiplication]] operator <math>\times</math> can be defined via {{math|''a'' × 0 {{=}} 0}} and {{math|''a'' × S(''b'') {{=}} (''a'' × ''b'') + ''a''}}. This turns <math>(\mathbb{N}^*, \times)</math> into a [[free commutative monoid]] with identity element&nbsp;1; a generator set for this monoid is the set of [[prime number]]s.

===Relationship between addition and multiplication===
Addition and multiplication are compatible, which is expressed in the [[distributivity|distribution law]]: {{math|''a'' × (''b'' + ''c'') {{=}} (''a'' × ''b'') + (''a'' × ''c'')}}. 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 <math>\mathbb{N}</math> is not [[closure (mathematics)|closed]] under subtraction (that is, subtracting one natural from another does not always result in another natural), means that <math>\mathbb{N}</math> is ''not'' a [[ring (mathematics)|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 {{math|''a'' + 1 {{=}} ''S''(''a'')}} and {{math|''a'' × 1 {{=}} ''a''}}. Furthermore, <math>(\mathbb{N^*}, +)</math> has no identity element.

===Order===
In this section, juxtaposed variables such as {{math|''ab''}} indicate the product {{math|''a'' × ''b''}},<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Multiplication |url=https://mathworld.wolfram.com/Multiplication.html |access-date=27 July 2020 |website=mathworld.wolfram.com |language=en}}</ref> and the standard [[order of operations]] is assumed.

A [[total order]] on the natural numbers is defined by letting {{math|''a'' ≤ ''b''}} if and only if there exists another natural number {{math|''c''}} where {{math|''a'' + ''c'' {{=}} ''b''}}. This order is compatible with the [[arithmetical operations]] in the following sense: if {{math|''a''}}, {{math|''b''}} and {{math|''c''}} are natural numbers and {{math|''a'' ≤ ''b''}}, then {{math|''a'' + ''c'' ≤ ''b'' + ''c''}} and {{math|''ac'' ≤ ''bc''}}.

An important property of the natural numbers is that they are [[well-order]]ed: every non-empty set of natural numbers has a least element. The rank among well-ordered sets is expressed by an [[ordinal number]]; for the natural numbers, this is denoted as {{math|[[omega (ordinal)|''ω'']]}} (omega).

===Division===
In this section, juxtaposed variables such as {{math|''ab''}} indicate the product {{math|''a'' × ''b''}}, and the standard [[order of operations]] 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 {{math|''a''}} and {{math|''b''}} with {{math|''b'' ≠ 0}} there are natural numbers {{math|''q''}} and {{math|''r''}} such that
:<math>a = bq + r \text{ and } r < b. </math>

The number {{math|''q''}} is called the ''[[quotient]]'' and {{math|''r''}} is called the ''[[remainder]]'' of the division of {{math|''a''}} by&nbsp;{{math|''b''}}. The numbers {{math|''q''}} and {{math|''r''}} are uniquely determined by {{math|''a''}} and&nbsp;{{math|''b''}}. This Euclidean division is key to the several other properties ([[divisibility]]), algorithms (such as the [[Euclidean algorithm]]), 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 (mathematics)|Closure]] under addition and multiplication: for all natural numbers {{math|''a''}} and {{math|''b''}}, both {{math|''a'' + ''b''}} and {{math|''a'' × ''b''}} are natural numbers.<ref>{{cite book |last1=Fletcher |first1=Harold |last2=Howell |first2=Arnold A. |date=9 May 2014 |title=Mathematics with Understanding |publisher=Elsevier |isbn=978-1-4832-8079-0 |page=116 |language=en |url=https://books.google.com/books?id=7cPSBQAAQBAJ&q=Natural+numbers+closed&pg=PA116 |quote=...the set of natural numbers is closed under addition... set of natural numbers is closed under multiplication}}</ref>
* [[Associativity]]: for all natural numbers {{math|''a''}}, {{math|''b''}}, and {{math|''c''}}, {{math|''a'' + (''b'' + ''c'') {{=}} (''a'' + ''b'') + ''c''}} and {{math|''a'' × (''b'' × ''c'') {{=}} (''a'' × ''b'') × ''c''}}.<ref>{{cite book |last=Davisson |first=Schuyler Colfax |title=College Algebra |date=1910 |publisher=Macmillian Company |page=2 |language=en |url=https://books.google.com/books?id=E7oZAAAAYAAJ&q=Natural+numbers+associative&pg=PA2 |quote=Addition of natural numbers is associative.}}</ref>
* [[Commutativity]]: for all natural numbers {{math|''a''}} and {{math|''b''}}, {{math|''a'' + ''b'' {{=}} ''b'' + ''a''}} and {{math|''a'' × ''b'' {{=}} ''b'' × ''a''}}.<ref>{{cite book |last1=Brandon |first1=Bertha (M.) |last2=Brown |first2=Kenneth E. |last3=Gundlach |first3=Bernard H. |last4=Cooke |first4=Ralph J. |date=1962 |title=Laidlaw mathematics series |publisher=Laidlaw Bros. |volume=8 |page=25 |language=en |url=https://books.google.com/books?id=xERMAQAAIAAJ&q=Natural+numbers+commutative}}</ref>
* Existence of [[identity element]]s: for every natural number {{Math|''a''}}, {{math|''a'' + 0 {{=}} ''a''}} and {{math|''a'' × 1 {{=}} ''a''}}. 
** If the natural numbers are taken as "excluding 0", and "starting at 1", then for every natural number {{Math|''a''}}, {{math|''a'' × 1 {{=}} ''a''}}. However, the "existence of additive identity element" property is not satisfied
* [[Distributivity]] of multiplication over addition for all natural numbers {{math|''a''}}, {{math|''b''}}, and {{math|''c''}}, {{math|''a'' × (''b'' + ''c'') {{=}} (''a'' × ''b'') + (''a'' × ''c'')}}.
* No nonzero [[zero divisor]]s: if {{math|''a''}} and {{math|''b''}} are natural numbers such that {{math|''a'' × ''b'' {{=}} 0}}, then {{math|''a'' {{=}} 0}} or {{math|''b'' {{=}} 0}} (or both).

==Generalizations==
Natural numbers are broadly used in two ways: to quantify and to order. A number used to represent the quantity of objects in a collection ("there are 6 coins on the table") is called a [[cardinal numeral]], while a number used to order individual objects within a collection ("she finished 6th in the race") is an [[ordinal numeral]]. 

These two uses of natural numbers apply only to [[finite set]]s. [[Georg Cantor]] discovered at the end of the 19th century that both uses of natural numbers can be generalized to [[infinite set]]s, but that they lead to two different concepts of "infinite" numbers, the [[cardinal number]]s and the [[ordinal number]]s.

Other generalizations of natural numbers are discussed in {{section link|Number#Extensions of the concept}}.

==See also==
{{Portal|Mathematics}}
* {{annotated link|Canonical representation of a positive integer}}
* {{annotated link|Countable set}}
* [[Sequence]]&nbsp;– Function of the natural numbers in another set
* {{annotated link|Ordinal number}}
* {{annotated link|Cardinal number}}
* {{annotated link|Set-theoretic definition of natural numbers}}
{{Classification of numbers}}

==Notes==
{{Notelist}}

==References==
{{Reflist|25em}}

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{{refend}}

==External links==
{{Commons category|Natural numbers}}
* {{springer|title=Natural number|id=p/n066090}}
* {{cite web
 |title=Axioms and construction of natural numbers
 |website=apronus.com
 |url=http://www.apronus.com/provenmath/naturalaxioms.htm
}}

{{Number systems}}
{{Classes of natural numbers}}
{{Authority control}}

[[Category:Cardinal numbers]]
[[Category:Elementary mathematics]]
[[Category:Integers]]
[[Category:Number theory]]
[[Category:Sets of real numbers]]