View source for Natural number

{{Short description|Number used for counting}}
[[File:Example of a natural number.png|thumb|right|Example of a natural number: 6. There are 6 apples in this picture and 6 is shown as an Arabic numeral.]]

In [[Mathematics|mathematics]], the '''natural numbers''' are the [[Number|number]]s 1, 2, 3, etc., possibly including 0 as well. Some definitions, including the standard ISO 80000-2,<ref name=ISO80000/> begin the natural numbers with {{math|0}}, corresponding to the '''non-negative integers''' {{math|1=0, 1, 2, 3, ...}}, whereas others start with {{math|1}}, corresponding to the '''positive integers''' {{math|1, 2, 3, ...}}<ref>{{cite dictionary
| url = http://www.merriam-webster.com/dictionary/natural%20number
| title = natural number
| dictionary = Merriam-Webster.com
| publisher = Merriam-Webster
| access-date = 4 October 2014
| archive-url = https://web.archive.org/web/20191213133201/https://www.merriam-webster.com/dictionary/natural%20number
| archive-date = 13 December 2019
| url-status = live
}}</ref>{{efn|{{harvtxt|Carothers|2000|page=3}} says, "<math>\N</math> is the set of natural numbers (positive integers)." Both definitions are acknowledged whenever convenient, and there is no general consensus on whether zero should be included in the natural numbers.<ref name=":1"/>}} 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).<ref>{{cite dictionary |first1=Jack G. |last1=Ganssle |first2=Michael |last2=Barr |name-list-style=amp |year=2003 |dictionary=Embedded Systems Dictionary |isbn=978-1-57820-120-4 |title=integer |pages=138 (integer), 247 (signed integer), & 276 (unsigned integer) |publisher=Taylor & Francis |via=Google Books |url=https://books.google.com/books?id=zePGx82d_fwC |access-date=28 March 2017 |url-status=live |archive-url=https://web.archive.org/web/20170329150719/https://books.google.com/books?id=zePGx82d_fwC |archive-date=29 March 2017}}</ref> In common language, particularly in primary school education, natural numbers may be called '''counting numbers'''<ref name=MathWorld_CountingNumber>{{MathWorld|title=Counting Number|id=CountingNumber}}</ref> to intuitively exclude the negative integers and zero, and also to contrast the [[Discrete mathematics|discreteness]] of [[Counting|counting]] to the [[List of continuity-related mathematical topics|continuity]] of [[Measurement|measurement]]—a hallmark characteristic of [[Real number|real number]]s.

The natural numbers can be used for counting<!-- Please, do not link this word that is used in its common language meaning, and not in any technical meaning --> (as in "there are ''six'' coins on the table"), in which case they serve as ''[[Cardinal number|cardinal number]]s''. They may also be used for ordering<!-- Please, do not link this word that is used in its common language meaning, and not in any technical meaning --> (as in "this is the ''third'' largest city in the country"), in which case they serve as ''[[Ordinal number|ordinal number]]s''. Natural numbers are sometimes used as labels{{mdash}}also known as ''[[Nominal number|nominal number]]s'', (e.g. jersey numbers in sports){{mdash}}which do not have the properties of numbers in a mathematical sense.<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><ref>{{cite web |title=Natural Numbers |website=Brilliant Math & Science Wiki |url=https://brilliant.org/wiki/natural-numbers/ |access-date=11 August 2020 |language=en-us}}</ref>

The natural numbers form a [[Set (mathematics)|set]], often symbolized as <math display="inline">\mathbb{N}</math>. Many other number sets are built by successively extending the set of natural numbers: the [[Integer|integer]]s, by including an [[Additive identity|additive identity]] 0 (if not yet in) and an [[Additive inverse|additive inverse]] {{math|−''n''}} for each nonzero natural number {{mvar|n}}; the [[Rational number|rational number]]s, by including a [[Multiplicative inverse|multiplicative inverse]] <math>1/n</math> for each nonzero integer {{mvar|n}} (and also the product of these inverses by integers); the [[Real number|real number]]s by including the [[Limit (mathematics)|limits]] of [[Construction of the real numbers#Construction from Cauchy sequences|Cauchy sequences]]{{efn|Any Cauchy sequence in the Reals converges,}} of rationals; the [[Complex number|complex number]]s, by adjoining to the real numbers a [[Imaginary unit|square root of {{math|−1}}]] (and also the sums and products thereof); and so on.{{efn|{{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."}}{{efn|{{harvtxt|Bluman|2010|page=1}}: "Numbers make up the foundation of mathematics."}} This chain of extensions canonically [[Embedding|embeds]] the natural numbers in the other number systems.

Properties of the natural numbers, such as divisibility and the distribution of [[Prime number|prime number]]s, are studied in [[Number theory|number theory]]. Problems concerning counting and ordering, such as [[Partition (number theory)|partitioning]] and [[Enumerative combinatorics|enumerations]], are studied in [[Combinatorics|combinatorics]].

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

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

[[File:Ishango bone (cropped).jpg|thumb|The [[Ishango bone|Ishango bone]] (on exhibition at the Royal Belgian Institute of Natural Sciences)<ref name=RBINS_intro>{{cite web |title=Introduction |series=[[Ishango bone|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|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 Egyptians developed a powerful system of numerals with distinct [[Social: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 [[History:Babylonia|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 civilizations used&nbsp;0 as a separate number as early as the {{nowrap|1st century BCE}}, but this usage did not spread beyond [[Place:Mesoamerica|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 [[Religion:Computus|computus]] (the calculation of the date of Easter), beginning with [[Biography:Dionysius Exiguus|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 [[Philosophy:Abstraction|abstraction]]s is usually credited to the {{wipe|Ancient Greece}} philosophers [[Biography:Pythagoras|Pythagoras]] and [[Biography:Archimedes|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 |editor-first=D. |editor-last=Joyce |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>}} [[Biography:Euclid|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|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 {{wipe|India}}, China, and [[Place:Mesoamerica|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>

===Modern definitions===
In 19th century Europe, there was mathematical and philosophical discussion about the exact nature of the natural numbers. [[Biography:Henri Poincaré|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> [[Biography:Leopold Kronecker|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
 |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 constructivists saw a need to improve upon the logical rigor in the [[Philosophy:Foundations of mathematics|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, [[Biography:Hermann Grassmann|Hermann Grassmann]] suggested a [[Recursive definition|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 [[Biography: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>

The second class of definitions was introduced by [[Biography:Charles Sanders Peirce|Charles Sanders Peirce]], refined by [[Biography:Richard Dedekind|Richard Dedekind]], and further explored by [[Biography:Giuseppe Peano|Giuseppe Peano]]; this approach is now called Peano arithmetic. It is based on an axiomatization of the properties of [[Ordinal number|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|set theory]]. One such system is ZFC with the [[Axiom of infinity|axiom of infinity]] replaced by its negation.{{Citation needed|date=October 2023}} 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>

With all these definitions, it is convenient to include&nbsp;0 (corresponding to the [[Empty set|empty set]]) as a natural number. Including&nbsp;0 is now the common convention among set theorists<ref>{{cite book |last1=Bagaria |first1=Joan |title=Set Theory |url=http://plato.stanford.edu/entries/set-theory/ |publisher=The Stanford Encyclopedia of Philosophy |edition=Winter 2014 |year=2017 |access-date=13 February 2015 |archive-url=https://web.archive.org/web/20150314173026/http://plato.stanford.edu/entries/set-theory/ |archive-date=14 March 2015 |url-status=live}}</ref> and logicians.<ref>{{cite book |last1=Goldrei |first1=Derek |title=Classic Set Theory: A guided independent study |url=https://archive.org/details/classicsettheory00gold |url-access=limited |date=1998 |publisher=Chapman & Hall/CRC |location=Boca Raton, Fla. [u.a.] |isbn=978-0-412-60610-6 |page=[https://archive.org/details/classicsettheory00gold/page/n39 33] |edition=1. ed., 1. print|chapter=3}}</ref> Other mathematicians also include&nbsp;0,{{efn|name=MacLaneBirkhoff1999p15|{{harvtxt|Mac Lane|Birkhoff|1999|page=15}} include zero in the natural numbers: 'Intuitively, the set <math>\N=\{0,1,2,\ldots\}</math> of all ''natural numbers'' may be described as follows: <math>\N</math> contains an "initial" number {{math|0}}; ...'. They follow that with their version of the Peano's axioms.}} and [[Computer language|computer language]]s often [[Zero-based numbering|start from zero]] when enumerating items like [[For loop|loop counters]] and [[String (computer science)|string-]] or [[Array data structure|array-elements]].<ref>{{cite journal |last1=Brown |first1=Jim |title=In defense of index origin 0 |journal=ACM SIGAPL APL Quote Quad |date=1978 |volume=9 |issue=2 |page=7 |doi=10.1145/586050.586053|s2cid=40187000 }}</ref><ref>{{cite web |last1=Hui |first1=Roger |title=Is index origin 0 a hindrance? |url=http://www.jsoftware.com/papers/indexorigin.htm |website=jsoftware.com |access-date=19 January 2015 |archive-url=https://web.archive.org/web/20151020195547/http://www.jsoftware.com/papers/indexorigin.htm |archive-date=20 October 2015 |url-status=live}}</ref> On the other hand, many mathematicians have kept the older tradition to take&nbsp;1 to be the first natural number.<ref>This is common in texts about [[Real analysis]]. See, for example, {{harvtxt|Carothers|2000|p=3}} or {{harvtxt|Thomson|Bruckner|Bruckner|2008|p=2}}.</ref>

==Notation==
The [[Set (mathematics)|set]] of all natural numbers is standardly denoted {{math|'''N'''}} or <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> 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>

Since natural numbers may contain {{math|0}} 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:<ref name="ISO80000"/><ref name="Grimaldi">{{cite book |last1=Grimaldi |first1=Ralph P. |title=Discrete and Combinatorial Mathematics: An applied introduction |publisher=Pearson Addison Wesley |isbn=978-0-201-72634-3 |edition=5th |year=2004}}</ref>
* Naturals without zero: <math>\{1,2,...\}=\mathbb{N}^*= \mathbb N^+=\mathbb{N}_0\smallsetminus\{0\} = \mathbb{N}_1</math>
* Naturals with zero: <math>\;\{0,1,2,...\}=\mathbb{N}_0=\mathbb N^0=\mathbb{N}^*\cup\{0\}</math>

Alternatively, since the natural numbers naturally form a [[Subset|subset]] of the [[Integer|integer]]s (often {{nowrap|denoted <math>\mathbb Z</math>),}} they may be referred to as the positive, or the non-negative integers, respectively.<ref>{{cite book |last1=Grimaldi |first1=Ralph P. |title=A review of discrete and combinatorial mathematics |date=2003 |publisher=Addison-Wesley |location=Boston |isbn=978-0-201-72634-3 |page=133 |edition=5th}}</ref> To be unambiguous about whether 0 is included or not, sometimes a superscript "<math>*</math>" or "+" is added in the former case, and a subscript (or superscript) "0" is added in the latter case:<ref name=ISO80000 >{{cite book |title=ISO 80000-2:2019 |chapter-url=https://cdn.standards.iteh.ai/samples/64973/329519100abd447ea0d49747258d1094/ISO-80000-2-2019.pdf#page=10 |publisher=[[Organization:International Organization for Standardization|International Organization for Standardization]]| chapter = Standard number sets and intervals | date=19 May 2020 |page=4|url=https://www.iso.org/standard/64973.html}}</ref>

:<math>\{1, 2, 3,\dots\} = \{x \in \mathbb Z : x > 0\}=\mathbb Z^+= \mathbb{Z}_{>0}</math>
:<math>\{0, 1, 2,\dots\} = \{x \in \mathbb Z : x \ge 0\}=\mathbb Z^{+}_{0}=\mathbb{Z}_ {\ge 0}</math>

==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|successor function]] <math>S \colon \mathbb{N} \to \mathbb{N}</math> sending each natural number to the next one, one can define 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|algebraic structure]] <math>(\mathbb{N}, +)</math> is a commutative [[Monoid|monoid]] with [[Identity element|identity element]]&nbsp;0. It is a [[Free object|free monoid]] on one generator. This commutative monoid satisfies the [[Cancellation property|cancellation property]], so it can be embedded in a [[Group (mathematics)|group]]. The smallest group containing the natural numbers is the [[Integer|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|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|prime number]]s.

===Relationship between addition and multiplication===
Addition and multiplication are compatible, which is expressed in the 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|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|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|order of operations]] is assumed.

A [[Total order|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|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|ordinal number]]; for the natural numbers, this is denoted as {{math|''ω''}} (omega).

===Division===
In this section, juxtaposed variables such as {{math|''ab''}} indicate the product {{math|''a'' × ''b''}}, and the standard [[Order of operations|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|quotient]]'' and {{math|''r''}} is called the ''[[Remainder|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|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|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).
** 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|cardinal number]]s and [[Ordinal number|ordinal number]]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 [[Equinumerosity|the same size]], exactly if there exists a [[Bijection|bijection]] between them. The set of natural numbers itself, and any bijective image of it, is said to be ''[[Countable set|countably infinite]]'' and to have [[Cardinality|cardinality]] [[Aleph number#Aleph-null|aleph-null]] ({{math|{{not a typo|ℵ}}<sub>0</sub>}}).
* 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|well-order]]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|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 {{math|ω}}; this is also the ordinal number of the set of natural numbers itself.

The least ordinal of cardinality {{math|{{not a typo|ℵ}}<sub>0</sub>}} (that is, the [[Von Neumann cardinal assignment|initial ordinal]] of {{math|{{not a typo|ℵ}}<sub>0</sub>}}) is {{math|ω}} but many well-ordered sets with cardinal number {{math|{{not a typo|ℵ}}<sub>0</sub>}} have an ordinal number greater than {{math|ω}}.

For [[Finite set|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|sequence]].

A countable [[Non-standard model of arithmetic|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.

[[Biography:Georges Reeb|Georges Reeb]] used to claim provocatively that "The naïve integers don't fill up" <math>\mathbb{N}</math>. Other generalizations are discussed in the article on numbers.

==Formal definitions==

There are two standard methods for formally defining natural numbers. The first one, named for [[Biography:Giuseppe Peano|Giuseppe Peano]], consists of an autonomous axiomatic theory called Peano arithmetic, based on few axioms called [[Peano axioms]].

The second definition is based on [[Set theory|set theory]]. It defines the natural numbers as specific [[Set (mathematics)|set]]s. More precisely, each natural number {{mvar|n}} is defined as an explicitly defined set, whose elements allow counting the elements of other sets, in the sense that the sentence "a set {{mvar|S}} has {{mvar|n}} elements" means that there exists a one to one correspondence between the two sets {{mvar|n}} and {{mvar|S}}.

The sets used to define natural numbers satisfy Peano axioms. It follows that every [[Theorem|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]].

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 be contradictory, and every theorem of set theory would be both true and wrong. 

===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=[[Company: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|1960|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}}

Intuitively, the natural number {{mvar|n}} is the common property of all [[Set (mathematics)|set]]s that have {{mvar|n}} elements. So, it seems natural to define {{mvar|n}} as an [[Equivalence class|equivalence class]] under the relation "can be made in one to one correspondence". Unfortunately, this does not work in [[Set theory|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 {{mvar|n}} elements that will be called the natural number {{mvar|n}}.

The following definition was first published by [[Biography:John von Neumann|John von Neumann]],<ref name="vonNeumann1923pp199-208">{{Harvp|von&nbsp;Neumann|1923}}</ref> although Levy attributes the idea to unpublished work of Zermelo in 1916.<ref name="Levy">{{harvp|Levy|1979|page=52}}</ref> As this definition extends to [[Infinite set|infinite set]] as a definition of [[Ordinal number|ordinal number]], the sets considered below are sometimes called von Neumann ordinals.

The definition proceeds as follows:
* Call {{math|0 {{=}} {{mset| }}}}, the [[Empty set|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|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''.

It follows that the natural numbers are defined iteratively as follows:
:*{{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| }}}}, ...}}}}}}, 
:* etc.

It can be checked that the natural numbers satisfies the [[Peano axioms]].

With this definition, 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|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|subset]] of {{math|''m''}}. In other words, the set inclusion defines the usual [[Total order|total order]] on the natural numbers. This order is a [[Well-order|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 for defining all [[Ordinal number|ordinal number]]s, including the infinite ones: "each ordinal is the well-ordered set of all smaller ordinals."

If one [[Finitism|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, [[Biography:Ernst Zermelo|Ernst Zermelo]] provided a construction that is nowadays only of historical interest, and is sometimes referred to as '''{{vanchor|Zermelo ordinals}}'''.<ref name="Levy"/> It consists in defining {{math|0}} as the empty set, and {{math|''S''(''a'') {{=}} {{mset|''a''}}}}.

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 {{mvar|n}}th element of a sequence) is immediate. Unlike von&nbsp;Neumann's construction, the Zermelo ordinals do not extend to infinite ordinals.

==See also==
* {{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}}

==Notes==
{{Notelist}}

==References==
{{Reflist}}

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{{Refend}}

==External links==
* {{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}}

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

{{Sourceattribution|Natural number}}