Natural number# Order

{{further|Prehistoric counting}}

File:Ishango bone (cropped).jpg (on exhibition at the Royal Belgian Institute of Natural Sciences){{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}}{{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}}{{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. is believed to have been used 20,000 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 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 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.{{cite book |first=Georges |last=Ifrah |year=2000 |title=The Universal History of Numbers |publisher=Wiley |isbn=0-471-37568-3}}

A much later advance was the development of the idea that {{num|0}} can be considered as a number, with its own numeral. The use of a 0 digit in place-value notation (within other numbers) dates back as early as 700 BCE by the Babylonians, who omitted such a digit when it would have been the last symbol in the number.{{efn| A tablet found at Kish ... thought to date from around 700 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.{{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}}}} The Olmec and Maya civilizations used 0 as a separate number as early as the {{nowrap|1st century BCE}}, but this usage did not spread beyond Mesoamerica.{{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}}{{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}} 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 do not have a symbol for 0; instead, nulla (or the genitive form nullae) from {{Lang|la|nullus}}, the Latin word for "none", was employed to denote a 0 value.{{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 }}

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.{{efn|This convention is used, for example, in Euclid's Elements, see D. Joyce's web edition of Book VII.{{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=Elements |publisher=Clark University |chapter-url=http://aleph0.clarku.edu/~djoyce/java/elements/bookVII/defVII1.html }}}} 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).{{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}} However, in the definition of perfect number which comes shortly afterward, Euclid treats 1 as a number like any other.{{cite book |author=Euclid |author-link=Euclid |editor-first=D. |editor-last=Joyce |chapter=Book VII, definition 22 |title=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.

Independent studies on numbers also occurred at around the same time in India, China, and Mesoamerica.{{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}}

Nicolas Chuquet used the term progression naturelle (natural progression) in 1484.{{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}} The earliest known use of "natural number" as a complete English phrase is in 1763.{{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}} The 1771 Encyclopaedia Britannica defines natural numbers in the logarithm article.{{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}}

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.{{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}} In 1889, Giuseppe Peano used N for the positive integers and started at 1,{{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}} but he later changed to using N₀ and N₁.{{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}} Historically, most definitions have excluded 0,{{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}}{{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}} 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.{{cite web |title=Natural Number |url=https://archive.lib.msu.edu/crcmath/math/math/n/n035.htm |website=archive.lib.msu.edu}}

Mathematicians have noted tendencies in which definition is used, such as algebra texts including 0,{{efn|name=MacLaneBirkhoff1999p15|{{harvtxt|Mac Lane|Birkhoff|1999|page=15}} include zero in the natural numbers: 'Intuitively, the set $\backslash N=\backslash \{0,1,2,\backslash ldots\backslash \}$ of all natural numbers may be described as follows: $\backslash N$ contains an "initial" number {{math|0}}; ...'. They follow that with their version of the Peano's axioms.}} number theory and analysis texts excluding 0,{{cite book |last1=Křížek |first1=Michal |last2=Somer |first2=Lawrence |last3=Šolcová |first3=Alena |title=From Great Discoveries in Number Theory to Applications |date=21 September 2021 |publisher=Springer Nature |isbn=978-3-030-83899-7 |page=6 |url=https://books.google.com/books?id=tklEEAAAQBAJ&dq=natural%20numbers%20zero&pg=PA6 |language=en}}See, for example, {{harvtxt|Carothers|2000|p=3}} or {{harvtxt|Thomson|Bruckner|Bruckner|2008|p=2}} logic and set theory texts including 0,{{cite book |last1=Gowers |first1=Timothy |title=The Princeton companion to mathematics |date=2008 |publisher=Princeton university press |location=Princeton |isbn=978-0-691-11880-2 |page=17}}{{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}}{{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}} dictionaries excluding 0,{{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}} school books (through high-school level) excluding 0, and upper-division college-level books including 0.{{cite book |last1=Enderton |first1=Herbert B. |title=Elements of set theory |date=1977 |publisher=Academic Press |location=New York |isbn=0122384407 |page=66}} There are exceptions to each of these tendencies and as of 2023 no formal survey has been conducted. Arguments raised include division by zero and the size of the empty set. Computer languages often start from zero when enumerating items like loop counters and string- or array-elements.{{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 }}{{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}} Including 0 began to rise in popularity in the 1960s. The ISO 31-11 standard included 0 in the natural numbers in its first edition in 1978 and this has continued through its present edition as ISO 80000-2.

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.{{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}} 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."{{cite book

|last=Gray |first=Jeremy |author-link=Jeremy Gray

|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

}}{{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. 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

}}}}

The 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 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.{{harvnb|Eves|1990|loc=Chapter 15}}

In 1881, Charles Sanders Peirce provided the first axiomatization of natural-number arithmetic.{{cite journal|last= Peirce|first= C. 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}}{{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}} In 1888, Richard Dedekind proposed another axiomatization of natural-number arithmetic,{{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}} and in 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 numbers: 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.{{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}} Theorems that can be proved in ZFC but cannot be proved using the Peano Axioms include Goodstein's theorem.{{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}}

The set of all natural numbers is standardly denoted {{math|N}} or $\backslash mathbb\; N.${{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}} Older texts have occasionally employed {{math|J}} as the symbol for this set.{{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}}

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:{{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}}

• Naturals without zero: $\backslash \{1,2,...\backslash \}=\backslash mathbb\{N\}^*=\; \backslash mathbb\; N^+=\backslash mathbb\{N\}\_0\backslash smallsetminus\backslash \{0\backslash \}\; =\; \backslash mathbb\{N\}\_1$
• Naturals with zero: $\backslash ;\backslash \{0,1,2,...\backslash \}=\backslash mathbb\{N\}\_0=\backslash mathbb\; N^0=\backslash mathbb\{N\}^*\backslash cup\backslash \{0\backslash \}$

Alternatively, since the natural numbers naturally form a subset of the integers (often {{nowrap|denoted $\backslash mathbb\; Z$),}} they may be referred to as the positive, or the non-negative integers, respectively.{{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}} To be unambiguous about whether 0 is included or not, sometimes a superscript "$*$" or "+" is added in the former case, and a subscript (or superscript) "0" is added in the latter case:{{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=International Organization for Standardization| chapter = Standard number sets and intervals | date=19 May 2020 |page=4|url=https://www.iso.org/standard/64973.html}}

:$\backslash \{1,\; 2,\; 3,\backslash dots\backslash \}\; =\; \backslash \{x\; \backslash in\; \backslash mathbb\; Z\; :\; x\; >\; 0\backslash \}=\backslash mathbb\; Z^+=\; \backslash mathbb\{Z\}\_\{>0\}$

:$\backslash \{0,\; 1,\; 2,\backslash dots\backslash \}\; =\; \backslash \{x\; \backslash in\; \backslash mathbb\; Z\; :\; x\; \backslash ge\; 0\backslash \}=\backslash mathbb\; Z^\{+\}\_\{0\}=\backslash mathbb\{Z\}\_\; \{\backslash ge\; 0\}$

This section uses the convention $\backslash mathbb\{N\}=\backslash mathbb\{N\}\_0=\backslash mathbb\{N\}^*\backslash cup\backslash \{0\backslash \}$.

Given the set $\backslash mathbb\{N\}$ of natural numbers and the successor function $S\; \backslash colon\; \backslash mathbb\{N\}\; \backslash to\; \backslash mathbb\{N\}$ 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 $(\backslash mathbb\{N\},\; +)$ is a commutative monoid with identity element 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 {{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}}.

Analogously, given that addition has been defined, a multiplication operator $\backslash times$ can be defined via {{math|a × 0 {{=}} 0}} and {{math|a × S(b) {{=}} (a × b) + a}}. This turns $(\backslash mathbb\{N\}^*,\; \backslash times)$ into a free commutative monoid with identity element 1; a generator set for this monoid is the set of prime numbers.

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. Semirings are an algebraic generalization of the natural numbers where multiplication is not necessarily commutative. The lack of additive inverses, which is equivalent to the fact that $\backslash mathbb\{N\}$ is not closed under subtraction (that is, subtracting one natural from another does not always result in another natural), means that $\backslash mathbb\{N\}$ 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 {{math|a + 1 {{=}} S(a)}} and {{math|a × 1 {{=}} a}}. Furthermore, $(\backslash mathbb\{N^*\},\; +)$ has no identity element.

In this section, juxtaposed variables such as {{math|ab}} indicate the product {{math|a × b}},{{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}} 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-ordered: 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|ω}} (omega).

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

:

The number {{math|q}} is called the quotient and {{math|r}} is called the remainder of the division of {{math|a}} by {{math|b}}. The numbers {{math|q}} and {{math|r}} are uniquely determined by {{math|a}} and {{math|b}}. This Euclidean division is key to the several other properties (divisibility), algorithms (such as the Euclidean algorithm), and ideas in number theory.

The addition (+) and multiplication (×) operations on natural numbers as defined above have several algebraic properties:

• 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.{{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}}
• 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}}.{{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.}}
• Commutativity: for all natural numbers {{math|a}} and {{math|b}}, {{math|a + b {{=}} b + a}} and {{math|a × b {{=}} b × a}}.{{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}}
• Existence of identity elements: 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 divisors: 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).

Two important generalizations of natural numbers arise from the two uses of counting and ordering: cardinal numbers and ordinal numbers.

• 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. The numbering of cardinals usually begins at zero, to accommodate the empty set $\backslash emptyset$. 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 ({{math|{{not a typo|ℵ}}₀}}).
• Natural numbers are also used as linguistic ordinal numbers: "first", "second", "third", and so forth. The numbering of ordinals usually begins at zero, to accommodate the order type of the empty set $\backslash emptyset$. 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 without limit points. 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 {{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}}₀}} (that is, the initial ordinal of {{math|{{not a typo|ℵ}}₀}}) is {{math|ω}} but many well-ordered sets with cardinal number {{math|{{not a typo|ℵ}}₀}} have an ordinal number greater than {{math|ω}}.

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

Georges Reeb used to claim provocatively that "The naïve integers don't fill up $\backslash mathbb\{N\}$".{{cite journal |title=Approaches To Analysis With Infinitesimals Following Robinson, Nelson, And Others |journal=Real Analysis Exchange |date=2017 |volume=42 |issue=2 |pages=193–253 |doi=10.14321/realanalexch.42.2.0193|doi-access=free|arxiv=1703.00425 |last1=Fletcher |first1=Peter |last2=Hrbacek |first2=Karel |last3=Kanovei |first3=Vladimir |last4=Katz |first4=Mikhail G. |last5=Lobry |first5=Claude |last6=Sanders |first6=Sam }}

There are two standard methods for formally defining natural numbers. The first one, named for 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. It defines the natural numbers as specific sets. 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 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.

{{Main|Peano axioms}}

The five Peano axioms are the following:{{cite encyclopedia

|editor-first=G.E. |editor-last=Mints

|title=Peano axioms

|encyclopedia=Encyclopedia of Mathematics

|publisher=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

}}{{efn|{{harvtxt|Hamilton|1988|pages=117 ff}} calls them "Peano's Postulates" and begins with "1.{{spaces|2}}0 is a natural number."

{{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).

{{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)

}}

1. 0 is a natural number.
2. Every natural number has a successor which is also a natural number.
3. 0 is not the successor of any natural number.
4. If the successor of $x$ equals the successor of $y$, then $x$ equals $y$.
5. 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 $x\; +\; 1$.

{{Main|Set-theoretic definition of natural numbers}}

Intuitively, the natural number {{mvar|n}} is the common property of all sets that have {{mvar|n}} elements. So, it seems natural to define {{mvar|n}} as an equivalence class under the relation "can be made in one to one correspondence". This does not work in all set theories, as such an equivalence class would not be a set{{efn|In some set theories, e.g., New Foundations, a universal set exists and Russel's paradox cannot be formulated.}} (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 John von Neumann,{{Harvp|von Neumann|1923}} although Levy attributes the idea to unpublished work of Zermelo in 1916.{{harvp|Levy|1979|page=52}} As this definition extends to infinite set as a definition of ordinal number, the sets considered below are sometimes called von Neumann ordinals.

The definition proceeds as follows:

• 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 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}}}} {{math|{{=}} {{mset|{{mset| }}, {{mset|{{mset| }}}}, {{mset|{{mset| }}, {{mset|{{mset| }}}}}}}}}},

:*{{math|n {{=}} n−1 ∪ {{mset|n−1}} {{=}} {{mset|0, 1, ..., n−1}}}} {{math|{{=}} {{mset|{{mset| }}, {{mset|{{mset| }}}}, ..., {{mset|{{mset| }}, {{mset|{{mset| }}}}, ...}}}}}},

:* etc.

It can be checked that the natural numbers satisfy 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 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.

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 numbers, 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 {{vanchor|Zermelo ordinals}}. It consists in defining {{math|0}} as the empty set, and {{math|S(a) {{=}} {{mset|a}}}}.

With this definition each nonzero 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 Neumann's construction, the Zermelo ordinals do not extend to infinite ordinals.

{{Portal|Mathematics}}

• {{annotated link|Canonical representation of a positive integer}}
• {{annotated link|Countable set}}
• Sequence – 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}}

{{Notelist}}

{{Reflist|25em}}

{{refbegin|25em|small=yes}}

• {{cite book

|last=Bluman |first=Allan

|year=2010 |edition=Second

|title=Pre-Algebra DeMYSTiFieD

|publisher=McGraw-Hill Professional

|isbn=978-0-07-174251-1

|url=https://books.google.com/books?id=oRLc9r4bmSgC

|via=Google Books

}}

• {{cite book

|last=Carothers |first=N.L.

|year=2000

|title=Real Analysis

|publisher=Cambridge University Press

|isbn=978-0-521-49756-5

|via=Google Books

|url=https://books.google.com/books?id=4VFDVy1NFiAC&q=%22natural+numbers%22

}}

• {{cite book

|last1=Clapham |first1=Christopher

|last2=Nicholson |first2=James

|year=2014 |edition=Fifth

|title=The Concise Oxford Dictionary of Mathematics

|publisher=Oxford University Press

|isbn=978-0-19-967959-1

|via=Google Books

|url=https://books.google.com/books?id=c69GBAAAQBAJ

}}

• {{cite book

|last=Dedekind |first=Richard |author-link=Richard Dedekind

|translator-last=Beman |translator-first=Wooster Woodruff

|orig-year=1901 |year=1963 |edition=reprint

|title=Essays on the Theory of Numbers

|publisher=Dover Books

|isbn=978-0-486-21010-0

|via=Archive.org

|url=https://archive.org/details/essaysontheoryof0000dede

}}

• {{cite book

|author-first=Richard |author-last=Dedekind |author-link=Richard Dedekind

|translator-first=Wooster Woodruff |translator-last=Beman

|year=1901

|title=Essays on the Theory of Numbers

|publisher=Open Court Publishing Company

|place=Chicago, IL

|via=Project Gutenberg

|url=https://www.gutenberg.org/ebooks/21016

|access-date=13 August 2020

}}

• {{cite book

|last=Dedekind |first=Richard |author-link=Richard Dedekind

|orig-year=1901 |year=2007

|title=Essays on the Theory of Numbers

|publisher=Kessinger Publishing, LLC

|isbn=978-0-548-08985-9

}}

• {{cite book

|last=Eves |first=Howard |author-link=Howard Eves

|year=1990 |edition=6th

|title=An Introduction to the History of Mathematics

|publisher=Thomson

|isbn=978-0-03-029558-4

|via=Google Books

|url=https://books.google.com/books?id=PXvwAAAAMAAJ

}}

• {{cite book

|last=Halmos |first=Paul |author-link=Paul Halmos

|year=1960

|title=Naive Set Theory

|publisher=Springer Science & Business Media

|isbn=978-0-387-90092-6

|via=Google Books

|url=https://books.google.com/books?id=x6cZBQ9qtgoC&q=peano+axioms&pg=PP1

}}

• {{cite book

|last=Hamilton |first=A.G.

|year=1988 |edition=Revised

|title=Logic for Mathematicians

|publisher=Cambridge University Press

|isbn=978-0-521-36865-0

|via=Google Books

|url=https://books.google.com/books?id=TO098EjWT38C&q=peano%27s+postulates

}}

• {{cite book

|last1=James |first1=Robert C. |author-link1=Robert C. James

|last2=James |first2=Glenn

|year=1992 |edition=Fifth

|title=Mathematics Dictionary

|publisher=Chapman & Hall

|isbn=978-0-412-99041-0

|via=Google Books

|url=https://books.google.com/books?id=UyIfgBIwLMQC

}}

• {{cite book

|last=Landau |first=Edmund |author-link=Edmund Landau

|year=1966 |edition=Third

|title=Foundations of Analysis

|publisher=Chelsea Publishing

|isbn=978-0-8218-2693-5

|via=Google Books

|url=https://books.google.com/books?id=DvIJBAAAQBAJ

}}

• {{cite book

|last=Levy |first=Azriel |author-link=Azriel Levy

|year=1979

|title=Basic Set Theory

|publisher=Springer-Verlag Berlin Heidelberg

|isbn= 978-3-662-02310-5

}}

• {{cite book

|last1=Mac Lane |first1=Saunders |author1-link=Saunders Mac Lane

|last2=Birkhoff |first2=Garrett |author2-link=Garrett Birkhoff

|year=1999 |edition=3rd

|title=Algebra

|publisher=American Mathematical Society

|isbn=978-0-8218-1646-2

|via=Google Books

|url=https://books.google.com/books?id=L6FENd8GHIUC&q=%22the+natural+numbers%22&pg=PA15

}}

• {{cite book

|last=Mendelson |first=Elliott |author-link=Elliott Mendelson

|orig-year=1973 |year=2008

|title=Number Systems and the Foundations of Analysis

|publisher=Dover Publications

|isbn=978-0-486-45792-5

|via=Google Books

|url=https://books.google.com/books?id=3domViIV7HMC

}}

• {{cite book

|last=Morash |first=Ronald P.

|year=1991 |edition=Second

|title=Bridge to Abstract Mathematics: Mathematical proof and structures

|publisher=Mcgraw-Hill College

|isbn=978-0-07-043043-3

|via=Google Books

|url=https://books.google.com/books?id=fH9YAAAAYAAJ

}}

• {{cite book

|last1=Musser |first1=Gary L.

|last2=Peterson |first2=Blake E.

|last3=Burger |first3=William F.

|year=2013 |edition=10th

|title=Mathematics for Elementary Teachers: A contemporary approach

|publisher=Wiley Global Education

|isbn=978-1-118-45744-3

|via=Google Books

|url=https://books.google.com/books?id=b3dbAgAAQBAJ

}}

• {{cite book

|last1=Szczepanski |first1=Amy F.

|last2=Kositsky |first2=Andrew P.

|year=2008

|title=The Complete Idiot's Guide to Pre-algebra

|publisher=Penguin Group

|isbn=978-1-59257-772-9

|via=Google Books

|url=https://books.google.com/books?id=wLA2tlR_LYYC

}}

• {{cite book

|last1=Thomson |first1=Brian S.

|last2=Bruckner |first2=Judith B.

|last3=Bruckner |first3=Andrew M.

|year=2008 |edition=Second

|title=Elementary Real Analysis

|publisher=ClassicalRealAnalysis.com

|isbn=978-1-4348-4367-8

|via=Google Books

|url=https://books.google.com/books?id=vA9d57GxCKgC

}}

• {{cite journal

|last=von Neumann |first=John |author-link=John von Neumann

|year=1923

|title=Zur Einführung der transfiniten Zahlen

|trans-title=On the Introduction of the Transfinite Numbers

|journal=Acta Litterarum AC Scientiarum Ragiae Universitatis Hungaricae Francisco-Josephinae, Sectio Scientiarum Mathematicarum

|volume=1 |pages=199–208

|url=http://acta.fyx.hu/acta/showCustomerArticle.action?id=4981&dataObjectType=article&returnAction=showCustomerVolume&sessionDataSetId=39716d660ae98d02&style=

|url-status=dead |access-date=15 September 2013

|archive-url=https://web.archive.org/web/20141218090535/http://acta.fyx.hu/acta/showCustomerArticle.action?id=4981&dataObjectType=article&returnAction=showCustomerVolume&sessionDataSetId=39716d660ae98d02&style=

|archive-date=18 December 2014

}}

• {{cite book

|author-last=von Neumann |author-first=John |author-link=John von Neumann

|editor-first=Jean |editor-last=van Heijenoort

|orig-year=1923 |date=January 2002 |edition=3rd

|title=From Frege to Gödel: A source book in mathematical logic, 1879–1931

|chapter=On the introduction of transfinite numbers |pages=346–354

|isbn=978-0-674-32449-7

|publisher=Harvard University Press

|chapter-url=http://www.hup.harvard.edu/catalog.php?isbn=978-0674324497

}} – English translation of {{Harvnb|von Neumann|1923}}.

{{refend}}

{{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