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Natural number
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{{Short description|Number used for counting}}{{About|“positive integers” and “non-negative integers“|all the numbers ..., â2, â1, 0, 1, 2, ...|Integer}}{{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, etc., possibly excluding 0.{{under discussion inline|Is 0 a natural number?}} Some define the natural numbers as the non-negative integers {{math|1=0, 1, 2, 3, ...}}, while others define them as the positive integers {{math|1, 2, 3, ...}}.{{efn|See {{section link|#Emergence as a term}}}} Some authors acknowledge both definitions whenever convenient. Some texts define the whole numbers as the natural numbers together with zero, excluding zero from the natural numbers, while in other writings, the whole numbers refer to all of the integers (including negative integers).DICTIONARY, Jack G., Ganssle, Michael, Barr, amp, 2003, Embedded Systems Dictionary, 978-1-57820-120-4, integer, 138 (integer), 247 (signed integer), & 276 (unsigned integer), Taylor & Francis, Google Books,books.google.com/books?id=zePGx82d_fwC, 28 March 2017, live,web.archive.org/web/20170329150719/https://books.google.com/books?id=zePGx82d_fwC, 29 March 2017, The counting numbers refer to the natural numbers in common language, particularly in primary school education, and are similarly ambiguous although typically exclude zero.{{MathWorld|title=Counting Number|id=CountingNumber}}The natural numbers can be used for counting (as in “there are six coins on the table“), in which case they serve as cardinal numbers. They may also be used for ordering (as in “this is the third largest city in the country“), in which case they serve as ordinal numbers. Natural numbers are sometimes used as labels{{mdash}}also known as nominal numbers, (e.g. jersey numbers in sports){{mdash}}which do not have the properties of numbers in a mathematical sense.WEB, Weisstein, Eric W., Natural Number,mathworld.wolfram.com/NaturalNumber.html, 11 August 2020, mathworld.wolfram.com, en, WEB, Natural Numbers, Brilliant Math & Science Wiki,brilliant.org/wiki/natural-numbers/, 11 August 2020, en-us, The natural numbers form a set, often symbolized as mathbb{N}. Many other number sets are built by successively extending the set of natural numbers: the integers, by including an additive identity 0 (if not yet in) and an additive inverse {{math|ân}} for each nonzero natural number {{mvar|n}}; the rational numbers, by including a multiplicative inverse 1/n for each nonzero integer {{mvar|n}} (and also the product of these inverses by integers); the real numbers by including the limits of Cauchy sequences{{efn|Any Cauchy sequence in the Reals converges,}} of rationals; the complex numbers, by adjoining to the real numbers a square root of {{math|â1}} (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 embeds the natural numbers in the other number systems.Properties of the natural numbers, such as divisibility and the distribution of prime numbers, are studied in number theory. Problems concerning counting and ordering, such as partitioning and enumerations, are studied in combinatorics.- the content below is remote from Wikipedia
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History
Ancient roots
{{further|Prehistoric counting}}File:Ishango bone (cropped).jpg|thumb|The Ishango bone (on exhibition at the Royal Belgian Institute of Natural Sciences)WEB, Introduction, Ishango bone, Royal Belgian Institute of Natural Sciences, Brussels, Belgium,www.naturalsciences.be/expo/old_ishango/en/ishango/introduction.html,web.archive.org/web/20160304051733/https://www.naturalsciences.be/expo/old_ishango/en/ishango/introduction.html, 4 March 2016, WEB, Flash presentation, Ishango bone, Royal Belgian Institute of Natural Sciences, Brussels, Belgium,ishango.naturalsciences.be/Flash/flash_local/Ishango-02-EN.html,ishango.naturalsciences.be/Flash/flash_local/Ishango-02-EN.html," title="web.archive.org/web/20160527164619ishango.naturalsciences.be/Flash/flash_local/Ishango-02-EN.html,">web.archive.org/web/20160527164619ishango.naturalsciences.be/Flash/flash_local/Ishango-02-EN.html, 27 May 2016, WEB, The Ishango Bone, Democratic Republic of the Congo, UNESCO’s Portal to the Heritage of Astronomy,www2.astronomicalheritage.net/index.php/show-entity?identity=4&idsubentity=1,www2.astronomicalheritage.net/index.php/show-entity?identity=4&idsubentity=1," title="web.archive.org/web/20141110195426www2.astronomicalheritage.net/index.php/show-entity?identity=4&idsubentity=1,">web.archive.org/web/20141110195426www2.astronomicalheritage.net/index.php/show-entity?identity=4&idsubentity=1, 10 November 2014, , on permanent display at the Royal Belgian Institute of Natural SciencesRoyal Belgian Institute of Natural SciencesThe 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 shortageâby 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 oneâits value being determined from context.BOOK, Georges, Ifrah, 2000, The Universal History of Numbers, Wiley, 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.WEB, A history of Zero, MacTutor History of Mathematics,www-history.mcs.st-and.ac.uk/history/HistTopics/Zero.html, live, 23 January 2013,www-history.mcs.st-and.ac.uk/history/HistTopics/Zero.html," title="web.archive.org/web/20130119083234www-history.mcs.st-and.ac.uk/history/HistTopics/Zero.html,">web.archive.org/web/20130119083234www-history.mcs.st-and.ac.uk/history/HistTopics/Zero.html, 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.BOOK, Charles C., Mann, 2005, 1491: New Revelations of the Americas before Columbus, 19, Knopf, 978-1-4000-4006-3,books.google.com/books?id=Jw2TE_UNHJYC&pg=PA19, live, Google Books, 3 February 2015,web.archive.org/web/20150514105855/https://books.google.com/books?id=Jw2TE_UNHJYC&pg=PA19, 14 May 2015, BOOK, Brian, Evans, 2014, The Development of Mathematics Throughout the Centuries: A brief history in a cultural context, John Wiley & Sons, 978-1-118-85397-9, Chapter 10. Pre-Columbian Mathematics: The Olmec, Maya, and Inca Civilizations, Google Books,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 , the Latin word for “none”, was employed to denote a 0 value.WEB, Michael, Deckers, Cyclus Decemnovennalis Dionysii â Nineteen year cycle of Dionysius,hbar.phys.msu.ru/gorm/chrono/paschata.htm, Hbar.phys.msu.ru, 25 August 2003, 13 February 2012,hbar.phys.msu.ru/gorm/chrono/paschata.htm," title="web.archive.org/web/20190115083618hbar.phys.msu.ru/gorm/chrono/paschata.htm,">web.archive.org/web/20190115083618hbar.phys.msu.ru/gorm/chrono/paschata.htm, 15 January 2019, 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.BOOK, Euclid, Euclid, D., Joyce, Book VII, definitions 1 and 2, Euclid’s Elements, Elements, Clark University,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).BOOK, Mueller, Ian, 2006, Philosophy of mathematics and deductive structure in Euclid’s Elements, 58, Dover Publications, Mineola, New York, 978-0-486-45300-2, 69792712, However, in the definition of perfect number which comes shortly afterward, Euclid treats 1 as a number like any other.BOOK, Euclid, Euclid, D., Joyce, Book VII, definition 22, Euclid’s Elements, Elements, Clark University,aleph0.clarku.edu/~djoyce/java/elements/bookVII/defVII22.html, 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.BOOK, Morris, Kline, 1990, 1972, Mathematical Thought from Ancient to Modern Times, Oxford University Press, 0-19-506135-7,Emergence as a term
Nicolas Chuquet used the term progression naturelle (natural progression) in 1484.BOOK, Chuquet, Nicolas, Nicolas Chuquet, Le Triparty en la science des nombres, 1881, 1484,gallica.bnf.fr/ark:/12148/bpt6k62599266/f75.image, fr, The earliest known use of “natural number” as a complete English phrase is in 1763.BOOK, Emerson, William, The method of increments, 1763, 113,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.WEB, Earliest Known Uses of Some of the Words of Mathematics (N),mathshistory.st-andrews.ac.uk/Miller/mathword/n/, Maths History, 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.BOOK, Fontenelle, Bernard de, Eléments de la géométrie de l’infini, 1727, 3,gallica.bnf.fr/ark:/12148/bpt6k64762n/f31.item, fr, In 1889, Giuseppe Peano used N for the positive integers and started at 1,BOOK, Arithmetices principia: nova methodo, 1889, Fratres Bocca,archive.org/details/arithmeticespri00peangoog/page/n12/mode/2up, 12, Latin, but he later changed to using N0 and N1.BOOK, Peano, Giuseppe, Formulaire des mathematiques, 1901, Paris, Gauthier-Villars, 39,archive.org/details/formulairedesmat00pean/page/38/mode/2up, fr, Historically, most definitions have excluded 0,BOOK, Fine, Henry Burchard, A College Algebra, 1904, Ginn, 6,www.google.com/books/edition/A_College_Algebra/RR4PAAAAIAAJ?hl=en&gbpv=1&pg=PA6&printsec=frontcover&dq=%22natural%20number%22, en, BOOK, Advanced Algebra: A Study Guide to be Used with USAFI Course MC 166 Or CC166, 1958, United States Armed Forces Institute, 12,www.google.com/books/edition/Advanced_Algebra/184i06Py1ZYC?hl=en&gbpv=1&dq=%22natural%20number%22%201&pg=PA12&printsec=frontcover, 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.WEB, Natural Number,archive.lib.msu.edu/crcmath/math/math/n/n035.htm, 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 N={0,1,2,ldots} of all natural numbers may be described as follows: 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,BOOK, KÅÞek, Michal, Somer, Lawrence, Å olcová, Alena, From Great Discoveries in Number Theory to Applications, 21 September 2021, Springer Nature, 978-3-030-83899-7, 6,www.google.com/books/edition/From_Great_Discoveries_in_Number_Theory/tklEEAAAQBAJ?hl=en&gbpv=1&dq=natural%20numbers%20zero&pg=PA6&printsec=frontcover, en, See, for example, {{harvtxt|Carothers|2000|p=3}} or {{harvtxt|Thomson|Bruckner|Bruckner|2008|p=2}} logic and set theory texts including 0,BOOK, Gowers, Timothy, The Princeton companion to mathematics, 2008, Princeton university press, Princeton, 978-0-691-11880-2, 17, BOOK, Bagaria, Joan, Set Theory,plato.stanford.edu/entries/set-theory/, The Stanford Encyclopedia of Philosophy, Winter 2014, 2017, 13 February 2015,plato.stanford.edu/entries/set-theory/," title="web.archive.org/web/20150314173026plato.stanford.edu/entries/set-theory/,">web.archive.org/web/20150314173026plato.stanford.edu/entries/set-theory/, 14 March 2015, live, BOOK, Goldrei, Derek, Classic Set Theory: A guided independent study,archive.org/details/classicsettheory00gold, limited, 1998, Chapman & Hall/CRC, Boca Raton, Fla. [u.a.], 978-0-412-60610-6, 33, 1. ed., 1. print, 3, dictionaries excluding 0,DICTIONARY,www.merriam-webster.com/dictionary/natural%20number, natural number, Merriam-Webster.com, Merriam-Webster, 4 October 2014,web.archive.org/web/20191213133201/https://www.merriam-webster.com/dictionary/natural%20number, 13 December 2019, live, school books (through high-school level) excluding 0, and upper-division college-level books including 0.BOOK, Enderton, Herbert B., Elements of set theory, 1977, Academic Press, New York, 0122384407, 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.JOURNAL, Brown, Jim, In defense of index origin 0, ACM SIGAPL APL Quote Quad, 1978, 9, 2, 7, 10.1145/586050.586053, 40187000, WEB, Hui, Roger, Is index origin 0 a hindrance?,www.jsoftware.com/papers/indexorigin.htm, jsoftware.com, 19 January 2015,www.jsoftware.com/papers/indexorigin.htm," title="web.archive.org/web/20151020195547www.jsoftware.com/papers/indexorigin.htm,">web.archive.org/web/20151020195547www.jsoftware.com/papers/indexorigin.htm, 20 October 2015, 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.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.BOOK, Poincaré, Henri, William John, Greenstreet, La Science et l’hypothèse, Science and Hypothesis, 1902, 1905, On the nature of mathematical reasoning,en.wikisource.org/wiki/Science_and_Hypothesis/Chapter_1, 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 1891â1892, 19, quoting from a lecture of Kronecker’s of 1886.“BOOK, Gray, Jeremy, Jeremy Gray
, 2008
, Plato’s Ghost: The modernist transformation of mathematics
, 153
, Princeton University Press
, 978-1-4008-2904-0
, Google Books
,books.google.com/books?id=ldzseiuZbsIC&q=%22God+made+the+integers%2C+all+else+is+the+work+of+man.%22
, live
,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
, 29 March 2017
, BOOK
, 2008
, Plato’s Ghost: The modernist transformation of mathematics
, 153
, Princeton University Press
, 978-1-4008-2904-0
, Google Books
,books.google.com/books?id=ldzseiuZbsIC&q=%22God+made+the+integers%2C+all+else+is+the+work+of+man.%22
, live
,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
, 29 March 2017
, Weber, Heinrich L.
, 1891â1892
, Kronecker
,www.digizeitschriften.de/dms/img/?PPN=PPN37721857X_0002&DMDID=dmdlog6
,www.digizeitschriften.de/dms/img/?PPN=PPN37721857X_0002&DMDID=dmdlog6" title="web.archive.org/web/20180809110042www.digizeitschriften.de/dms/img/?PPN=PPN37721857X_0002&DMDID=dmdlog6">web.archive.org/web/20180809110042www.digizeitschriften.de/dms/img/?PPN=PPN37721857X_0002&DMDID=dmdlog6
, 9 August 2018
, Jahresbericht der Deutschen Mathematiker-Vereinigung
, Annual report of the German Mathematicians Association
, 2:5â23. (The quote is on p. 19)
, ;
, WEB
, 1891â1892
, Kronecker
,www.digizeitschriften.de/dms/img/?PPN=PPN37721857X_0002&DMDID=dmdlog6
,www.digizeitschriften.de/dms/img/?PPN=PPN37721857X_0002&DMDID=dmdlog6" title="web.archive.org/web/20180809110042www.digizeitschriften.de/dms/img/?PPN=PPN37721857X_0002&DMDID=dmdlog6">web.archive.org/web/20180809110042www.digizeitschriften.de/dms/img/?PPN=PPN37721857X_0002&DMDID=dmdlog6
, 9 August 2018
, Jahresbericht der Deutschen Mathematiker-Vereinigung
, Annual report of the German Mathematicians Association
, 2:5â23. (The quote is on p. 19)
, ;
, access to Jahresbericht der Deutschen Mathematiker-Vereinigung
,www.digizeitschriften.de/dms/toc/?PPN=PPN37721857X_0002
,www.digizeitschriften.de/dms/toc/?PPN=PPN37721857X_0002" title="web.archive.org/web/20170820201100www.digizeitschriften.de/dms/toc/?PPN=PPN37721857X_0002">web.archive.org/web/20170820201100www.digizeitschriften.de/dms/toc/?PPN=PPN37721857X_0002
, 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 naturalâbut a consequence of definitions. Later, two classes of such formal definitions were constructed; later still, they were shown to be equivalent in most practical applications.Set-theoretical definitions of natural numbers were initiated by Frege. He initially defined a natural number as the class of all sets that are in one-to-one correspondence with a particular set. However, this definition turned out to lead to paradoxes, including Russell’s paradox. 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 within this second class of definitions.JOURNAL, Peirce, C. S., Charles Sanders Peirce, 1881, On the Logic of Number,archive.org/details/jstor-2369151, American Journal of Mathematics, 4, 1, 85â95, 10.2307/2369151, 1507856, 2369151, BOOK, Shields, Paul, 1997, Studies in the Logic of Charles Sanders Peirce,archive.org/details/studiesinlogicof00nath, registration, 3. Peirce’s Axiomatization of Arithmetic,books.google.com/books?id=pWjOg-zbtMAC&pg=PA43, Houser,www.digizeitschriften.de/dms/toc/?PPN=PPN37721857X_0002
,www.digizeitschriften.de/dms/toc/?PPN=PPN37721857X_0002" title="web.archive.org/web/20170820201100www.digizeitschriften.de/dms/toc/?PPN=PPN37721857X_0002">web.archive.org/web/20170820201100www.digizeitschriften.de/dms/toc/?PPN=PPN37721857X_0002
, 20 August 2017
editor2-last= Roberts | editor3-last= Van Evra | publisher= Indiana University Press | pages= 43â52, In 1888, Richard Dedekind proposed another axiomatization of natural-number arithmetic,WAS SIND UND WAS SOLLEN DIE ZAHLEN? >DATE=1893 | URL=HTTPS://ARCHIVE.ORG/DETAILS/WASSINDUNDWASSO00DEDEGOOG/PAGE/N42/MODE/2UP | 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 (). 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.JOURNAL
, Baratella, Stefano
, Ferro, Ruggero , 10.1002/malq.19930390138 , 3 , Mathematical Logic Quarterly , 1270381 , 338â352 , A theory of sets with the negation of the axiom of infinity , 39 , 1993, Theorems that can be proved in ZFC but cannot be proved using the Peano Axioms include Goodstein’s theorem.JOURNAL, Kirby, Laurie, Paris, Jeff, Accessible Independence Results for Peano Arithmetic, Bulletin of the London Mathematical Society, Wiley, 14, 4, 1982, 0024-6093, 10.1112/blms/14.4.285, 285â293, NotationThe set of all natural numbers is standardly denoted {{math|N}} or mathbb N.WEB, Listing of the Mathematical Notations used in the Mathematical Functions Website: Numbers, variables, and functions,functions.wolfram.com/Notations/1/, 27 July 2020, functions.wolfram.com, Older texts have occasionally employed {{math|J}} as the symbol for this set.BOOK,archive.org/details/1979RudinW, Principles of Mathematical Analysis, Rudin, W., McGraw-Hill, 1976, 978-0-07-054235-8, New York, 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:BOOK, Grimaldi, Ralph P., Discrete and Combinatorial Mathematics: An applied introduction, Pearson Addison Wesley, 978-0-201-72634-3, 5th, 2004,
{1, 2, 3,dots} = {x in mathbb Z : x > 0}=mathbb Z^+= mathbb{Z}_{>0}
{0, 1, 2,dots} = {x in mathbb Z : x ge 0}=mathbb Z^{+}_{0}=mathbb{Z}_ {ge 0}
PropertiesThis section uses the convention mathbb{N}=mathbb{N}_0=mathbb{N}^*cup{0}.AdditionGiven the set mathbb{N} of natural numbers and the successor function S colon mathbb{N} to 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 (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}}.MultiplicationAnalogously, given that addition has been defined, a multiplication operator times can be defined via {{math|a à 0 {{=}} 0}} and {{math|a à S(b) {{=}} (a à b) + a}}. This turns (mathbb{N}^*, times) into a free commutative monoid with identity element 1; a generator set for this monoid is the set of prime numbers.Relationship between addition and multiplicationAddition and multiplication are compatible, which is expressed in the distribution law: {{math|a à (b + c) {{=}} (a à b) + (a à 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 mathbb{N} is not closed under subtraction (that is, subtracting one natural from another does not always result in another natural), means that 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 à are as above, except that they begin with {{math|a + 1 {{=}} S(a)}} and {{math|a à 1 {{=}} a}}. Furthermore, (mathbb{N^*}, +) has no identity element.OrderIn this section, juxtaposed variables such as {{math|ab}} indicate the product {{math|a à b}},WEB, Weisstein, Eric W., Multiplication,mathworld.wolfram.com/Multiplication.html, 27 July 2020, mathworld.wolfram.com, 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|Ï}} (omega).DivisionIn this section, juxtaposed variables such as {{math|ab}} indicate the product {{math|a à 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 â 0}} there are natural numbers {{math|q}} and {{math|r}} such that
a = bq + r text{ and } r
| < b. 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.