Editing Natural number (section)

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

Starting at 0 or 1 has long been a matter of definition. In 1727, [[Bernard Le Bovier de Fontenelle]] wrote that his notions of distance and element led to defining the natural numbers as including or excluding 0.<ref>{{cite book |last1=Fontenelle |first1=Bernard de |title=Eléments de la géométrie de l'infini |date=1727 |page=3 |url=https://gallica.bnf.fr/ark:/12148/bpt6k64762n/f31.item |language=fr}}</ref> In 1889, [[Giuseppe Peano]] used N for the positive integers and started at 1,<ref>{{cite book |title=Arithmetices principia: nova methodo |date=1889 |publisher=Fratres Bocca |url=https://archive.org/details/arithmeticespri00peangoog/page/n12/mode/2up|page=12 |language=Latin}}</ref> but he later changed to using N<sub>0</sub> and N<sub>1</sub>.<ref>{{cite book |last1=Peano |first1=Giuseppe |title=Formulaire des mathematiques |date=1901 |publisher=Paris, Gauthier-Villars |page=39 |url=https://archive.org/details/formulairedesmat00pean/page/38/mode/2up|language=fr}}</ref> Historically, most definitions have excluded 0,<ref name="MacTutor"/><ref>{{cite book |last1=Fine |first1=Henry Burchard |title=A College Algebra |date=1904 |publisher=Ginn |page=6 |url=https://books.google.com/books?id=RR4PAAAAIAAJ&dq=%22natural%20number%22&pg=PA6 |language=en}}</ref><ref>{{cite book |title=Advanced Algebra: A Study Guide to be Used with USAFI Course MC 166 Or CC166 |date=1958 |publisher=United States Armed Forces Institute |page=12 |url=https://books.google.com/books?id=184i06Py1ZYC&dq=%22natural%20number%22%201&pg=PA12 |language=en}}</ref> but many mathematicians such as [[George A. Wentworth]], [[Bertrand Russell]], [[Nicolas Bourbaki]], [[Paul Halmos]], [[Stephen Cole Kleene]], and [[John Horton Conway]] have preferred to include 0.<ref>{{cite web |title=Natural Number |url=https://archive.lib.msu.edu/crcmath/math/math/n/n035.htm |website=archive.lib.msu.edu}}</ref><ref name="MacTutor"/>

Mathematicians have noted tendencies in which definition is used, such as algebra texts including 0,<ref name="MacTutor"/>{{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]].}} number theory and analysis texts excluding 0,<ref name="MacTutor"/><ref name="Křížek">{{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}}</ref><ref>See, for example, {{harvtxt|Carothers|2000|p=3}} or {{harvtxt|Thomson|Bruckner|Bruckner|2008|p=2}}</ref> logic and set theory texts including 0,<ref>{{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}}</ref><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><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> dictionaries excluding 0,<ref name="MacTutor"/><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> school books (through high-school level) excluding 0, and upper-division college-level books including 0.<ref name="Enderton">{{cite book |last1=Enderton |first1=Herbert B. |title=Elements of set theory |date=1977 |publisher=Academic Press |location=New York |isbn=0122384407 |page=66}}</ref> There are exceptions to each of these tendencies and as of 2023 no formal survey has been conducted. Arguments raised include [[division by zero]]<ref name="Křížek"/> and the size of the [[empty set]]. [[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> Including 0 began to rise in popularity in the 1960s.<ref name="MacTutor"/> 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/IEC 80000|ISO 80000-2]].<ref name=ISO80000/>