Editing Natural number (section)

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

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

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

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

A much later advance was the development of the idea that&nbsp;{{num|0}} can be considered as a number, with its own numeral. The use of a&nbsp;0 [[numerical digit|digit]] in place-value notation (within other numbers) dates back as early as 700&nbsp;BCE by the Babylonians, who omitted such a digit when it would have been the last symbol in the number.{{efn| A tablet found at Kish ... thought to date from around 700&nbsp;BC, uses three hooks to denote an empty place in the positional notation. Other tablets dated from around the same time use a single hook for an empty place.<ref>{{cite web |title=A history of Zero |website=MacTutor History of Mathematics |url=http://www-history.mcs.st-and.ac.uk/history/HistTopics/Zero.html |url-status=live |access-date=23 January 2013 |archive-url=https://web.archive.org/web/20130119083234/http://www-history.mcs.st-and.ac.uk/history/HistTopics/Zero.html |archive-date=19 January 2013}}</ref>}} The [[Olmec]] and [[Maya civilization]]s used&nbsp;0 as a separate number as early as the {{nowrap|1st century BCE}}, but this usage did not spread beyond [[Mesoamerica]].<ref>{{cite book |first=Charles C. |last=Mann |year=2005 |title=1491: New Revelations of the Americas before Columbus |page=19 |publisher=Knopf |isbn=978-1-4000-4006-3 |url=https://books.google.com/books?id=Jw2TE_UNHJYC&pg=PA19 |url-status=live |via=Google Books |access-date=3 February 2015 |archive-url=https://web.archive.org/web/20150514105855/https://books.google.com/books?id=Jw2TE_UNHJYC&pg=PA19 |archive-date=14 May 2015}}</ref><ref>{{cite book |first=Brian |last=Evans |year=2014 |title=The Development of Mathematics Throughout the Centuries: A brief history in a cultural context |publisher=John Wiley & Sons |isbn=978-1-118-85397-9 |chapter=Chapter 10. Pre-Columbian Mathematics: The Olmec, Maya, and Inca Civilizations |via=Google Books |chapter-url=https://books.google.com/books?id=3CPwAgAAQBAJ&pg=PT73}}</ref> The use of a numeral&nbsp;0 in modern times originated with the Indian mathematician [[Brahmagupta]] in 628&nbsp;CE. However, 0 had been used as a number in the medieval [[computus]] (the calculation of the date of Easter), beginning with [[Dionysius Exiguus]] in 525&nbsp;CE, without being denoted by a numeral. Standard [[Roman numerals]] do not have a symbol for&nbsp;0; instead, ''nulla'' (or the genitive form ''nullae'') from {{Lang|la|nullus}}, the Latin word for "none", was employed to denote a&nbsp;0 value.<ref>{{cite web |first=Michael |last=Deckers |title=Cyclus Decemnovennalis Dionysii – Nineteen year cycle of Dionysius |url=http://hbar.phys.msu.ru/gorm/chrono/paschata.htm |publisher=Hbar.phys.msu.ru |date=25 August 2003 |access-date=13 February 2012 |archive-url=https://web.archive.org/web/20190115083618/http://hbar.phys.msu.ru/gorm/chrono/paschata.htm |archive-date=15 January 2019 |url-status=live }}</ref>

The first systematic study of numbers as [[abstraction]]s is usually credited to the [[ancient Greece|Greek]] philosophers [[Pythagoras]] and [[Archimedes]]. Some Greek mathematicians treated the number&nbsp;1 differently than larger numbers, sometimes even not as a number at all.{{efn|This convention is used, for example, in [[Euclid's Elements]], see D.&nbsp;Joyce's web edition of Book VII.<ref name=EuclidVIIJoyce>{{cite book |author=Euclid |author-link=Euclid |editor-first=D. |editor-last=Joyce|editor-link=David E. Joyce (mathematician) |chapter=Book VII, definitions 1 and 2 |title=[[Euclid's Elements|Elements]] |publisher=Clark University |chapter-url=http://aleph0.clarku.edu/~djoyce/java/elements/bookVII/defVII1.html }}</ref>}} [[Euclid]], for example, defined a unit first and then a number as a multitude of units, thus by his definition, a unit is not a number and there are no unique numbers (e.g., any two units from indefinitely many units is a 2).<ref name="Mueller 2006 p. 58">{{cite book |last=Mueller |first=Ian |year=2006 |title=Philosophy of mathematics and deductive structure in [[Euclid's Elements]] |page=58 |publisher=Dover Publications |location=Mineola, New York |isbn=978-0-486-45300-2 |oclc=69792712}}</ref> However, in the definition of [[perfect number]] which comes shortly afterward, Euclid treats 1 as a number like any other.<ref>{{cite book |author=Euclid |author-link=Euclid |editor-first=D. |editor-last=Joyce |chapter=Book VII, definition 22 |title=[[Euclid's Elements|Elements]] |publisher=Clark University |chapter-url=http://aleph0.clarku.edu/~djoyce/java/elements/bookVII/defVII22.html |quote=A perfect number is that which is equal to the sum of its own parts. }} In definition VII.3 a "part" was defined as a number, but here 1 is considered to be a part, so that for example {{math|1=6 = 1 + 2 + 3}} is a perfect number.</ref>

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