Editing Natural number (section)

{{Short description|Number used for counting}}
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[[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]], and so on, possibly excluding 0.<ref name="Enderton"/> Some start counting with 0, defining the natural numbers as the '''non-negative integers''' {{math|1=0, 1, 2, 3, ...}}, while others start with 1, defining them as the '''positive integers''' {{nobr|{{math|1, 2, 3, ...}} .{{efn|See {{section link|#Emergence as a term}}}} }} Some authors acknowledge both definitions whenever convenient.<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> Sometimes, the '''whole numbers''' are the natural numbers as well as zero. In other cases, the ''whole numbers'' refer to all of the [[integer]]s, 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> The '''counting numbers''' are another term for the natural numbers, particularly in primary education, and are ambiguous as well although typically start at 1.<ref name=MathWorld_CountingNumber>{{MathWorld|title=Counting Number|id=CountingNumber}}</ref>

The natural numbers are used for counting<!-- Please, do not link this word that is used in its common language meaning, and not in any technical meaning --> things, like "there are ''six'' coins on the table", in which case they are called ''[[cardinal number]]s''. They are also used to put things in order,<!-- Please, do not link this word that is used in its common language meaning, and not in any technical meaning --> like "this is the ''third'' largest city in the country", which are called ''[[ordinal number]]s''. Natural numbers are also used as labels, like [[Number (sports)|jersey numbers]] on a sports team, where they serve as ''[[nominal number]]s'' and do not have mathematical properties.<ref>{{cite journal |last1=Woodin |first1=Greg |first2=Bodo |last2=Winter |title=Numbers in Context: Cardinals, Ordinals, and Nominals in American English |journal=Cognitive Science |volume=48|number=6 |year=2024 |article-number=e13471 |doi=10.1111/cogs.13471 |doi-access=free|pmid=38895756 |pmc=11475258 }}</ref>

The natural numbers form a [[set (mathematics)|set]], commonly symbolized as a bold {{math|'''N'''}} or [[blackboard bold]] {{tmath|\N}}. Many other [[number set]]s are built from the natural numbers. For example, the [[integer]]s are made by adding 0 and negative numbers. The [[rational number]]s add fractions, and the [[real number]]s add all infinite decimals. [[Complex number]]s add the [[Imaginary unit|square root of {{math|−1}}]]. This chain of extensions canonically [[Embedding|embeds]] the natural numbers in the other number systems.<ref>{{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."</ref><ref>{{harvtxt|Bluman|2010|page=1}}: "Numbers make up the foundation of mathematics."</ref>

Natural numbers are studied in different areas of math. [[Number theory]] looks at things like how numbers divide evenly ([[divisibility]]), or how [[prime number]]s are spread out. [[Combinatorics]] studies counting and arranging numbered objects, such as [[Partition (number theory)|partition]]s and [[Enumerative combinatorics|enumerations]].