Editing Number (section)

{{Short description|Used to count, measure, and label}}
{{Other uses}}
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[[File:NumberSetinC.svg|thumb|[[Set inclusion]]s between the [[natural number]]s {{bug workaround|(ℕ), the [[integer]]s (ℤ), the [[rational number]]s (ℚ), the [[real number]]s (ℝ), and the [[complex number]]s (ℂ)}}]]
A '''number''' is a [[mathematical object]] used to count, measure, and label. The most basic examples are the [[natural number]]s 1, 2, 3, 4, and so forth.<ref>{{Cite journal |title=number, n. |url=http://www.oed.com/view/Entry/129082 |journal=OED Online |language=en-GB |publisher=Oxford University Press |access-date=2017-05-16 |archive-url=https://web.archive.org/web/20181004081907/http://www.oed.com/view/Entry/129082 |archive-date=2018-10-04 |url-status=live }}</ref> Numbers can be represented in language with number words. More universally, individual numbers can be represented by symbols, called ''numerals''; for example, "5" is a numeral that represents the number five. As only a relatively small number of symbols can be memorized, basic numerals are commonly organized in a [[numeral system]], which is an organized way to represent any number. The most common numeral system is the [[Hindu–Arabic numeral system]], which allows for the representation of any [[Integer|non-negative integer]] using a combination of ten fundamental numeric symbols, called [[numerical digit|digit]]s.<ref>{{Cite journal |title=numeral, adj. and n. |url=http://www.oed.com/view/Entry/129111 |journal=OED Online |publisher=Oxford University Press |access-date=2017-05-16 |archive-date=2022-07-30 |archive-url=https://web.archive.org/web/20220730095156/https://www.oed.com/start;jsessionid=B9929F0647C8EE5D4FDB3A3C1B2CA3C3?authRejection=true&url=%2Fview%2FEntry%2F129111 |url-status=live }}</ref>{{efn|In [[linguistics]], a [[numeral (linguistics)|numeral]] can refer to a symbol like 5, but also to a word or a phrase that names a number, like "five hundred"; numerals include also other words representing numbers, like "dozen".}} In addition to their use in counting and measuring, numerals are often used for labels (as with telephone numbers), for ordering (as with [[serial number]]s), and for codes (as with [[ISBN]]s). In common usage, a ''numeral'' is not clearly distinguished from the ''number'' that it represents.

In mathematics, the notion of number has been extended over the centuries to include zero (0),<ref>{{Cite news |url=https://www.scientificamerican.com/article/history-of-zero/ |title=The Origin of Zero |last=Matson |first=John |work=Scientific American |access-date=2017-05-16 |language=en |archive-url=https://web.archive.org/web/20170826235655/https://www.scientificamerican.com/article/history-of-zero/ |archive-date=2017-08-26 |url-status=live }}</ref> [[negative number]]s,<ref name=":0">{{Cite book |url=https://books.google.com/books?id=f6HlhlBuQUgC&pg=PA88 |title=A History of Mathematics: From Mesopotamia to Modernity |last=Hodgkin |first=Luke |date=2005-06-02 |publisher=OUP Oxford |isbn=978-0-19-152383-0 |pages=85–88 |language=en |access-date=2017-05-16 |archive-url=https://web.archive.org/web/20190204012433/https://books.google.com/books?id=f6HlhlBuQUgC&pg=PA88 |archive-date=2019-02-04 |url-status=live }}</ref> [[rational number]]s such as [[one half]] <math>\left(\tfrac{1}{2}\right)</math>, [[real number]]s such as the [[square root of 2]] <math>\left(\sqrt{2}\right)</math> and [[pi|{{pi}}]],<ref>{{cite book |title=Mathematics across cultures : the history of non-western mathematics |date=2000 |publisher=Kluwer Academic |location=Dordrecht |isbn=1-4020-0260-2 |pages=410–411}}</ref> and [[complex number]]s<ref>{{Cite book |last=Descartes |first=René |title=La Géométrie: The Geometry of René Descartes with a facsimile of the first edition |url=https://archive.org/details/geometryofrenede00rend |year=1954 |author-link=René Descartes |orig-year=1637 |publisher=[[Dover Publications]] |isbn=0-486-60068-8 |access-date=20 April 2011 }}</ref> which extend the real numbers with a [[imaginary unit|square root of {{math|−1}}]] (and its combinations with real numbers by adding or subtracting its multiples).<ref name=":0" /> [[Calculation]]s with numbers are done with arithmetical operations, the most familiar being [[addition]], [[subtraction]], [[multiplication]], [[division (mathematics)|division]], and [[exponentiation]]. Their study or usage is called [[arithmetic]], a term which may also refer to [[number theory]], the study of the properties of numbers.

Besides their practical uses, numbers have cultural significance throughout the world.<ref name="Gilsdorf">{{Cite book |last=Gilsdorf |first=Thomas E. |url=https://books.google.com/books?id=IN8El-TTlSQC |title=Introduction to cultural mathematics : with case studies in the Otomies and the Incas |date=2012 |publisher=Wiley |isbn=978-1-118-19416-4 |location=Hoboken, N.J. |oclc=793103475}}</ref><ref name="Restivo">{{Cite book |last=Restivo |first=Sal P. |url=https://books.google.com/books?id=V0RuCQAAQBAJ&q=Mathematics+in+Society+and+History |title=Mathematics in society and history : sociological inquiries |date=1992 |isbn=978-94-011-2944-2 |location=Dordrecht |oclc=883391697}}</ref> For example, in Western society, the [[13 (number)|number 13]] is often regarded as [[unlucky]], and "[[One million|a million]]" may signify "a lot" rather than an exact quantity.<ref name="Gilsdorf" /> Though it is now regarded as [[pseudoscience]], belief in a mystical significance of numbers, known as [[numerology]], permeated ancient and medieval thought.<ref name="Ore">{{Cite book |last=Ore |first=Øystein |url=https://books.google.com/books?id=Sl_6BPp7S0AC |title=Number theory and its history |date=1988 |publisher=Dover |isbn=0-486-65620-9 |location=New York |oclc=17413345}}</ref> Numerology heavily influenced the development of [[Greek mathematics]], stimulating the investigation of many problems in number theory which are still of interest today.<ref name="Ore" />

During the 19th century, mathematicians began to develop many different abstractions which share certain properties of numbers, and may be seen as extending the concept. Among the first were the [[hypercomplex number]]s, which consist of various extensions or modifications of the [[complex number]] system. In modern mathematics, number systems are considered important special examples of more general algebraic structures such as [[ring (mathematics)|rings]] and [[field (mathematics)|fields]], and the application of the term "number" is a matter of convention, without fundamental significance.<ref>Gouvêa, Fernando Q. ''[[The Princeton Companion to Mathematics]], Chapter II.1, "The Origins of Modern Mathematics"'', p. 82. Princeton University Press, September 28, 2008. {{isbn|978-0-691-11880-2}}. "Today, it is no longer that easy to decide what counts as a 'number.' The objects from the original sequence of 'integer, rational, real, and complex' are certainly numbers, but so are the ''p''-adics. The quaternions are rarely referred to as 'numbers,' on the other hand, though they can be used to coordinatize certain mathematical notions."</ref>