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{{Other uses}}
[[File:Decimal_digit.png|thumb|upright=1.2|Place value of number in decimal system]]
The '''decimal''' [[numeral system]] (also called the '''base-ten''' [[positional numeral system]] and '''denary''' {{IPAc-en|ˈ|d|iː|n|ər|i}}<ref>{{OED|denary}}</ref> or '''decanary''') is the standard system for denoting [[integer]] and non-integer [[number]]s. It is the extension to non-integer numbers (''decimal fractions'') of the [[Hindu–Arabic numeral system]].
A '''decimal numeral''' (also often just ''decimal'' or, less correctly, ''decimal number''), refers generally to the notation of a number in the decimal numeral system. Decimals may sometimes be identified by a [[decimal separator]] (usually "." or "," as in {{math|25.9703}} or {{math|3,1415}}).<ref name=":1">{{Cite web |last=Weisstein |first=Eric W. |date=March 10, 2022 |title=Decimal Point |url=https://mathworld.wolfram.com/DecimalPoint.html |url-status=live |access-date=March 17, 2022 |website=Wolfram MathWorld |language=en |archive-date=March 21, 2022 |archive-url=https://web.archive.org/web/20220321195047/https://mathworld.wolfram.com/DecimalPoint.html }}</ref>
''Decimal'' may also refer specifically to the digits after the decimal separator, such as in "{{math|3.14}} is the approximation of {{pi}} to ''two decimals''". Zero-digits after a decimal separator serve the purpose of signifying the precision of a value. The numbers that may be represented in the decimal system are the [[#Decimal fractions|'''decimal fractions''']]. That is, [[fraction (mathematics)|fractions]] of the form {{math|''a''/10<sup>''n''</sup>}}, where {{math|''a''}} is an integer, and {{math|''n''}} is a [[non-negative integer]]. Decimal fractions also result from the addition of an integer and a ''[[fractional part]]''; the resulting sum sometimes is called a ''fractional number''.
{{anchor|terminating decimal}} Originally and in most uses, a decimal has only a finite number of digits after the decimal separator. However, the decimal system has been extended to ''infinite decimals'' for representing any [[real number]], by using an [[sequence (mathematics)|infinite sequence]] of digits after the decimal separator (see [[decimal representation]]). In this context, the ==Origin==
[[File:Two hand, ten fingers.jpg|thumb|right|Ten digits on two hands, the possible origin of decimal counting|upright=1.2]]
Many [[numeral system]]s of ancient civilizations use ten and its powers for representing numbers, possibly because there are ten fingers on two hands and people started counting by using their fingers. Examples are firstly the [[Egyptian numerals]], then the [[Brahmi numerals]], [[Greek numerals]], [[Hebrew numerals]], [[Roman numerals]], and [[Chinese numerals]].<ref name=":0">{{Cite book |last=Lockhart |first=Paul |title=Arithmetic |date=2017 |publisher=The Belknap Press of Harvard University Press |isbn=978-0-674-97223-0 |location=Cambridge, Massachusetts London, England}}</ref> Very large numbers were difficult to represent in these old numeral systems, and only the best mathematicians were able to multiply or divide large numbers. These difficulties were completely solved with the introduction of the [[Hindu–Arabic numeral system]] for representing [[integer]]s. This system has been extended to represent some non-integer numbers, called ''[[#Decimal fractions|decimal fractions]]'' or ''decimal numbers'', for forming the ''decimal numeral system''.<ref name=":0" />
== Decimal notation ==
For writing numbers, the decimal system uses ten [[decimal digit]]s, a [[decimal mark]], and, for [[negative number]]s, a [[minus sign]] "−". The decimal digits are [[0]], [[1]], [[2]], [[3]], [[4]], [[5]], [[6]], [[7]], [[8]], [[9]];<ref>In some countries, such as [[
For representing a [[non-negative number]], a decimal numeral consists of
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== Decimal fractions ==
{{Table Numeral Systems}}
'''Decimal fractions''' (sometimes called '''decimal numbers''', especially in contexts involving explicit fractions) are the [[rational number]]s that may be expressed as a [[Fraction (mathematics)|fraction]] whose [[denominator]] is a [[exponentiation|power]] of ten.<ref>{{cite encyclopedia|url=https://www.encyclopediaofmath.org/index.php/Decimal_fraction|title=Decimal Fraction|encyclopedia=[[Encyclopedia of Mathematics]]|access-date=2013-06-18|archive-date=2013-12-11|archive-url=https://web.archive.org/web/20131211035917/http://www.encyclopediaofmath.org/index.php/Decimal_fraction|url-status=live}}</ref> For example, the
More generally, a decimal with {{math|''n''}} digits after the [[Decimal separator|separator]] (a point or comma) represents the fraction with denominator {{math|10<sup>''n''</sup>}}, whose numerator is the integer obtained by removing the separator.
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:<math>1=2^0\cdot 5^0, 2=2^1\cdot 5^0, 4=2^2\cdot 5^0, 5=2^0\cdot 5^1, 8=2^3\cdot 5^0, 10=2^1\cdot 5^1, 16=2^4\cdot 5^0, 20=2^2\cdot5^1, 25=2^0\cdot 5^2, \ldots</math>
===Approximation using decimal numbers===
Decimal numerals do not allow an exact representation for all [[real number]]s
▲Decimal numerals do not allow an exact representation for all [[real number]]s, e.g. for the real number [[pi|{{pi}}]]. Nevertheless, they allow approximating every real number with any desired accuracy, e.g., the decimal 3.14159 approximates the real {{pi}}, being less than 10<sup>−5</sup> off; so decimals are widely used in [[science]], [[engineering]] and everyday life.
More precisely, for every real number {{Mvar|x}} and every positive integer {{Mvar|n}}, there are two decimals {{Mvar|''L''}} and {{Mvar|''u''}} with at most ''{{Mvar|n}}'' digits after the decimal mark such that {{Math|''L'' ≤ ''x'' ≤ ''u''}} and {{Math|1=(''u'' − ''L'') = 10<sup>−''n''</sup>}}.
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==Infinite decimal expansion==
{{main|Decimal representation}}
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== History ==
[[File:Qinghuajian, Suan Biao.jpg|thumb|upright|The world's earliest decimal multiplication table was made from bamboo slips, dating from 305 BCE, during the [[Warring States]] period in China.]]
Many ancient cultures calculated with numerals based on ten,
The Egyptian hieratic numerals, the Greek alphabet numerals, the Hebrew alphabet numerals, the Roman numerals, the Chinese numerals and early Indian Brahmi numerals are all non-positional decimal systems, and required large numbers of symbols. For instance, Egyptian numerals used different symbols for 10, 20 to 90, 100, 200 to 900, 1000, 2000, 3000, 4000, to 10,000.<ref>[[Lam Lay Yong]] et al. The Fleeting Footsteps pp. 137–39</ref>
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=== History of decimal fractions ===
[[File:Rod fraction.jpg|thumb|right|150px|counting rod decimal fraction 1/7]]
Historians of Chinese science have speculated that the idea of decimal fractions may have been transmitted from China to the Middle East.<ref name=Lam>[[Lam Lay Yong]], "The Development of Hindu–Arabic and Traditional Chinese Arithmetic", ''Chinese Science'', 1996 p. 38, Kurt Vogel notation</ref>
▲:::::{{lang|zh|寸}}
▲:::::[[File:Counting rod 0.png]] [[File:Counting rod h9 num.png]] [[File:Counting rod v6.png]] [[File:Counting rod h6.png]] [[File:Counting rod v4.png]] [[File:Counting rod h4.png]], meaning
<div style="float: right;">[[File:Stevin-decimal notation.svg]]</div>▼
Positional decimal fractions appear for the first time in a book by the Arab mathematician [[Abu'l-Hasan al-Uqlidisi]] written in the 10th century.<ref name=Berggren>{{cite book | first=J. Lennart | last=Berggren | title=The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook | chapter=Mathematics in Medieval Islam |editor-first=Victor J.|editor-last=Katz|publisher=Princeton University Press | year=2007 | isbn=978-0-691-11485-9 | page=530 }}</ref> The Jewish mathematician [[Immanuel Bonfils]] used decimal fractions around 1350 but did not develop any notation to represent them.<ref>[[Solomon Gandz|Gandz, S.]]: The invention of the decimal fractions and the application of the exponential calculus by Immanuel Bonfils of Tarascon (c. 1350), Isis 25 (1936), 16–45.</ref> The Persian mathematician [[Jamshid al-Kashi]] used, and claimed to have discovered, decimal fractions in the 15th century.<ref name=Berggren />
A forerunner of modern European decimal notation was introduced by [[Simon Stevin]] in the 16th century.<ref name=van>{{Cite book | author = B. L. van der Waerden | author-link = Bartel Leendert van der Waerden | year = 1985 | title = A History of Algebra. From Khwarizmi to Emmy Noether | publisher = Springer-Verlag | place = Berlin}}</ref>▼
▲<div style="float: right;">[[File:Stevin-decimal notation.svg]]</div>
▲A forerunner of modern European decimal notation was introduced by [[Simon Stevin]] in the 16th century. Stevin's influential booklet ''[[De Thiende]]'' ("the art of tenths") was first published in Dutch in 1585 and translated into French as ''La Disme''.<ref name=van>{{Cite book | author = B. L. van der Waerden | author-link = Bartel Leendert van der Waerden | year = 1985 | title = A History of Algebra. From Khwarizmi to Emmy Noether | publisher = Springer-Verlag | place = Berlin}}</ref>
[[John Napier]] introduced using the period (.) to separate the integer part of a decimal number from the fractional part in his book on constructing tables of logarithms, published posthumously in 1620.<ref name=constructionIA>{{cite book|title=[[Commons:File:The_Construction_of_the_Wonderful_Canon_of_Logarithms.djvu|The Construction of the Wonderful Canon of Logarithms]]|first=John|last=Napier|translator-last1=Macdonald|translator-first1= William Rae|date=1889|orig-date=1620|publisher=Blackwood & Sons|publication-place=Edinburgh|via=Internet Archive|quote=In numbers distinguished thus by a period in their midst, whatever is written after the period is a fraction, the denominator of which is unity with as many cyphers after it as there are figures after the period.}}</ref>{{rp|p. 8, archive p. 32)}}
=== Natural languages ===
A method of expressing every possible [[natural number]] using a set of ten symbols emerged in India.<ref>{{cite web |url=https://mathshistory.st-andrews.ac.uk/HistTopics/Indian_numerals/ |title=Indian numerals|work=Ancient Indian mathematics }}</ref> Several Indian languages show a straightforward decimal system.
The [[Hungarian language]] also uses a straightforward decimal system. All numbers between 10 and 20 are formed regularly (e.g. 11 is expressed as "tizenegy" literally "one on ten"), as with those between 20 and 100 (23 as "huszonhárom" = "three on twenty").
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A straightforward decimal rank system with a word for each order (10 {{lang|zh|十}}, 100 {{lang|zh|百}}, 1000 {{lang|zh|千}}, 10,000 {{lang|zh|万}}), and in which 11 is expressed as ''ten-one'' and 23 as ''two-ten-three'', and 89,345 is expressed as 8 (ten thousands) {{lang|zh|万}} 9 (thousand) {{lang|zh|千}} 3 (hundred) {{lang|zh|百}} 4 (tens) {{lang|zh|十}} 5 is found in [[Chinese language|Chinese]], and in [[Vietnamese language|Vietnamese]] with a few irregularities. [[Japanese language|Japanese]], [[Korean language|Korean]], and [[Thai language|Thai]] have imported the Chinese decimal system. Many other languages with a decimal system have special words for the numbers between 10 and 20, and decades. For example, in English 11 is "eleven" not "ten-one" or "one-teen".
Incan languages such as [[
Some psychologists suggest irregularities of the English names of numerals may hinder children's counting ability.<ref>{{Cite journal| last=Azar| first=Beth| year=1999| title=English words may hinder math skills development| url=http://www.apa.org/monitor/apr99/english.html |journal=American Psychological Association Monitor| volume=30| issue=4 |archive-url = https://web.archive.org/web/20071021015527/http://www.apa.org/monitor/apr99/english.html |archive-date = 2007-10-21}}</ref>
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| pages = 487–95
| title = The cardinal numerals in pre-and proto-Germanic
| volume = 86}}.</ref> Where this counting system is known, it is based on the "[[long hundred]]" = 120, and a "long thousand" of 1200. The descriptions like "long" only appear after the "small hundred" of 100 appeared with the Christians. Gordon's [https://www.scribd.com/doc/49127454/Introduction-to-Old-Norse-by-E-V-Gordon Introduction to Old Norse] {{Webarchive|url=https://web.archive.org/web/20160415205641/https://www.scribd.com/doc/49127454/Introduction-to-Old-Norse-by-E-V-Gordon |date=2016-04-15 }} p. 293, gives number names that belong to this system. An expression cognate to 'one hundred and eighty' translates to 200, and the cognate to 'two hundred' translates to 240. [http://ads.ahds.ac.uk/catalogue/adsdata/arch-352-1/dissemination/pdf/vol_123/123_395_418.pdf Goodare]{{Dead link|date=January 2024 |bot=InternetArchiveBot |fix-attempted=yes }} details the use of the long hundred in Scotland in the Middle Ages, giving examples such as calculations where the carry implies i C (i.e. one hundred) as 120, etc. That the general population were not alarmed to encounter such numbers suggests common enough use. It is also possible to avoid hundred-like numbers by using intermediate units, such as stones and pounds, rather than a long count of pounds. Goodare gives examples of numbers like vii score, where one avoids the hundred by using extended scores. There is also a paper by W.H. Stevenson, on 'Long Hundred and its uses in England'.<ref>{{Cite journal|last=Stevenson|first=W.H.|date=1890|title=The Long Hundred and its uses in England|journal=Archaeological Review|volume=December 1889|pages=313–22}}</ref><ref>{{Cite book|last=Poole, Reginald Lane|title=The Exchequer in the twelfth century : the Ford lectures delivered in the University of Oxford in Michaelmas term, 1911|date=2006|publisher=Lawbook Exchange|isbn=1-58477-658-7|location=Clark, NJ|oclc=76960942}}</ref>
* Many or all of the [[Chumashan languages]] originally used a [[quaternary numeral system|base-4]] counting system, in which the names for numbers were structured according to multiples of 4 and [[hexadecimal|16]].<ref>There is a surviving list of [[Ventureño language]] number words up to 32 written down by a Spanish priest ca. 1819. "Chumashan Numerals" by Madison S. Beeler, in ''Native American Mathematics'', edited by Michael P. Closs (1986), {{isbn|0-292-75531-7}}.</ref>
* Many languages<ref name="Hammarstrom 2010">{{Cite book
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* [[Scientific notation]]
* [[Serial decimal]]
* [[
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