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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]].<ref>{{Cite journal |last=Cajori |first=Florian |date=Feb 1926 |title=The History of Arithmetic. Louis Charles Karpinski |url=https://www.journals.uchicago.edu/doi/10.1086/358384 |journal=Isis |publisher=[[University of Chicago Press]] |volume=8 |issue=1 |pages=231–232 |doi=10.1086/358384 |issn=0021-1753 |access-date=2022-03-17 |archive-date=2022-03-17 |archive-url=https://web.archive.org/web/20220317152517/https://www.journals.uchicago.edu/doi/10.1086/358384 |url-status=live }}</ref> The way of denoting numbers in the decimal system is often referred to as ''decimal notation''.<ref>{{Cite book |lastlast1=Yong |firstfirst1=Lam Lay |url=http://dx.doi.org/10.1142/5425 |title=Fleeting Footsteps |last2=Se |first2=Ang Tian |date=April 2004 |publisher=[[World Scientific]] |isbn=978-981-238-696-0 |at=268 |doi=10.1142/5425 |access-date=March 17, 2022 |archive-date=April 1, 2023 |archive-url=https://web.archive.org/web/20230401132256/https://www.worldscientific.com/worldscibooks/10.1142/5425 |url-status=live }}</ref>
 
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''.
 
TheDecimals are commonly used to [[approximation (mathematics)|approximate]] real numbers. By increasing the number of digits after the decimal separator, one can make the [[approximation error]]s as small as one wants, when one has a method for computing the new digits.

{{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 decimalusual numeralsdecimals, with a finite number of non-zero digits after the decimal separator, are sometimes called '''terminating decimals'''. A ''[[repeating decimal]]'' is an infinite decimal that, after some place, repeats indefinitely the same sequence of digits (e.g., {{math|1=5.123144144144144... = 5.123{{overline|144}}}}).<ref>The [[Vinculum (symbol)|vinculum (overline)]] in 5.123<span style="text-decoration: overline;">144</span> indicates that the '144' sequence repeats indefinitely, i.e. {{val|5.123144144144144|s=...}}.</ref> An infinite decimal represents a [[rational number]], the [[quotient]] of two integers, if and only if it is a repeating decimal or has a finite number of non-zero digits.
 
==Origin==
{{unreferenced section|date=May 2022}}
[[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 [[Arab language|Arab speakingArabic]]-speaking ones, other [[glyph]]s are used for the digits</ref> the [[decimal separator]] is the dot "{{math|.}}" in many countries (mostly English-speaking),<ref>{{Cite web|last=Weisstein|first=Eric W.|title=Decimal|url=https://mathworld.wolfram.com/Decimal.html|access-date=2020-08-22|website=mathworld.wolfram.com|language=en|archive-date=2020-03-18|archive-url=https://web.archive.org/web/20200318204545/https://mathworld.wolfram.com/Decimal.html|url-status=live}}</ref> and a comma "{{math|,}}" in other countries.<ref name=":1" />
 
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 decimalsdecimal expressions <math>0.8, 14.89, 0.00079, 1.618, 3.14159</math> represent the fractions {{math|{{sfrac|4|5}}}}, {{math|{{sfrac|1489|100}}}}, {{math|{{sfrac|79|100000}}}}, {{Math|{{sfrac||809|500}}}} and {{Math|{{sfrac||314159|100000}}}}, and are therefore denote decimal numbersfractions. An example of a fraction that cannot be represented by a decimal expression (with a finite number of digits) is {{math|{{sfrac|1|3}}}}, 3 not being a power of 10.
 
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===
==Real number approximation==
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.
{{unreferenced section|date=April 2020}}
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==
{{unreferenced section|date=April 2020}}
{{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, sometimesperhaps arguedbecause due totwo human hands typically havinghave ten fingers/digits.<ref>{{citation|first=Tobias|last=Dantzig|title=Number / The Language of Science |edition=4th |year=1954|publisher=The Free Press (Macmillan Publishing Co.) |isbn=0-02-906990-4|page=12}}</ref> Standardized weights used in the [[Indus Valley civilizationCivilisation]] ({{circa|3300–1300 BCE}}) were based on the ratios: 1/20, 1/10, 1/5, 1/2, 1, 2, 5, 10, 20, 50, 100, 200, and 500, while their standardized ruler – the ''Mohenjo-daro ruler'' – was divided into ten equal parts.<ref>Sergent, Bernard (1997), ''Genèse de l'Inde'' (in French), Paris: Payot, p. 113, {{ISBN|2-228-89116-9}}</ref><ref>{{cite journal | last1 = Coppa | first1 = A. | display-authors = etal | year = 2006 | title = Early Neolithic tradition of dentistry: Flint tips were surprisingly effective for drilling tooth enamel in a prehistoric population | bibcode = 2006Natur.440..755C | journal = Nature | volume = 440 | issue = 7085| pages = 755–56 | doi = 10.1038/440755a | pmid = 16598247 | s2cid = 6787162 }}</ref><ref>Bisht, R. S. (1982), "Excavations at Banawali: 1974–77", in Possehl, Gregory L. (ed.), Harappan ''Civilisation: A Contemporary Perspective'', New Delhi: Oxford and IBH Publishing Co., pp. 113–24</ref> [[Egyptian hieroglyphs]], in evidence since around 3000 BCE, used a purely decimal system,<ref>Georges Ifrah: ''From One to Zero. A Universal History of Numbers'', Penguin Books, 1988, {{isbn|0-14-009919-0}}, pp.&nbsp;200–13 (Egyptian Numerals)</ref> as did the [[Linear A]] script ({{circa|1800–1450 BCE}}) of the [[Minoan civilization|Minoans]]<ref>Graham Flegg: Numbers: their history and meaning, Courier Dover Publications, 2002, {{isbn|978-0-486-42165-0}}, p.&nbsp;50</ref><ref>Georges Ifrah: ''From One to Zero. A Universal History of Numbers'', Penguin Books, 1988, {{isbn|0-14-009919-0}}, pp. 213–18 (Cretan numerals)</ref> and the [[Linear B]] script (c. 1400–1200 BCE) of the [[Mycenaean Greece|Mycenaeans]]. The [[Únětice culture]] in central Europe (2300-1600 BC) used standardised weights and a decimal system in trade.<ref>{{cite book |last1=Krause |first1=Harald |url=https://www.academia.edu/34550316 |title=Spangenbarrenhort Oberding |last2=Kutscher |first2=Sabrina |date=2017 |publisher=Museum Erding |isbn=978-3-9817606-5-1 |pages=238–243 |chapter=Spangenbarrenhort Oberding: Zusammenfassung und Ausblick}}</ref> The number system of [[classical Greece]] also used powers of ten, including an intermediate base of 5, as did [[Roman numerals]].<ref name="Greek numerals">{{Cite web |url=http://www-history.mcs.st-and.ac.uk/HistTopics/Greek_numbers.html |title=Greek numbers |access-date=2019-07-21 |archive-date=2019-07-21 |archive-url=https://web.archive.org/web/20190721085640/http://www-history.mcs.st-and.ac.uk/HistTopics/Greek_numbers.html |url-status=live }}</ref> Notably, the polymath [[Archimedes]] (c. 287–212 BCE) invented a decimal positional system in his [[The Sand Reckoner|Sand Reckoner]] which was based on 10<sup>8</sup>.<ref name="Greek numerals"/> and later led the German mathematician [[Carl Friedrich Gauss]] to lament what heights science would have already reached in his days if Archimedes had fully realized the potential of his ingenious discovery.<ref>[[Karl Menninger (mathematics)|Menninger, Karl]]: ''Zahlwort und Ziffer. Eine Kulturgeschichte der Zahl'', Vandenhoeck und Ruprecht, 3rd. ed., 1979, {{isbn|3-525-40725-4}}, pp.&nbsp;150–53</ref> [[Hittites|Hittite]] hieroglyphs (since 15th century BCE) were also strictly decimal.<ref>Georges Ifrah: ''From One to Zero. A Universal History of Numbers'', Penguin Books, 1988, {{isbn|0-14-009919-0}}, pp. 218f. (The Hittite hieroglyphic system)</ref>
 
Some non-mathematical ancient texts such as the [[Vedas]], dating back to 1700–900 BCE make use of decimals and mathematical decimal fractions.<ref>(Atharva Veda 5.15, 1–11)</ref>
 
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]]
DecimalStarting fractions were first developed and used byfrom the Chinese in the end of 4th2nd century BCE,<ref>{{cite websome |url=http://www.kaogu.cn/en/News/New_discoveries/2017/0425/57954.html |title=Ancient bambooChinese slipsunits for calculationlength enterwere worldbased recordson bookdivisions |language=eninto |website=Theten; Instituteby ofthe Archaeology,3rd Chinesecentury AcademyCE ofthese Socialmetrological Sciencesunits |access-date=10were May 2017 |archive-date=1 May 2017 |archive-url=https://web.archive.org/web/20170501055416/http://www.kaogu.cn/en/News/New_discoveries/2017/0425/57954.html |url-status=live }}</ref> and then spreadused to theexpress Middledecimal Eastfractions andof fromlengths, there to Europenon-positionally.<ref name=Lam/><ref name=jnfractn1>{{Cite book | author=Joseph Needham | author-link=Joseph Needham | chapter = Decimal19.2 SystemDecimals, Metrology, and the Handling of Large Numbers |pages=82–90 | title = Science and Civilisation in China, Volume |volume=III, "Mathematics and the Sciences of the Heavens and the Earth" | title-link=Science and Civilisation in China | year = 1959 | publisher = Cambridge University Press}}</ref> TheCalculations written Chinesewith decimal fractions wereof non-positional.<reflengths name=jnfractn1/> However,were [[Rod calculus#FractionsDecimal fraction|performed using positional counting rod fractionsrods]], wereas positional.<refdescribed name=Lam>in the 3rd–5th century CE ''[[LamSunzi Lay YongSuanjing]],''. "The Development5th century CE mathematician [[Zu Chongzhi]] calculated a 7-digit [[approximations of Hindu–Arabicπ|approximation andof Traditional{{mvar|π}}]]. Chinese[[Qin Arithmetic",Jiushao]]'s book ''Chinese[[Mathematical ScienceTreatise in Nine Sections]]'' (1247) explicitly writes a decimal fraction representing a number rather than a measurement, 1996using pcounting rods.<ref>Jean-Claude 38Martzloff, KurtA VogelHistory notationof Chinese Mathematics, Springer 1997 {{isbn|3-540-33782-2}}</ref> The number 0.96644 is denoted
 
:::::{{lang|zh|寸}}
[[Qin Jiushao]] in his book [[Mathematical Treatise in Nine Sections]] (1247<ref>Jean-Claude Martzloff, A History of Chinese Mathematics, Springer 1997 {{isbn|3-540-33782-2}}</ref>) denoted 0.96644 by
:::::[[File:Counting rod 0.png|frameless|18px]] [[File:Counting rod h9 num.png|frameless|18px]] [[File:Counting rod v6.png|frameless|18px]] [[File:Counting rod h6.png|frameless|18px]] [[File:Counting rod v4.png|frameless|18px]] [[File:Counting rod h4.png|frameless|18px]], meaning.
 
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
:::::{{lang|zh|寸}}
:::::096644
 
J. Lennart Berggren notes that positional decimal fractions appear for the first time in a book by the Arab mathematician [[Abu'lAl-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, anticipating [[Simon Stevin]], 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 [[Jamshīd al-Kāshī]] claimed to have discovered decimal fractions himself in the 15th century.<ref name=Berggren /> [[Al Khwarizmi]] introduced fractionfractions to Islamic countries in the early 9th century; aCE, Chinesewritten authorwith has alleged that his fraction presentation was an exact copy of traditional Chinese mathematical fraction from [[Sunzi Suanjing]].<ref name=Lam/> This form of fraction witha numerator on topabove and denominator at bottombelow, without a horizontal bar. wasThis alsoform usedof byfraction al-Uqlidisi and by al-Kāshīremained in hisuse work "Arithmeticfor Key"centuries.<ref name=Lam/><ref>{{cite journal | last1 = Lay Yong | first1 = Lam | author-link = Lam Lay Yong | title = A Chinese Genesis, Rewriting the history of our numeral system | journal = Archive for History of Exact Sciences | volume = 38 | pages = 101–08 }}</ref>
<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. Many [[Indo-Aryan languages|Indo-Aryan]] and [[Dravidian languages]] have numbers between 10 and 20 expressed in a regular pattern of addition to 10.<ref>{{cite webcn|urldate=http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Indian_numerals.html|title=IndianMarch numerals|work=Ancient Indian mathematics|access-date=2015-05-22|archive-date=2007-09-29|archive-url=https://web.archive.org/web/20070929131009/http://www-gap.dcs.st-and.ac.uk/%7Ehistory/HistTopics/Indian_numerals.html|url-status=dead2024}}</ref>
 
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 [[QuechuaQuechuan languages|Quechua]] and [[Aymara language|Aymara]] have an almost straightforward decimal system, in which 11 is expressed as ''ten with one'' and 23 as ''two-ten with three''.
 
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.&nbsp;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]]
* [[SIMetric prefix]]
}}