Editing Numerical digit (section)

==History of modern numbers==
In [[China]], armies and provisions were counted using modular tallies of [[prime number]]s. Unique numbers of troops and measures of rice appear as unique combinations of these tallies. A great convenience of [[modular arithmetic]] is that it is easy to multiply.<ref>{{Cite book|author=Knuth, Donald Ervin|title=The art of computer programming|year=1998 |publisher=Addison-Wesley Pub. Co|isbn=0-201-03809-9|location=Reading, Mass.|oclc=823849|quote=The advantages of a modular representation are that addition, subtraction, and multiplication are very simple}}</ref> This makes use of modular arithmetic for provisions especially attractive. Conventional tallies are quite difficult to multiply and divide. In modern times modular arithmetic is sometimes used in [[digital signal processing]].<ref>{{Cite book|last1=Echtle|first1=Klaus|url=https://books.google.com/books?id=bzw15Ew_iOoC&q=modern+times+modular+arithmetic++digital+signal+processing.&pg=PA439|title=Dependable Computing - EDCC-1: First European Dependable Computing Conference, Berlin, Germany, October 4-6, 1994. Proceedings|last2=Hammer|first2=Dieter|last3=Powell|first3=David|date=1994-09-21|publisher=Springer Science & Business Media|isbn=978-3-540-58426-1|pages=439|language=en}}</ref>

The oldest Greek system was that of the [[Attic numerals]],<ref>{{Cite book|author=Woodhead, A. G. (Arthur Geoffrey)|title=The study of Greek inscriptions|date=1981|publisher=Cambridge University Press|isbn=0-521-23188-4|edition=2nd|location=Cambridge|pages=109–110|oclc=7736343}}</ref> but in the 4th century BC they began to use a quasidecimal alphabetic system (see [[Greek numerals]]).<ref>{{Cite book|last=Ushakov|first=Igor|url=https://books.google.com/books?id=4cXOAwAAQBAJ&q=quasidecimal+alphabetic+system+greek&pg=PA17|title=In the Beginning Was the Number (2)|date=22 June 2012 |publisher=Lulu.com|isbn=978-1-105-88317-0|language=en}}</ref> Jews began using a similar system ([[Hebrew numerals]]), with the oldest examples known being coins from around 100&nbsp;BC.<ref>{{Cite book|author=Chrisomalis, Stephen|title=Numerical notation : a comparative history|date=2010|publisher=Cambridge University Press|isbn=978-0-511-67683-3|location=Cambridge|pages=157|oclc=630115876|quote=The first safely dated instance in which the use of Hebrew alphabetic numerals is certain is on coins from the reign of Hasmonean king Alexander Janneus(103 to 76 BC)...}}</ref>

The Roman empire used tallies written on wax, papyrus and stone, and roughly followed the Greek custom of assigning letters to various numbers. The [[Roman numerals|Roman numerals system]] remained in common use in Europe until [[positional notation]] came into common use in the 16th&nbsp;century.<ref>{{Cite book|last=Silvercloud|first=Terry David|url=https://books.google.com/books?id=Zy-ODwAAQBAJ&q=Roman+numerals+system+remained+in+common+use&pg=PA152|title=The Shape of God: Secrets, Tales, and Legends of the Dawn Warriors|date=2007|publisher=Terry David Silvercloud|isbn=978-1-4251-0836-6|pages=152|language=en}}</ref>

The [[Maya numerals|Maya]] of Central America used a mixed base 18 and base 20 system, possibly inherited from the [[Olmec]], including advanced features such as positional notation and a [[zero]].<ref>{{citation |title=Modern Mathematics |first1=Ruric E. |last1=Wheeler |first2=Ed R. |last2=Wheeler |publisher=Kendall Hunt |year=2001 |isbn=9780787290627 |page=130 |url=https://books.google.com/books?id=azSPh9SBwwEC&pg=PA130}}.</ref> They used this system to make advanced astronomical calculations, including highly accurate calculations of the length of the solar year and the orbit of [[Venus]].<ref>{{Cite book|last=Swami|first=Devamrita|url=https://books.google.com/books?id=5JRdIkxETUsC&q=Maya+length+of+the+solar+year+and+the+orbit+of+Venus&pg=PT304|title=Searching for Vedic India|date=2002|publisher=The Bhaktivedanta Book Trust|isbn=978-0-89213-350-5|language=en|quote=Maya astronomy finely calculated both the duration of the solar year and the synodical revolution of Venus}}</ref>

The Incan Empire ran a large command economy using [[quipu]], tallies made by knotting colored fibers.<ref>{{Cite web|title=Quipu {{!}} Incan counting tool|url=https://www.britannica.com/technology/quipu|access-date=2020-07-23|website=Encyclopedia Britannica|language=en}}</ref> Knowledge of the encodings of the knots and colors was suppressed by the [[Spain|Spanish]] [[conquistador]]s in the 16th&nbsp;century, and has not survived although simple quipu-like recording devices are still used in the [[Andes|Andean]] region.

Some authorities believe that positional arithmetic began with the wide use of [[counting rods]] in China.<ref>{{Cite book|last=Chen|first=Sheng-Hong|url=https://books.google.com/books?id=K3lhDwAAQBAJ&q=positional+arithmetic+began+with+the+wide+use+of+counting+rods+in+China&pg=PA8|title=Computational Geomechanics and Hydraulic Structures|date=2018-06-21|publisher=Springer|isbn=978-981-10-8135-4|pages=8|language=en|quote=… definitely before 400 BC they possessed a similar positional notation based on the ancient counting rods.}}</ref> The earliest written positional records seem to be [[rod calculus]] results in China around 400. Zero was first used in India in the 7th century CE by [[Brahmagupta]].<ref>{{Cite web|title=Foundations of mathematics – The reexamination of infinity|url=https://www.britannica.com/science/foundations-of-mathematics|access-date=2020-07-23|website=Encyclopædia Britannica|language=en}}</ref>

The modern positional Arabic numeral system was developed by [[Indian mathematics|mathematicians in India]], and passed on to [[Islamic mathematics|Muslim mathematicians]], along with astronomical tables brought to [[Baghdad]] by an Indian ambassador around 773.<ref>{{Cite book|url=https://books.google.com/books?id=uM0sRPoABq8C&q=astronomical+tables+brought+to+Baghdad+by+an+Indian+ambassador+around+773&pg=PA626|title=The Encyclopedia Britannica|date=1899|pages=626|language=en}}</ref>

From [[India subcontinent|India]], the thriving trade between Islamic sultans and Africa carried the concept to [[Cairo]]. Arabic mathematicians extended the system to include [[Decimal|decimal fractions]], and [[Muḥammad ibn Mūsā al-Ḵwārizmī]] wrote an important work about it in the 9th&nbsp;&nbsp;century.<ref>{{Cite book|author=Struik, Dirk J. (Dirk Jan)|title=A concise history of mathematics|date=1967|publisher=Dover Publications|isbn=0-486-60255-9|edition=3d rev.|location=New York|oclc=635553}}</ref> The modern [[Arabic numerals]] were introduced to Europe with the translation of this work in the 12th&nbsp;century in Spain and [[Leonardo of Pisa]]'s ''Liber Abaci'' of 1201.<ref>{{Cite book|last=Sigler|first=Laurence|url=https://books.google.com/books?id=PilhoGJeKBUC&q=Leonardo+of+Pisa's+Liber+Abaci+of+1201|title=Fibonacci's Liber Abaci: A Translation into Modern English of Leonardo Pisano's Book of Calculation|date=2003-11-11|publisher=Springer Science & Business Media|isbn=978-0-387-40737-1|language=en}}</ref> In Europe, the complete Indian system with the zero was derived from the Arabs in the 12th&nbsp;century.<ref>{{Cite book|author=Deming, David|title=Science and technology in world history. Volume 1, The ancient world and classical civilization|date=2010|publisher=McFarland & Co|isbn=978-0-7864-5657-4|location=Jefferson, N.C.|pages=86|oclc=650873991}}</ref>

The [[binary numeral system|binary system]] (base 2) was propagated in the 17th&nbsp;century by [[Gottfried Leibniz]].<ref name=":1">{{Cite book|last=Yanushkevich|first= Svetlana N.|author-link=Svetlana Yanushkevich|title=Introduction to logic design|date=2008|publisher=CRC Press|others=Shmerko, Vlad P.|isbn=978-1-4200-6094-2|location=Boca Raton|pages=56|oclc=144226528}}</ref> Leibniz had developed the concept early in his career, and had revisited it when he reviewed a copy of the ''[[I Ching]]'' from China.<ref>{{Cite book|author=Sloane, Sarah|title=The I Ching for writers : finding the page inside you|date=2005|publisher=New World Library|isbn=1-57731-496-4|location=Novato, Calif.|pages=9|oclc=56672043}}</ref> Binary numbers came into common use in the 20th&nbsp;century because of computer applications.<ref name=":1" />

===<span id="popular"></span>Numerals in most popular systems===
{| class="wikitable" summary="Numerals in many different writing systems"
!West Arabic
! 0
! 1
! 2
! 3
! 4
! 5
! 6
! 7
! 8
! 9
|-
!Asomiya (Assamese); [[Bengali language|Bengali]]
| ০
| ১
| ২
| ৩
| ৪
| ৫
| ৬
| ৭
| ৮
| ৯
|-
! [[Devanagari]]
| ०
| १
| २
| ३
| ४
| ५
| ६
| ७
| ८
| ९
|-
!East Arabic
| ٠
| ١
| ٢
| ٣
| ٤
| ٥
| ٦
| ٧
| ٨
| ٩
|-
![[Persian language|Persian]]
| ٠
| ١
| ٢
| ٣
| ۴
| ۵
| ۶
| ٧
| ٨
| ٩
|-
! [[Gurmukhi]]
| ੦
| ੧
| ੨
| ੩
| ੪
| ੫
| ੬
| ੭
| ੮
| ੯
|-
! [[Urdu]]
| {{Urdu numeral||15}}
| {{Urdu numeral|1|15}}
| {{Urdu numeral|2|15}}
| {{Urdu numeral|3|15}}
| {{Urdu numeral|4|15}}
| {{Urdu numeral|5|15}}
| {{Urdu numeral|6|15}}
| {{Urdu numeral|7|15}}
| {{Urdu numeral|8|15}}
| {{Urdu numeral|9|15}}
|-
! [[Chinese language|Chinese]] (everyday)
| 〇
| 一
| 二
| 三
| 四
| 五
| 六
| 七
| 八
| 九
|-
! Chinese (Traditional)
| 零
| 壹
| 貳
| 叄
| 肆
| 伍
| 陸
| 柒
| 捌
| 玖
|-
!Chinese (Simplified)
|零
|壹
|贰
|叁
|肆
|伍
|陆
|柒
|捌
|玖
|-
! Chinese (Suzhou)
| 〇
| 〡
| 〢
| 〣
| 〤
| 〥
| 〦
| 〧
| 〨
| 〩
|-

! Ge'ez (Ethiopic)
|
| ፩
| ፪
| ፫
| ፬
| ፭
| ፮
| ፯
| ፰
| ፱
|-
! [[Gujarati language|Gujarati]]
| ૦
| ૧
| ૨
| ૩
| ૪
| ૫
| ૬
| ૭
| ૮
| ૯
|-
! Hieroglyphic Egyptian
| 
| 𓏺
| 𓏻
| 𓏼
| 𓏽
| 𓏾
| 𓏿
| 𓐀
| 𓐁
| 𓐂
|-
! [[Japanese numerals|Japanese]] (everyday)
| {{lang|ja|〇}}
| {{lang|ja|一}}
| {{lang|ja|二}}
| {{lang|ja|三}}
| {{lang|ja|四}}
| {{lang|ja|五}}
| {{lang|ja|六}}
| {{lang|ja|七}}
| {{lang|ja|八}}
| {{lang|ja|九}}
|-
!Japanese (formal)
|零
|壱
|弐
|参
|四
|五
|六
|七
|八
|九
|-
! [[Kannada]]
| ೦
| ೧
| ೨
| ೩
| ೪
| ೫
| ೬
| ೭
| ೮
| ೯
|-
! [[Khmer language|Khmer]] (Cambodia)
| ០
| ១
| ២
| ៣
| ៤
| ៥
| ៦
| ៧
| ៨
| ៩

|-
! [[Lao language|Lao]]
| ໐
| ໑
| ໒
| ໓
| ໔
| ໕
| ໖
| ໗
| ໘
| ໙
|-
! [[Limbu language|Limbu]]
| {{lang|lif|᥆}}
| {{lang|lif|᥇}}
| {{lang|lif|᥈}}
| {{lang|lif|᥉}}
| {{lang|lif|᥊}}
| {{lang|lif|᥋}}
| {{lang|lif|᥌}}
| {{lang|lif|᥍}}
| {{lang|lif|᥎}}
| {{lang|lif|᥏}}
|-
! [[Malayalam]]
| ൦
| ൧
| ൨
| ൩
| ൪
| ൫
| ൬
| ൭
| ൮
| ൯
|-
! [[Mongolian alphabet|Mongolian]]
| ᠐
| ᠑
| ᠒
| ᠓
| ᠔
| ᠕
| ᠖
| ᠗
| ᠘
| ᠙
|-
! [[Burmese script|Burmese]]
| ၀
| ၁
| ၂
| ၃
| ၄
| ၅
| ၆
| ၇
| ၈
| ၉
|-
! [[Oriya alphabet|Oriya]]
| ୦
| ୧
| ୨
| ୩
| ୪
| ୫
| ୬
| ୭
| ୮
| ୯
|-
! [[Roman numerals|Roman]]
|
| I
| II
| III
| IV
| V
| VI
| VII
| VIII
| IX
|-
|-
! [[Shan language|Shan]]
| ႐
| ႑
| ႒
| ႓
| ႔
| ႕
| ႖
| ႗
| ႘
| ႙
|-
! [[Sinhala numerals|Sinhala]]
|
| 𑇡
| 𑇢
| 𑇣
| 𑇤
| 𑇥
| 𑇦
| 𑇧
| 𑇨
| 𑇩
|-
! [[Tamil language|Tamil]]
| ௦
| ௧
| ௨
| ௩
| ௪
| ௫
| ௬
| ௭
| ௮
| ௯
|-
! [[Telugu language|Telugu]]
| ౦
| ౧
| ౨
| ౩
| ౪
| ౫
| ౬
| ౭
| ౮
| ౯
|-
! [[Thai numerals|Thai]]
| ๐
| ๑
| ๒
| ๓
| ๔
| ๕
| ๖
| ๗
| ๘
| ๙
|-
! [[Tibetan alphabet|Tibetan]]
| ༠
| ༡
| ༢
| ༣
| ༤
| ༥
| ༦
| ༧
| ༨
| ༩
|-
! [[New Tai Lue alphabet|New Tai Lue]]
| ᧐	
| ᧑
| 	᧒
| 	᧓
| 	᧔
| 	᧕
| 	᧖
| 	᧗
| 	᧘
| 	᧙
|-
! [[Javanese script|Javanese]]
| ꧐
| ꧑
| ꧒
| ꧓
| ꧔
| ꧕
| ꧖
| ꧗
| ꧘
| ꧙
|-
|}

===Additional numerals===
{| class="wikitable" summary="Additional numerals used in Chinese"
!
! 1
! 5
! 10
! 20
! 30
! 40
! 50
! 60
! 70
! 80
! 90
! 100
! 500
! 1000
! 10000
! 10<sup>8</sup>
|-
! [[Chinese numerals|Chinese (ordinary)]]
| 一
| 五
| 十
| 二十
| 三十
| 四十
| 五十
| 六十
| 七十
| 八十
| 九十
| 百
| 五百
| 千
| 万
| 亿
|-
! [[Chinese numerals|Chinese (financial)]]
| 壹
| 伍
| 拾
| 贰拾
| 叁拾
| 肆拾
| 伍拾
| 陆拾
| 柒拾
| 捌拾
| 玖拾
| 佰
| 伍佰
| 仟
| 萬
| 億
|-
! [[Geʽez script|Geʽez]]
| ፩ 
| ፭ 
| ፲
| ፳
| ፴
| ፵
| ፶
| ፷
| ፸
| ፹
| ፺
| ፻
| ፭፻ 
| ፲፻
| ፼
| ፼፼
|-
! [[Roman numerals|Roman]]
| I
| V
| X
| XX
| XXX
| XL
| L
| LX
| LXX
| LXXX
| XC
| C
| D
| M
| <span style="text-decoration:overline;">X</span>
|
|}