{{Short description|Most common system for writing numbers}}
{{Use dmy dates|date=November 2024 ||cs1-dates=sy }}
{{Numeral systems|expand=Place-value notation|expand2=Hindu-Arabic numeral system}}
[[File:EgyptphoneKeypad.svg|thumb|Modern-day Arab telephone keypad with two forms of Arabic numerals: Western Arabic numerals on the left and Eastern Arabic numerals on the right]]
The '''Hindu–Arabic numeral system''' (also known as the '''Indo-Arabic numeral system''',[{{Cite book |last=Holme |first=Audun |url=https://books.google.com/books?id=zXwQGo8jyHUC&dq=%22indo+arabic+numeral+system%22&pg=PA188 |title=Geometry: Our Cultural Heritage |date=2010-09-23 |publisher=Springer Science & Business Media |isbn=978-3-642-14441-7 |language=en}}] '''Hindu numeral system''', and '''Arabic numeral system''')[{{cite book|url=https://books.google.com/books?id=uIgxAQAAIAAJ&q=%22empire+was+expanding+and+contact+was+made+with+India%22|title=Collier's Encyclopedia, with bibliography and index|author=William Darrach Halsey, Emanuel Friedman|year=1983|quote=When the Arabian empire was expanding and contact was made with India, the Hindu numeral system and the early algorithms were adopted by the Arabs}}]{{refn|group=note|[[Hindustan|Hindu]] was the Persian name for "Indian" in the 10th century, when the Arabs adopted the number system. The use of "[[Hinduism|Hindu]]" to refer to a religion was a later development.}} is a [[base ten]] (decimal) [[positional notation|positional numeral system]]. It is presently the most common decimal system.
The system was invented between the 1st and 4th centuries by [[Indian mathematics|Indian mathematicians]]. By the 9th century, the system was adopted by [[Arabic mathematics|Arabic mathematicians]] who extended it to include [[fraction (mathematics)|fractions]]. It became more widely known through the writings in [[Arabic]] of the Persian mathematician [[Al-Khwārizmī]][{{citation|last=Brezina|first=Corona|title=Al-Khwarizmi: The Inventor of Algebra|url=https://books.google.com/books?id=3Sfrxde0CXIC&pg=PA39|year=2006|publisher=The Rosen Publishing Group|isbn=978-1-4042-0513-0|pages=39–40}}] (''On the Calculation with Hindu Numerals'', {{circa|825}}) and Arab mathematician [[Al-Kindi]] (''On the Use of the Hindu Numerals'', {{circa|830}}). The system had spread to medieval Europe by the [[High Middle Ages]], notably following [[Fibonacci]]'s 13th century ''[[Liber Abaci]]''; until the evolution of the [[printing press]] in the 15th century, use of the system in Europe was mainly confined to [[Northern Italy]].[{{cite journal | last=Danna | first=Raffaele | title=Figuring Out: The Spread of Hindu-Arabic Numerals in the European Tradition of Practical Mathematics (13th–16th Centuries) | journal=Nuncius | volume=36 | issue=1 | date=13 January 2021 | issn=0394-7394 | doi=10.1163/18253911-bja10004 | pages=5–48| doi-access=free }}]
It is based upon ten [[glyph]]s representing the numbers from zero to nine, and allows representing any [[natural number]] by a unique sequence of these glyphs. The symbols (glyphs) used to represent the system are in principle independent of the system itself. The glyphs in actual use are descended from [[Brahmi numerals]] and have split into various typographical variants since the [[Middle Ages]].
These symbol sets can be divided into three main families: [[Western Arabic numerals]] used in the [[Greater Maghreb]] and in [[Europe]]; [[Eastern Arabic numerals]] used in the [[Middle East]]; and the Indian numerals in various scripts used in the [[Indian subcontinent]].
== Positional notation ==
{{main|Positional notation|0 (number)}}
The Hindu–Arabic system is designed for [[positional notation]] in a [[decimal]] system. In a more developed form, positional notation also uses a [[Decimal separator|decimal marker]] (at first a mark over the ones digit but now more commonly a decimal point or a decimal comma which separates the ones place from the tenths place), and also a symbol for "these digits recur ''[[ad infinitum]]''". In modern usage, this latter symbol is usually a [[vinculum (symbol)|vinculum]] (a horizontal line placed over the repeating digits). In this more developed form, the numeral system can symbolize any [[rational number]] using only 13 symbols (the ten digits, decimal marker, vinculum, and a prepended [[minus sign]] to indicate a [[negative number]]).
Although generally found in text written with the Arabic [[abjad]], which is written right-to-left, numbers written with these numerals place the most-significant digit to the left, so they read from left to right (though digits are not always said in order from most to least significant[In German and Dutch, a number like 21 is said like "one and twenty", as though being read from right to left. In Biblical Hebrew, this is sometimes done even with larger numbers, as in Esther 1:1, which literally says, "Ahasuerus which reigned from India even unto Ethiopia, over seven and twenty and a hundred provinces".]). The requisite changes in reading direction are found in text that mixes left-to-right writing systems with right-to-left systems.
== Symbols ==
Various symbol sets are used to represent numbers in the Hindu–Arabic numeral system, most of which developed from the [[Brahmi numeral]]s.
The symbols used to represent the system have split into various typographical variants since the [[Middle Ages]], arranged in three main groups:
* The widespread Western "[[Arabic numerals]]" used with the [[Latin alphabet|Latin]], [[Cyrillic script|Cyrillic]], and [[Greek alphabet]]s in the table, descended from the "West Arabic numerals" which were developed in [[al-Andalus]] and the [[Maghreb]] (there are two [[Typography|typographic]] styles for rendering western Arabic numerals, known as lining figures and [[text figures]]).
* The "Arabic–Indic" or "[[Eastern Arabic numerals]]" used with Arabic script, developed primarily in what is now [[Iraq]].{{Citation needed|date=February 2020}} A variant of the Eastern Arabic numerals is used in Persian and Urdu.
* The [[Indian numerals]] in use with scripts of the [[Brahmic family]] in India and Southeast Asia. Each of the roughly dozen major scripts of India has its own numeral glyphs (as one will note when perusing Unicode character charts).
=== Glyph comparison ===
{| class="wikitable sortable" style=text-align:center;
|-
! colspan="10" class=unsortable | Symbol || Used with scripts || Numerals
|-
| 0 || 1 || 2 || 3 || 4 || 5 || 6 || 7 || 8 || 9 || [[Arabic alphabet|Arabic]], [[Cyrillic script|Cyrillic]], [[Greek alphabet|Greek]], and [[Latin alphabet|Latin]] among others || [[Arabic numerals]]
|-
| ٠ || ١ || ٢ || ٣ || ٤ || ٥ || ٦ || ٧ || ٨ || ٩ || [[Arabic]] || rowspan="3" | [[Eastern Arabic numerals]]
|-
| ۰ || ۱ || ۲ || ۳ || ۴ || ۵ || ۶ || ۷ || ۸ || ۹ || [[Dari]]/[[Pashto alphabet|Pashto]]/[[Persian alphabet|Persian]]
|-
| ۰ || ۱ || ۲ || ۳ || ۴ || ۵ || ۶ || ۷ || ۸ || ۹ || [[Shahmukhi]]/[[Urdu alphabet|Urdu]]
|-
| {{Braille cell|#0}} || {{Braille cell|#1}} || {{Braille cell|#2}} || {{Braille cell|#3}} || {{Braille cell|#4}} || {{Braille cell|#5}} || {{Braille cell|#6}} || {{Braille cell|#7}} || {{Braille cell|#8}} || {{Braille cell|#9}} || [[Braille]]|| [[Braille#Derivation|Braille numerals]]
|-
| 〇 || 一 || 二 || 三 || 四 || 五 || 六 || 七 || 八 || 九 || [[Written Chinese|Chinese]]/[[Japanese writing system|Japanese]]
|scope="row"| [[Chinese numerals|Chinese]] and [[Japanese numerals]]
|-
| 영 || 일 || 이 || 삼 || 사 || 오 || 육 || 칠 || 팔 || 구 || [[Korean language|Korean]]
|[[Korean numerals]] (Sino cardinals){{refn|group=note|6 is 륙 in North Korea.}}
|-
| 𑁦 || 𑁧 || 𑁨 || 𑁩 || 𑁪 || 𑁫 || 𑁬 || 𑁭 || 𑁮 || 𑁯 || [[Brahmi script|Brahmi]] || [[Brahmi numerals]]
|-
| ० || १ || २ || ३ || ४ || ५ || ६ || ७ || ८ || ९ || [[Devanagari]] || [[Devanagari numerals]]
|-
| ௦ || ௧ || ௨ || ௩ || ௪ || ௫ || ௬ || ௭ || ௮ || ௯ || [[Tamil script|Tamil]] || [[Tamil numerals]]
|-
| ০ || ১ || ২ || ৩ || ৪ || ৫ || ৬ || ৭ || ৮ || ৯ || [[Bengali–Assamese script|Eastern Nagari]] || [[Bengali numerals]]
|-
| 𐴰 || 𐴱 || 𐴲 || 𐴳 || 𐴴 || 𐴵 || 𐴶 || 𐴷 || 𐴸 || 𐴹 || [[Hanifi Rohingya script|Hanifi Rohingya]] || {{slink|Hanifi Rohingya script|Numbers}}
|-
| ੦ || ੧ || ੨ || ੩ || ੪ || ੫ || ੬ || ੭ || ੮ || ੯ || [[Gurmukhi]] || [[Gurmukhi numerals]]
|-
| ૦ || ૧ || ૨ || ૩ || ૪ || ૫ || ૬ || ૭ || ૮ || ૯ || [[Gujarati script|Gujarati]] || [[Gujarati numerals]]
|-
| 𑙐 || 𑙑 || 𑙒 || 𑙓 || 𑙔 || 𑙕 || 𑙖 || 𑙗 || 𑙘 || 𑙙 || [[Modi script|Modi]] || [[Modi script#Unicode|Modi numerals]]
|-
| 𑋰 || 𑋱 || 𑋲 || 𑋳 || 𑋴 || 𑋵 || 𑋶 || 𑋷 || 𑋸 || 𑋹 || [[Khudabadi script|Khudabadi]] || {{slink|Khudabadi script
|Numerals}}
|-
| ୦ || ୧ || ୨ || ୩ || ୪ || ୫ || ୬ || ୭ || ୮ || ୯ || [[Odia script|Odia]] || [[Odia numerals]]
|-
| ᱐ || ᱑ || ᱒ || ᱓ || ᱔ || ᱕ || ᱖ || ᱗ || ᱘ || ᱙ || [[Ol Chiki script|Santali]] || [[Ol Chiki script#Digits|Santali numerals]]
|-
| 𑇐 || 𑇑 || 𑇒 || 𑇓 || 𑇔 || 𑇕 || 𑇖 || 𑇗 || 𑇘 || 𑇙 || [[Sharada script|Sharada]] || [[Sharada script#Numerals|Sharada numerals]]
|-
| ౦ || ౧ || ౨ || ౩ || ౪ || ౫ || ౬ || ౭ || ౮ || ౯ || [[Telugu script|Telugu]] || {{slink|Telugu script|Numerals}}
|-
| ೦ || ೧ || ೨ || ೩ || ೪ || ೫ || ೬ || ೭ || ೮ || ೯ || [[Kannada script|Kannada]] || {{slink|Kannada script|Numerals}}
|-
| ൦ || ൧ || ൨ || ൩ || ൪ || ൫ || ൬ || ൭ || ൮ || ൯ || [[Malayalam script|Malayalam]] || [[Malayalam numerals]]
|-
| ꯰ || ꯱ || ꯲ || ꯳ || ꯴ || ꯵ || ꯶ || ꯷ || ꯸ || ꯹ || [[Meitei script|Meitei]] ||{{slink|Meitei script|Numerals}}
|-
| ෦ || ෧ || ෨ || ෩ || ෪ || ෫ || ෬ || ෭ || ෮ || ෯ || [[Sinhala script|Sinhala]] || [[Sinhala numerals]]
|-
| 𑓐 || 𑓑 || 𑓒 || 𑓓 || 𑓔 || 𑓕 || 𑓖 || 𑓗 || 𑓘 || 𑓙 || [[Tirhuta script|Tirhuta Mithilakshar]] || [[Tirhuta script#Numerals|Maithili numerals]]
|-
| ༠ || ༡ || ༢ || ༣ || ༤ || ༥ || ༦ || ༧ || ༨ || ༩ || [[Tibetan script|Tibetan]] || [[Tibetan numerals]]
|-
| ᠐ || ᠑ || ᠒ || ᠓ || ᠔ || ᠕ || ᠖ || ᠗ || ᠘ || ᠙ || [[Mongolian script|Mongolian]] || [[Mongolian numerals]]
|-
| ᥆ || ᥇ || ᥈ || ᥉ || ᥊ || ᥋ || ᥌ || ᥍ || ᥎ || ᥏ || [[Limbu script|Limbu]] || {{slink|Limbu script|Digits}}
|-
| ၀ || ၁ || ၂ || ၃ || ၄ || ၅ || ၆ || ၇ || ၈ || ၉ || [[Burmese script|Burmese]] || [[Burmese numerals]]
|-
| ႐ || ႑ || ႒ || ႓ || ႔ || ႕ || ႖ || ႗ || ႘ || ႙ || [[Shan alphabet|Shan]] || {{slink|Shan alphabet|Numerals}}
|-
| ០ || ១ || ២ || ៣ || ៤ || ៥ || ៦ || ៧ || ៨ || ៩ || [[Khmer script|Khmer]] || [[Khmer numerals]]
|-
| ๐ || ๑ || ๒ || ๓ || ๔ || ๕ || ๖ || ๗ || ๘ || ๙ || [[Thai script|Thai]] || [[Thai numerals]]
|-
| ໐ || ໑ || ໒ || ໓ || ໔ || ໕ || ໖ || ໗ || ໘ || ໙ || [[Lao script|Lao]] || {{slink|Lao script|Numerals}}
|-
| ᧐ || ᧑/᧚ || ᧒ || ᧓ || ᧔ || ᧕ || ᧖ || ᧗ || ᧘ || ᧙ || [[New Tai Lue script|New Tai Lue]] || {{slink|New Tai Lue script|Digits}}
|-
| ꩐ || ꩑ || ꩒ || ꩓ || ꩔ || ꩕ || ꩖ || ꩗ || ꩘ || ꩙ || [[Cham script|Cham]] || {{slink|Cham script|Numerals}}
|-
| 𑽐 || 𑽑 || 𑽒 || 𑽓 || 𑽔 || 𑽕 || 𑽖 || 𑽗 || 𑽘 || 𑽙 || [[Kawi script|Kawi]] || {{slink|Kawi script|Digits}}
|-
| ꧐ || ꧑ || ꧒ || ꧓ || ꧔ || ꧕ || ꧖ || ꧗ || ꧘ || ꧙ || [[Javanese script|Javanese]] || [[Javanese numerals]]
|-
| ᭐ || ᭑ || ᭒ || ᭓ || ᭔ || ᭕ || ᭖ || ᭗ || ᭘ || ᭙ || [[Balinese script|Balinese]] || [[Balinese numerals]]
|-
| ᮰ || ᮱ || ᮲ || ᮳ || ᮴ || ᮵ || ᮶ || ᮷ || ᮸ || ᮹ || [[Sundanese script|Sundanese]] || [[Sundanese numerals]]
|}
==History==
{{main|History of the Hindu–Arabic numeral system}}
Sometime around 600 CE, a change began in the writing of dates in the [[Brahmi numerals|Brāhmī]]-derived scripts of India and Southeast Asia, transforming from an additive system with separate numerals for numbers of different magnitudes to a positional place-value system with a single set of glyphs for 1–9 and a dot for zero, gradually displacing additive expressions of numerals over the following several centuries.{{sfn|Chrisomalis|2010|pp=194–197}}
When this system was adopted and extended by medieval Arabs and Persians, they called it ''al-ḥisāb al-hindī'' ("Indian arithmetic"). These numerals were gradually adopted in Europe starting around the 10th century, probably transmitted by Arab merchants;{{sfn|Smith|Karpinski|1911|loc=[https://archive.org/details/hinduarabicnumer00smitrich/page/99 Ch. 7, {{pgs|99–127}}]}} medieval and Renaissance European mathematicians generally recognized them as Indian in origin,{{sfn|Smith|Karpinski|1911|loc=[https://archive.org/details/hinduarabicnumer00smitrich/page/2/ {{pgs|2}}]}} however a few influential sources credited them to the Arabs, and they eventually came to be generally known as "Arabic numerals" in Europe.[Of particular note is [[Johannes de Sacrobosco]]'s 13th century ''Algorismus'', which was extremely popular and influential. See {{harvnb|Smith| Karpinski|1911|loc=[https://archive.org/details/hinduarabicnumer00smitrich/page/134/ {{pgs|134–135}}]}}.] According to some sources, this number system may have originated in Chinese [[Shang dynasty|Shang]] numerals (1200 BCE), which was also a [[decimal]] [[positional notation|positional]] numeral system.[{{cite book |last=Swetz |first=Frank |year=1984 |chapter=The Evolution of Mathematics in Ancient China |chapter-url=https://books.google.com/books?id=PFNsm_IaymYC&pg=PA28 |title=Mathematics: People, Problems, Results |editor1-last=Campbell |editor1-first=Douglas M. |editor2-last=Higgins |editor2-first=John C. |publisher=Taylor & Francis |isbn=978-0-534-02879-4}} {{pb}} {{Cite journal |last=Lam |first=Lay Yong |date=1988 |title=A Chinese Genesis: Rewriting the History of Our Numeral System |journal=Archive for History of Exact Sciences |volume=38 |issue=2 |pages=101–108 |doi=10.1007/BF00348453 |jstor=41133830 }} {{pb}} {{cite encyclopedia |last=Lam |first=Lay Yong |author-link=Lam Lay Yong |year=2008 |url=https://books.google.com/books?id=kt9DIY1g9HYC&pg=RA1-PA197 |title=Computation: Chinese Counting Rods |encyclopedia=Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures |editor-last=Selin |editor-first=Selaine |publisher=Springer |isbn=978-1-4020-4559-2}}]
===Predecessors===
[[File:Edicts of Ashoka numerals.jpg|thumb|The first [[Brahmi numerals]], ancestors of Hindu-Arabic numerals, used by [[Ashoka]] in his [[Edicts of Ashoka]] {{circa|250 BCE}}]]
The [[Brahmi numerals]] at the basis of the system predate the [[Common Era]]. They replaced the earlier [[Kharosthi numerals]] used since the 4th century BCE. Brahmi and Kharosthi numerals were used alongside one another in the [[Maurya Empire]] period, both appearing on the 3rd century BCE [[edicts of Ashoka]].{{sfn|Flegg|1984|loc=[https://archive.org/details/numberstheirhist0000fleg/page/67 {{pgs|67ff.}}]}}
[[File:Hindustani_numerals.svg|thumb|Nagari and Devanagari numerals with handwritten variants]]
[[Buddhist]] inscriptions from around 300 BCE use the symbols that became 1, 4, and 6. One century later, their use of the symbols that became 2, 4, 6, 7, and 9 was recorded. These [[Brahmi numeral]]s are the ancestors of the Hindu–Arabic glyphs 1 to 9, but they were not used as a [[positional system]] with a [[0 (number)|zero]], and there were rather{{Clarify|reason=This word is vague.|date=October 2024}} separate numerals for each of the tens (10, 20, 30, etc.).
The modern numeral system, including positional notation and use of zero, is in principle independent of the glyphs used, and significantly younger than the Brahmi numerals.
===Development===
The place-value system is used in the [[Bakhshali manuscript]], the earliest leaves being radiocarbon dated to the period 224–383 CE.[
{{cite web|title=The Bakhshali manuscript|author=Pearce, Ian|publisher=The MacTutor History of Mathematics archive|url=http://www-history.mcs.st-andrews.ac.uk/HistTopics/Bakhshali_manuscript.html|date=May 2002|access-date=24 July 2007}}] The development of the positional decimal system {{Clarify|reason=The meaning of this phrase is unclear.|date=October 2024|text=takes its origins in}} [[Indian mathematics]] during the [[Gupta period]]. Around 500, the astronomer [[Aryabhata]] uses the word ''kha'' ("emptiness") to mark "zero" in tabular arrangements of digits. The 7th century ''[[Brahmasphutasiddhanta|Brahmasphuta Siddhanta]]'' contains a comparatively advanced understanding of the mathematical role of [[0 (number)|zero]]. The Sanskrit translation of the lost 5th century Prakrit [[Jain cosmology|Jaina cosmological]] text ''[[Lokavibhaga]]'' may preserve an early instance of the positional use of zero.[Ifrah, G. (2000). ''The Universal History of Numbers: From Prehistory to the Invention of the Computer''. Translated from the French by David Bellos, E. F. Harding, Sophie Wood and Ian Monk. John Wiley and Sons.]
The first dated and undisputed inscription showing the use of a symbol for zero appears on a stone inscription found at the [[Chaturbhuj Temple, Gwalior|Chaturbhuja Temple]] at [[Gwalior]] in India, dated 876 CE.[{{cite web|url=https://www.ams.org/featurecolumn/archive/india-zero.html |title=All for Nought |work=Feature Column |author=Bill Casselman |author-link=Bill Casselman (mathematician) |publisher=AMS |date=February 2007}}]
====Medieval Islamic world====
These Indian developments were taken up in [[Islamic mathematics]] in the 8th century, as recorded in [[al-Qifti]]'s ''Chronology of the scholars'' (early 13th century).[[[al-Qifti]]'s ''Chronology of the scholars'' (early 13th century):
]... a person from India presented himself before the [[Caliph]] [[al-Mansur]] in the year 776 who was well versed in the siddhanta method of calculation related to the movement of the heavenly bodies, and having ways of calculating equations based on the half-chord [essentially the sine] calculated in half-degrees ... Al-Mansur ordered this book to be translated into Arabic, and a work to be written, based on the translation, to give the Arabs a solid base for calculating the movements of the planets ...
In 10th century [[Mathematics in medieval Islam|Islamic mathematics]], the system was extended to include fractions, as recorded in a treatise by [[Abbasid Caliphate]] mathematician [[Abu'l-Hasan al-Uqlidisi]], who was the first to describe positional decimal fractions.[{{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|editor-link=Victor J. Katz|publisher=Princeton University Press | year=2007 | isbn=978-0-691-11485-9 | page=530 }}] According to J. L. Berggren, the Muslims were the first to represent numbers as we do since they were the ones who initially extended this system of numeration to represent parts of the unit by decimal fractions, something that the Hindus did not accomplish. Thus, the system is accurately described as "Hindu–Arabic".[{{Cite book |last=Berggren |first=J. L. |url=https://books.google.com/books?id=I-jwDQAAQBAJ&dq=However%2C+the+Hindus+did+not+extend+this+system+to+represent+parts+of+the+unit+by&pg=PA30 |title=Episodes in the Mathematics of Medieval Islam |date=18 January 2017 |publisher=Springer |isbn=978-1-4939-3780-6 |language=en}}][{{cite book | first=J. Lennart | last=Berggren | title=The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook | chapter=Mathematics in Medieval Islam | publisher=Princeton University Press | year=2007 | isbn=978-0-691-11485-9 | page=518 }}]
The numeral system came to be known to both the [[Persian people|Persian]] mathematician [[Khwarizmi]], who wrote a book, ''On the Calculation with Hindu Numerals'' in about 825 CE, and the [[Arab people|Arab]] mathematician [[Al-Kindi]], who wrote a book, ''On the Use of the Hindu Numerals'' ({{lang|ar|كتاب في استعمال العداد الهندي}} [''kitāb fī isti'māl al-'adād al-hindī'']) around 830 CE. [[Persian people|Persian]] scientist [[Kushyar Gilani]], who wrote ''Kitab fi usul hisab al-hind ([[Principles of Hindu Reckoning]])'' is one of the oldest surviving manuscripts using the Hindu numerals.[{{cite book |first=Kūshyār |last=Ibn Labbān |author-link=Kushyar Gilani |translator-first1=Martin |translator-last1=Levey |translator-link1=Martin Levey |translator-first2=Marvin |translator-last2=Petruck |trans-title=[[Principles of Hindu Reckoning]] |title=Kitab fi usul hisab al-hind |page=3 |location=Madison |publisher=University of Wisconsin Press |date=1965 |isbn=978-0-299-03610-2 |lccn=65012106 |ol=OL5941486M |url=https://archive.org/details/kushyaribnlabban0000mart/page/n5 |url-access=registration}}] These books are principally responsible for the diffusion of the Hindu system of numeration throughout the [[Islamic world]], and ultimately also to Europe.
===Adoption in Europe===
{{Main|Arabic numerals}}
[[File:Codex Vigilanus Primeros Numeros Arabigos.jpg|thumb|right|The Arabic numeral system first appeared in Europe in the Spanish ''[[Codex Vigilanus]]'', year 976.]]
In Christian Europe, the first mention and representation of Hindu–Arabic numerals (from one to nine, without zero), is in the ''{{lang|la|[[Codex Vigilanus]]}}'' (aka ''Albeldensis''), an [[illuminated manuscript|illuminated]] compilation of various historical documents from the [[Visigoths|Visigothic]] period in [[Hispania|Spain]], written in the year 976 CE by three monks of the [[La Rioja (Spain)|Riojan]] monastery of [[San Martín de Albelda]]. Between 967 and 969 CE, [[Pope Sylvester II|Gerbert of Aurillac]] discovered and studied Arab science in the Catalan abbeys. Later he obtained from these places the book ''{{lang|la|De multiplicatione et divisione}}'' (''On Multiplication and Division''). After becoming [[Pope Sylvester II]] in the year 999 CE, he introduced a new model of [[abacus]], the so-called [[Pope Sylvester II#Abacus and numerals|Abacus of Gerbert]], by adopting tokens representing Hindu–Arabic numerals, from one to nine.
[[Fibonacci|Leonardo Fibonacci]] brought this system to Europe. His book ''{{lang|la|[[Liber Abaci]]}}'' introduced [[Liber Abaci#Modus Indorum|Modus Indorum]] (the method of the Indians), today known as Hindu–Arabic numeral system or base-10 positional notation, the use of zero, and the decimal place system to the Latin world. The numeral system came to be called "Arabic" by the Europeans. It was used in European mathematics from the 12th century, and entered common use from the 15th century to replace [[Roman numerals]].[{{cite web|url=http://www.halexandria.org/dward093.htm|title=Fibonacci Numbers|publisher=The Library of Halexandria}][{{Cite web |title=Fibonacci {{!}} Biography, Sequence, & Facts |url=https://www.britannica.com/biography/Fibonacci |access-date=2025-04-19 |website=[[Encyclopædia Britannica]] |date=2025-04-02 |language=en}}]
The familiar shape of the Western Arabic glyphs as now used with the Latin alphabet (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) are the product of the late 15th to early 16th century, when they entered early [[typesetting]]. Muslim scientists used the [[Babylonian numerals|Babylonian numeral system]], and merchants used the [[Abjad numerals]], a system similar to the [[Greek numerals|Greek numeral system]] and the [[Hebrew numerals|Hebrew numeral system]]. Similarly, Fibonacci's introduction of the system to Europe was restricted to learned circles. The credit for first establishing widespread understanding and usage of the decimal positional notation among the general population goes to [[Adam Ries]], an author of the [[German Renaissance]], whose 1522 ''{{lang|de|Rechenung auff der linihen und federn}}'' (Calculating on the Lines and with a Quill) was targeted at the apprentices of businessmen and craftsmen.
File:Houghton Typ 520.03.736 - Margarita philosophica.jpg|{{lang|de|[[Gregor Reisch]]}}, ''{{lang|la|Madame Arithmatica}}'', 1503
File:Rechentisch.png|A {{Interlanguage link|calculation table|de|3=Rechnen auf Linien}}, used for arithmetic using [[Roman numerals]]
File:Rechnung auff der Linihen und Federn.JPG|{{lang|de|[[Adam Ries]]}}, ''{{lang|de|Rechenung auff der linihen und federn}}'', 1522
File:Köbel Böschenteyn 1514.jpg|Two arithmetic books published in 1514 – {{lang|de|Köbel}} (left) using a calculation table and {{lang|de|Böschenteyn}} using numerals
File:Rechnung auff der linihen 1525 Adam Ries.PNG|{{lang|de|[[Adam Ries]]}}, ''{{lang|de|Rechenung auff der linihen und federn}}'' (2nd Ed.), 1525
File:1543 Robert Recorde.PNG|[[Robert Recorde]], ''The ground of artes'', 1543
File:Peter Apian 1544.PNG|{{lang|de|[[Petrus Apianus|Peter Apian]]}}, ''{{lang|de|Kaufmanns Rechnung}}'', 1527
File:Adam riesen.jpg|{{lang|de|[[Adam Ries]]}}, ''{{lang|de|Rechenung auff der linihen und federn}}'' (2nd Ed.), 1525
=== Adoption in East Asia ===
The '〇' is used to write zero in [[Suzhou numerals]], which is the only surviving variation of the [[rod numeral]] system. The ''[[Mathematical Treatise in Nine Sections]]'', written by [[Qin Jiushao]] in 1247, is the oldest surviving Chinese mathematical text to use the character ‘〇’ for zero.[{{Cite web |title=Mathematics in the Near and Far East |url=http://grmath4.phpnet.us/istoria/the_history_of%20math_greece/the_history_of%20math_greece_3-5.pdf |url-status=live |archive-url=https://web.archive.org/web/20131104120005/http://grmath4.phpnet.us/istoria/the_history_of%20math_greece/the_history_of%20math_greece_3-5.pdf |archive-date=4 November 2013 |access-date=7 June 2012 |website=grmath4.phpnet.us |page=262}}]
The origin of using the character '〇' to represent zero is unknown. [[Gautama Siddha]] introduced Hindu numerals with zero in 718 CE, but [[Chinese mathematics|Chinese mathematicians]] did not find them useful, as they already had the decimal positional [[counting rods]].[{{Citation |last=Qian |first=Baocong |title=Zhongguo Shuxue Shi (The history of Chinese mathematics) |year=1964 |place=Beijing |publisher=Kexue Chubanshe}}][{{Citation
| title=Sangi o koeta otoko (The man who exceeded counting rods)
| last=Wáng
| first=Qīngxiáng
| isbn=4-88595-226-3
| publisher=Tōyō Shoten
| place=Tokyo
| year=1999
}}] Some historians suggest that the use of '〇' for zero was influenced by Indian numerals imported by Gautama,[{{Citation |last=Mak |first=Bill M. |chapter=An 8th-Century CE Indian Astronomical Treatise in Chinese: The Nine Seizers Canon by Qutan Xida |date=7 April 2023 |title=Plurilingualism in Traditional Eurasian Scholarship |pages=352–362 |chapter-url=https://brill.com/display/book/9789004527256/BP000030.xml |access-date=27 March 2025 |publisher=Brill |language=en |doi=10.1163/9789004527256_031 |isbn=978-90-04-52725-6|doi-access=free }}]
An alternative hypothesis proposes that the use of '〇' to represent zero arose from a modification of the Chinese text space filler "□", making its resemblance to Indian numeral systems purely coincidental. Others think that the Indians acquired the symbol '〇' from China, because it resembles a [[Confucianism|Confucian]] philosophical symbol for "nothing".