Hindu–Arabic numeral system

Sometime around 600 CE, a change began in the writing of dates in the 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 numerals (1200 BCE), which was also a decimal 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}}

{{main article|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 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 (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 ("alphabet"), 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 significantIn German, 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.

Various symbol sets are used to represent numbers in the Hindu–Arabic numeral system, most of which developed from the Brahmi numerals.

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, Cyrillic, and Greek alphabets in the table, descended from the "West Arabic numerals" which were developed in al-Andalus and the Maghreb (there are two 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).

{{main article|History of the Hindu–Arabic numeral system}}

File:Edicts of Ashoka numerals.jpg, ancestors of Hindu-Arabic numerals, used by Ashoka in his Edicts of Ashoka {{circa|250 BC}}]]

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

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 numerals are the ancestors of the Hindu–Arabic glyphs 1 to 9, but they were not used as a positional system with a 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.

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 Brahmasphuta Siddhanta contains a comparatively advanced understanding of the mathematical role of zero. The Sanskrit translation of the lost 5th century Prakrit Jaina cosmological text Lokavibhaga may preserve an early instance of the positional use of zero.Ifrah, G. The Universal History of Numbers: From prehistory to the invention of the computer. John Wiley and Sons Inc., 2000. Translated from the French by David Bellos, E.F. Harding, Sophie Wood and Ian Monk

The first dated and undisputed inscription showing the use of a symbol for zero appears on a stone inscription found at the 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}}

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 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|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, we refer to the system as "Hindu–Arabic" rather appropriately.{{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 mathematician Khwarizmi, who wrote a book, On the Calculation with Hindu Numerals in about 825 CE, and the 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 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.

{{main article|Arabic numerals}}

File:Codex Vigilanus Primeros Numeros Arabigos.jpg, 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 compilation of various historical documents from the Visigothic period in Spain, written in the year 976 CE by three monks of the Riojan monastery of San Martín de Albelda. Between 967 and 969 CE, 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 Abacus of Gerbert, by adopting tokens representing Hindu–Arabic numerals, from one to nine.

Leonardo Fibonacci brought this system to Europe. His book {{lang|la|Liber Abaci}} introduced 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|website=www.halexandria.org}}{{Cite web |date=2025-04-02 |title=Fibonacci {{!}} Biography, Sequence, & Facts {{!}} Britannica |url=https://www.britannica.com/biography/Fibonacci |access-date=2025-04-19 |website=www.britannica.com |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 numeral system, and merchants used the Abjad numerals, a system similar to the Greek numeral system and the 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 multi|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|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

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 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, but Gautama’s numeral system represented zero with a dot rather than a hollow circle, similar to the Bakhshali manuscript.{{Citation |last=Mak |first=Bill M. |title=An 8th-Century CE Indian Astronomical Treatise in Chinese: The Nine Seizers Canon by Qutan Xida |date=7 April 2023 |work=Plurilingualism in Traditional Eurasian Scholarship |pages=352–362 |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 Confucian philosophical symbol for "nothing".

Chinese and Japanese finally adopted the Hindu–Arabic numerals in the 19th century, abandoning counting rods.

The "Western Arabic" numerals as they were in common use in Europe since the Baroque period have secondarily found worldwide use together with the Latin alphabet, and even significantly beyond the contemporary spread of the Latin alphabet, intruding into the writing systems in regions where other variants of the Hindu–Arabic numerals had been in use, but also in conjunction with Chinese and Japanese writing (see Chinese numerals, Japanese numerals).

• History of mathematics
• Numeral system
• {{annotated link|List of books on history of number systems}}

{{Reflist|group=note}}

{{Reflist|30em}}

• {{cite book |last=Chrisomalis |first=Stephen |year=2010 |title=Numerical Notation: A Comparative History |publisher=Cambridge University Press |isbn=978-0-521-87818-0 }}
• {{cite book |last=Flegg |first=Graham |year=1984 |title=Numbers: Their History and Meaning |publisher=Penguin |isbn= 978-0-14-022564-8|url=https://archive.org/details/numberstheirhist0000fleg |url-access=limited }}
• {{MacTutor |mode=cs1 |year=2001 |class=HistTopics |id=Arabic_numerals |title=The Arabic numeral system}}
• {{MacTutor |mode=cs1 |year=2000 |class=HistTopics |id=Indian_numerals |title=Indian numerals}}
• {{cite book |last1=Smith |first1=David Eugene |author1-link=David Eugene Smith |last2=Karpinski |first2=Louis Charles |author2-link=Louis Charles Karpinski |title=The Hindu–Arabic Numerals |year=1911 |place=Boston |publisher=Ginn |url=https://archive.org/details/hinduarabicnumer00smitrich }}

• {{cite book |last=Menninger |first=Karl W. |year=1969 |title=Number Words and Number Symbols: A Cultural History of Numbers |publisher=MIT Press |isbn=0-262-13040-8 }}
• [http://digital.nls.uk/early-gaelic-book-collections/pageturner.cfm?id=77845307 On the genealogy of modern numerals] by Edward Clive Bayley

{{list of writing systems}}

{{Indian mathematics}}

{{Islamic mathematics}}

{{DEFAULTSORT:Hindu-Arabic Numeral System}}

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