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Hindu–Arabic numeral system

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The Hindu–Arabic numeral system (also known as the Indo-Arabic numeral system,[1] Hindu numeral system, Arabic numeral system)[2][note 1] is a positional base-ten numeral system for representing integers; its extension to non-integers is the decimal numeral system, which is presently the most common numeral system.

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 system was invented between the 1st and 4th centuries by Indian mathematicians. The system was adopted in Arabic mathematics by the 9th century. It became more widely known through the writings in Arabic of the Persian mathematician Al-Khwārizmī[3] (On the Calculation with Hindu Numerals, c. 825) and Arab mathematician Al-Kindi (On the Use of the Hindu Numerals, c. 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.[4]

It is based upon ten glyphs 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.

Origins

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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.[5]

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;[6] medieval and Renaissance European mathematicians generally recognized them as Indian in origin,[7] however a few influential sources credited them to the Arabs, and they eventually came to be generally known as "Arabic numerals" in Europe.[8] According to some sources, this number system may have originated in Chinese Shang numerals (1200 BCE), which was also a decimal positional numeral system.[9]

Positional notation

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Main articles: Positional notation and 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 significant[10]). The requisite changes in reading direction are found in text that mixes left-to-right writing systems with right-to-left systems.

Symbols

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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:

Glyph comparison

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Symbol Used with scripts Numerals
0 1 2 3 4 5 6 7 8 9 Arabic, Latin, Cyrillic, and Greek Arabic numerals
٠ ١ ٢ ٣ ٤ ٥ ٦ ٧ ٨ ٩ Arabic Eastern Arabic numerals
۰ ۱ ۲ ۳ ۴ ۵ ۶ ۷ ۸ ۹ Persian / Dari / Pashto
۰ ۱ ۲ ۳ ۴ ۵ ۶ ۷ ۸ ۹ Urdu / Shahmukhi
⠚ ⠁ ⠃ ⠉ ⠙ ⠑ ⠋ ⠛ ⠓ ⠊ Braille Braille numerals
〇 一 二 三 四 五 六 七 八 九 Chinese / Japanese Chinese and Japanese numerals
영 일 이 삼 사 오 육 칠 팔 구 Korean Korean numerals (Sino cardinals)[note 2]
𑁦 𑁧 𑁨 𑁩 𑁪 𑁫 𑁬 𑁭 𑁮 𑁯 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 Tamil numerals
০ ১ ২ ৩ ৪ ৫ ৬ ৭ ৮ ৯ Eastern Nagari Bengali numerals
𐴰 𐴱 𐴲 𐴳 𐴴 𐴵 𐴶 𐴷 𐴸 𐴹 Hanifi Rohingya 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 Gujarati numerals
𑙐 𑙑 𑙒 𑙓 𑙔 𑙕 𑙖 𑙗 𑙘 𑙙 Modi Modi numerals
𑋰 𑋱 𑋲 𑋳 𑋴 𑋵 𑋶 𑋷 𑋸 𑋹 Khudabadi Khudabadi script § Numerals
୦ ୧ ୨ ୩ ୪ ୫ ୬ ୭ ୮ ୯ Odia Odia numerals
᱐ ᱑ ᱒ ᱓ ᱔ ᱕ ᱖ ᱗ ᱘ ᱙ Santali Santali numerals
𑇐 𑇑 𑇒 𑇓 𑇔 𑇕 𑇖 𑇗 𑇘 𑇙 Sharada Sharada numerals
౦ ౧ ౨ ౩ ౪ ౫ ౬ ౭ ౮ ౯ Telugu Telugu script § Numerals
೦ ೧ ೨ ೩ ೪ ೫ ೬ ೭ ೮ ೯ Kannada Kannada script § Numerals
൦ ൧ ൨ ൩ ൪ ൫ ൬ ൭ ൮ ൯ Malayalam Malayalam numerals
꯰ ꯱ ꯲ ꯳ ꯴ ꯵ ꯶ ꯷ ꯸ ꯹ Meitei Meitei script § Numerals
෦ ෧ ෨ ෩ ෪ ෫ ෬ ෭ ෮ ෯ Sinhala Sinhala numerals
𑓐 𑓑 𑓒 𑓓 𑓔 𑓕 𑓖 𑓗 𑓘 𑓙 Tirhuta Mithilakshar Maithili numerals
༠ ༡ ༢ ༣ ༤ ༥ ༦ ༧ ༨ ༩ Tibetan Tibetan numerals
᠐ ᠑ ᠒ ᠓ ᠔ ᠕ ᠖ ᠗ ᠘ ᠙ Mongolian Mongolian numerals
᥆ ᥇ ᥈ ᥉ ᥊ ᥋ ᥌ ᥍ ᥎ ᥏ Limbu Limbu script § Digits
၀ ၁ ၂ ၃ ၄ ၅ ၆ ၇ ၈ ၉ Burmese Burmese numerals
႐ ႑ ႒ ႓ ႔ ႕ ႖ ႗ ႘ ႙ Shan Shan alphabet § Numerals
០ ១ ២ ៣ ៤ ៥ ៦ ៧ ៨ ៩ Khmer Khmer numerals
๐ ๑ ๒ ๓ ๔ ๕ ๖ ๗ ๘ ๙ Thai Thai numerals
໐ ໑ ໒ ໓ ໔ ໕ ໖ ໗ ໘ ໙ Lao Lao script § Numerals
᧐ ᧑/᧚ ᧒ ᧓ ᧔ ᧕ ᧖ ᧗ ᧘ ᧙ New Tai Lue New Tai Lue script § Digits
꩐ ꩑ ꩒ ꩓ ꩔ ꩕ ꩖ ꩗ ꩘ ꩙ Cham Cham script § Numerals
𑽐 𑽑 𑽒 𑽓 𑽔 𑽕 𑽖 𑽗 𑽘 𑽙 Kawi Kawi script § Digits
꧐ ꧑ ꧒ ꧓ ꧔ ꧕ ꧖ ꧗ ꧘ ꧙ Javanese Javanese numerals
᭐ ᭑ ᭒ ᭓ ᭔ ᭕ ᭖ ᭗ ᭘ ᭙ Balinese Balinese numerals
᮰ ᮱ ᮲ ᮳ ᮴ ᮵ ᮶ ᮷ ᮸ ᮹ Sundanese Sundanese numerals

History

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Main article: History of the Hindu–Arabic numeral system

Predecessors

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The first Brahmi numerals, ancestors of Hindu-Arabic numerals, used by Ashoka in his Edicts of Ashoka c. 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.[11]

 
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 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[clarification needed] separate numerals for each of the tens (10, 20, 30, etc.).

The actual 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

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The place-value system is used in the Bakhshali manuscript, the earliest leaves being radiocarbon dated to the period 224–383 CE.[12] The development of the positional decimal system takes its origins in[clarification needed] 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.[13]

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.[14]

Medieval Islamic world

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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).[15]

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.[16] 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.[17][18]

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, and the Arab mathematician Al-Kindi, who wrote a book, On the Use of the Hindu Numerals (كتاب في استعمال العداد الهندي [kitāb fī isti'māl al-'adād al-hindī]) around 830. 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.[19] 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

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Main article: Arabic numerals
 
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 Codex Vigilanus (aka Albeldensis), an illuminated compilation of various historical documents from the Visigothic period in Spain, written in the year 976 by three monks of the Riojan monastery of San Martín de Albelda. Between 967 and 969, Gerbert of Aurillac discovered and studied Arab science in the Catalan abbeys. Later he obtained from these places the book De multiplicatione et divisione (On multiplication and division). After becoming Pope Sylvester II in the year 999, 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 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.[20][21]

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 Rechenung auff der linihen und federn (Calculating on the Lines and with a Quill) was targeted at the apprentices of businessmen and craftsmen.

Adoption in East Asia

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In AD 690, Empress Wu promulgated Zetian characters, one of which was "〇". The word is now used as a synonym for the number zero.

In China, Gautama Siddha introduced Hindu numerals with zero in 718, but Chinese mathematicians did not find them useful, as they had already had the decimal positional counting rods.[22][23]

In Chinese numerals, a circle (〇) is used to write zero in Suzhou numerals. Many historians think it was imported from Indian numerals by Gautama Siddha in 718, but some Chinese scholars think it was created from the Chinese text space filler "□".[22]

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

Spread of the Western Arabic variant

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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).

See also

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Notes

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  1. ^ Hindu was the Persian name for "Indian" in the 10th century, when the Arabs adopted the number system. The use of "Hindu" to refer to a religion was a later development.
  2. ^ 6 is 륙 in North Korea.

References

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  1. ^ Audun Holme, Geometry: Our Cultural Heritage, 2000
  2. ^ William Darrach Halsey, Emanuel Friedman (1983). Collier's Encyclopedia, with bibliography and index. 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
  3. ^ Brezina, Corona (2006), Al-Khwarizmi: The Inventor of Algebra, The Rosen Publishing Group, pp. 39–40, ISBN 978-1-4042-0513-0
  4. ^ Danna, Raffaele (2021-01-13). "Figuring Out: The Spread of Hindu-Arabic Numerals in the European Tradition of Practical Mathematics (13th–16th Centuries)". Nuncius. 36 (1): 5–48. doi:10.1163/18253911-bja10004. ISSN 0394-7394.
  5. ^ Chrisomalis 2010, pp. 194–197.
  6. ^ Smith & Karpinski 1911, Ch. 7, pp. 99–127.
  7. ^ Smith & Karpinski 1911, p. 2.
  8. ^ Of particular note is Johannes de Sacrobosco's 13th century Algorismus, which was extremely popular and influential. See Smith & Karpinski 1911, pp. 134–135.
  9. ^ Swetz, Frank (1984). "The Evolution of Mathematics in Ancient China". In Campbell, Douglas M.; Higgins, John C. (eds.). Mathematics: People, Problems, Results. Taylor & Francis. ISBN 978-0-534-02879-4.
    Lam, Lay Yong (1988). "A Chinese Genesis: Rewriting the History of Our Numeral System". Archive for History of Exact Sciences. 38 (2): 101–108. doi:10.1007/BF00348453. JSTOR 41133830.
    Lam, Lay Yong (2008). "Computation: Chinese Counting Rods". In Selin, Selaine (ed.). Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures. Springer. ISBN 978-1-4020-4559-2.
  10. ^ In 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".
  11. ^ Flegg 1984, p. 67ff..
  12. ^ Pearce, Ian (May 2002). "The Bakhshali manuscript". The MacTutor History of Mathematics archive. Retrieved 2007-07-24.
  13. ^ 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
  14. ^ Bill Casselman (February 2007). "All for Nought". Feature Column. AMS.
  15. ^ 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 ...
  16. ^ Berggren, J. Lennart (2007). "Mathematics in Medieval Islam". In Katz, Victor J. (ed.). The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook. Princeton University Press. p. 530. ISBN 978-0-691-11485-9.
  17. ^ Berggren, J. L. (2017-01-18). Episodes in the Mathematics of Medieval Islam. Springer. ISBN 978-1-4939-3780-6.
  18. ^ Berggren, J. Lennart (2007). "Mathematics in Medieval Islam". The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook. Princeton University Press. p. 518. ISBN 978-0-691-11485-9.
  19. ^ Ibn Labbān, Kūshyār (1965). Kitab fi usul hisab al-hind [Principles of Hindu Reckoning]. Translated by Levey, Martin; Petruck, Marvin. Madison: University of Wisconsin Press. p. 3. LCCN 65012106. OL 5941486M.
  20. ^ "Fibonacci Numbers". www.halexandria.org.
  21. ^ HLeonardo Pisano: "Contributions to number theory". Encyclopædia Britannica Online, 2006. p. 3. Retrieved 18 September 2006.
  22. ^ a b Qian, Baocong (1964), Zhongguo Shuxue Shi (The history of Chinese mathematics), Beijing: Kexue Chubanshe
  23. ^ Wáng, Qīngxiáng (1999), Sangi o koeta otoko (The man who exceeded counting rods), Tokyo: Tōyō Shoten, ISBN 4-88595-226-3

Bibliography

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Further reading

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