Hindu–Arabic numeral system

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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,

integers; its extension to non-integers is the decimal numeral system
, which is presently the most common numeral system.

The system was invented between the 1st and 4th centuries by

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:

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

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] This Indian numeral system was the first featuring the combination of ciphering, positional notation, zero, and a decimal base.

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 BC), which was also a decimal positional numeral system.[9]

Positional notation

Main articles:
0 (number)

The Hindu–Arabic system is designed for

minus sign to indicate a negative number
).

Although generally found in text written with the Arabic abjad ("alphabet"), numbers written with these numerals also 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

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] A variant of the Eastern Arabic numerals is used in Persian and Urdu.
  • 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

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
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
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
᠐ ᠑ ᠒ ᠓ ᠔ ᠕ ᠖ ᠗ ᠘ ᠙ Mongolian Mongolian 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

Main article: History of the Hindu–Arabic numeral system

Predecessors

The first Brahmi numerals, ancestors of Hindu-Arabic numerals, used by Ashoka in his Edicts of Ashoka c. 250 BC

The

Kharosthi numerals used since the 4th century BC. Brahmi and Kharosthi numerals were used alongside one another in the Maurya Empire period, both appearing on the 3rd century BC edicts of Ashoka.[11]

Nagari and Devanagari numerals with handwritten variants

zero
, and there were rather 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

The place-value system is used in the

zero. The Sanskrit translation of the lost 5th century Prakrit Jaina cosmological text Lokavibhaga
may preserve an early instance of 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.[14]

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).[15]

In 10th century

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. Breggren, 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

Islamic world
and ultimately also to Europe.

Adoption in Europe

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

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

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

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

The "Western Arabic" numerals as they were in common use in Europe since the

Japanese writing (see Chinese numerals, Japanese numerals
).

See also

Notes

  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.

References

  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. ISBN 978-1-4042-0513-0
  4. 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. 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. ISBN 978-0-691-11485-9
    .
  17. ISBN 978-1-4939-3780-6
    .
  18. ISBN 978-0-691-11485-9
    .
  19. ^ Martin Levey and Marvin Petruck, Principles of Hindu Reckoning, translation of Kushyar ibn Labban Kitab fi usul hisab al-hind, p. 3, University of Wisconsin Press, 1965
  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. ISBN 4-88595-226-3

Bibliography

Further reading

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