Hindu numeral system

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The Hindu–Arabic numeral system[1] (also called the Arabic numeral system or Hindu numeral system)[2] a positional decimal numeral system, is the most common system for the symbolic representation of numbers in the world. It was invented between the 1st and 4th centuries by Indian mathematicians. The system was adopted by Arabic mathematicians (Muḥammad ibn Mūsā al-Khwārizmī's c. 825 book On the Calculation with Hindu Numerals and Al-Kindi's c. 830 volumes On the Use of the Hindu Numerals) by the 9th century. It later spread to medieval Europe by the High Middle Ages.

The system is based upon ten (originally nine) different 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: Arabic numerals used in the Greater Maghreb and in Europe, Eastern Arabic numerals (also called "Indic numerals") used in the Middle East, and the Indian numerals used in the Indian subcontinent.

This numerical system is still used worldwide today.

Etymology

The Hindu-Arabic numerals were invented by mathematicians in India. Perso-Arabic mathematicians called them "Hindu numerals" (where "Hindu" meant Indian). Later they came to be called "Arabic numerals" in Europe, because they were introduced to the West by Arab merchants.[3]

Positional notation

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 usually 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 dash 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. 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. A variant of the Eastern Arabic numerals is used in Persian and Urdu.
  • The Hindu 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

# # # # # # # # # # Script See
0 1 2 3 4 5 6 7 8 9 Latin script Arabic numerals
〇/零 一 二 三 四 五 六 七 八 九 East Asia Chinese numerals, Japanese numerals, Korean numerals
ο/ō Αʹ Βʹ Γʹ Δʹ Εʹ Ϛʹ Ζʹ Ηʹ Θʹ Modern Greek Greek numerals
א ב ג ד ה ו ז ח ט Hebrew Hebrew numerals
० १ २ ३ ४ ५ ६ ७ ८ ९ Devanagari Indian numerals
૦ ૧ ૨ ૩ ૪ ૫ ૬ ૭ ૮ ૯ Gujarati
੦ ੧ ੨ ੩ ੪ ੫ ੬ ੭ ੮ ੯ Gurmukhi
༠ ༡ ༢ ༣ ༤ ༥ ༦ ༧ ༨ ༩ Tibetan
০ ১ ২ ৩ ৪ ৫ ৬ ৭ ৮ ৯ Assamese / Bengali / Sylheti Bengali-Assamese numerals
୦ ୧ ୨ ୩ ୪ ୫ ୬ ୭ ୮ ୯ Odia
൦ ൧ ൨ ൩ ൪ ൫ ൬ ൭ ൮ ൯ Malayalam
௦ ௧ ௨ ௩ ௪ ௫ ௬ ௭ ௮ ௯ Tamil Tamil numerals
0 ౧ ౨ ౩ ౪ ౫ ౬ ౭ ౮ ౯ Telugu
೦ ೧ ೨ ೩ ೪ ೫ ೬ ೭ ೮ ೯ Kannada
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
၀ ၁ ၂ ၃ ၄ ၅ ၆ ၇ ၈ ၉ Burmese
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 (Farsi) / Dari / Pashto
۰ ۱ ۲ ۳ ۴ ۵ ۶ ۷ ۸ ۹ Urdu / Shahmukhi

As in many numbering systems, the numbers 1, 2, and 3 represent simple tally marks; 1 being a single line, 2 being two lines (now connected by a diagonal) and 3 being three lines (now connected by two vertical lines). After three, numbers tend to become more complex symbols (examples are the Chinese numerals and Roman numerals). Theorists believe that this is because it becomes difficult to instantaneously count objects past three.[4]

History

Main article: History of the Hindu–Arabic numeral system

Despite being described as the "Hindu–Arabic numeral system", the corpus incorporates elements from earlier: Egyptian (decimal system with a glyph for Zero), Babylonian (Positional notation), Hellenistic (ο/ō glyphs for Zero), and Chinese (notation for fractions) works; in addition to a mathematical methodology developed by Indian mathematicians, and in use extensively throughout India, before being adopted by Perso-Arabic mathematicians in Baghdad.

Predecessors

The Brahmi numerals at the basis of the system predate the Common Era. They replaced the earlier 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.[5]

Buddhist inscriptions from around 300 BC 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 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.

Possible Chinese origin

File:Kushyar ibn Labban division.GIF

10th century Kushyar ibn Labban division
File:Sunzi division.GIF
Sunzi division algorithm of 400AD
[6]
File:AL Khwarizmi division.GIF

Khwarizmi division of 825AD, completely identical to Sun Zi division algorithm[7]

Singaporean historian of mathematics Lam Lay Yong (National University of Singapore) claims that the computation in Kitab al-Fusul fi al-Hisab al Hindi (925) by al-Uqlidisi, and another Latin translation of the Arab manuscript written by the Persian mathematician Khwarizmi (825), are completely identical to algorithms for square root extraction,[8] addition, subtraction,[9] multiplication and division[10] in the rod calculus described in Sunzi Suanjing, which was written five centuries earlier. Yong claims that these methods are too identical to be explained by independent development, and promotes a theory of Chinese origin of the Hindu-Arabic numerals.[11]

Development

File:Per galera.PNG
The "Galley" method of division.

The place-value system is used in the Bakhshali Manuscript. Although date of the composition of the manuscript is uncertain, the language used in the manuscript indicates that it could not have been composed any later than 400.[12] The development of the positional decimal system takes its origins in Hindu 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 positional use of zero.[13]

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).[14]

The numeral system came to be known to both the Perso-Arabic mathematician Khwarizmi, who wrote a book, On the Calculation with Hindu Numerals in about 825, and the Arab mathematician Al-Kindi, who wrote four volumes, On the Use of the Hindu Numerals (كتاب في استعمال العداد الهندي [kitāb fī isti'māl al-'adād al-hindī]) around 830. These earlier texts did not use the Hindu numerals. Kushyar ibn Labban who wrote Kitab fi usul hisab al-hind (Principles of Hindu Reckoning) is one of the oldest surviving manuscripts using the Hindu numerals.[15] These books are principally responsible for the diffusion of the Hindu system of numeration throughout the Islamic world and ultimately also to Europe.

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.[16]

In 10th century Islamic mathematics, the system was extended to include fractions, as recorded in a treatise by Syrian mathematician Abu'l-Hasan al-Uqlidisi in 952–953.[17]

Adoption in Europe

File:Nicolaus Kesler, about 1486.PNG
The bottom row shows the numeral glyphs as they appear in type in German incunabula (Nicolaus Kesler, Basel, 1486)
Main article: Arabic numerals

In Christian Europe, the first mention and representation of Hindu-Arabic numerals (from one to nine, without zero), is in the Codex Vigilanus, 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-Arab numerals, from one to nine.

Leonardo Fibonacci brought this system to Europe. His book Liber Abaci introduced Arabic numerals, 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. Robert of Chester translated the Latin into English.[citation needed]

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 enter 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 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.[18][19]

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 "□".[18]

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

Spread of the Western Arabic variant

File:EgyptphoneKeypad.jpg
An Arab telephone keypad with both the Western "Arabic numerals" and the Arabic "Arabic–Indic numerals" variants.

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).

Indian numerals

The most complex decimal system of all, where there is such complete fusion and inflection between the tens and unit numbers that it becomes necessary to learn individually the exact form of every number from 1-100, above the usual necessity of learning the unit words 1-9 and tens words 10-100. The only languages in the world which have a number system of this kind are the Indo-Aryan languages of North India.

This is in sharp contrast with all the other languages in the world other than the Indo-Aryan languages of North India. In all the other languages, it is necessary to learn by heart at the most the numbers from 1-10, or from 1-19, and the tens forms (20,30,40,50,60,70,80,90). All the numbers between 21 and 99 are formed from these numbers.

Large numbers

As early as the Vedic Samhitas, we had words in India for very high numbers. The Yajurveda (Yaj. 17.2) lists the following words for numbers from ten (101 or 10) to one trillion (1012 or 1,000,000,000,000): 101: daśa. 102: śata. 103: sahasra. 104: ayuta. 105: niyuta. 106: prayuta. 107: arbuda. 108: nyarbuda. 109: samudra. 1010: madhya. 1011: anta. 1012: parārdha.

For ritual purposes, the enumeration in this hymn stops at 1012.

The Lalitavistara, a Buddhist text, actually describes an even more elaborate system (where some of the above words from the Yajurveda are now replaced by other words, and all the names are given in multiples of hundred. Here in fact some of the above words, like ayuta and niyuta, are given higher values): 103: sahasra. 105: lakṣa. 107: koṭi. 109: ayuta. 1011: niyuta. 1013: kaṅkara. 1015: vivara. 1017: akṣobhya. 1019: vivāha. 1021: utsāṅga. 1023: bahula. 1025: nāgabala. 1027: tiṭilambha. 1029: vyavasthānaprajñāpti. 1031: hetuhila. 1033: karaphū. 1035: hetvindriya. 1037: samāptalambha. 1039: gaṇanāgati. 1041: niravadya. 1043: mudrābala. 1045: sarvabala. 1047: visaṁjñāgati. 1049: sarvasaṁjña. 1051: vibhūtaṅgamā. 1053. tallakṣaṇa.

The text does not stop there: it points out that this is just the first of a series of nine counting systems that can be expanded geometrically, and then goes on to mention the names of the culmination points of each of the nine systems (starting with the number 1053 above, as tallakṣaṇa, dhvajāgravatī, dhvajāgraniśāmaṇī, vāhanaprajñapti, iṅgā, kuruṭu, kuruṭāvi, sarvanikṣepa and agrasārā), culminating in a large number, 10421, or one followed by 421 zeroes! This text, and many other Sanskrit texts, go even further in indulging in flights of fantasy involving even higher numbers.

In modern American numbers, a number such as 123,456,789 would be "one-twenty-three million, four-fifty-six thousand, seven-eighty-nine". As per Indian numbers in Sanskrit, this would even now be put as 12,34,56,789 or "twelve crores, thirty-four lakhs, fifty-six thousand, seven (hundred and) eighty-nine".

Reference: [1] [archive]

Nubmers 21-100

Indian languages are probably the only languages in the world where anyone learning the language (any North Indian Indo-Aryan language) necessarily finds that he has to individually learn or memorize every single number from one to hundred. There is such complete fusion and inflection between the tens and unit numbers that it becomes necessary to learn individually the exact form of every number from 1-100.

The same irregularity or inflectional complexity can be seen in the formation of the numbers between 21 and 99 in all the Indo-Aryan languages of North India (right up to Kashmiri in the extreme north, and going so far westwards as to influence the Pashto language in the northwest which, although it belongs to the Iranian branch, has also been influenced by the Indo-Aryan cerebral sounds), but is found nowhere else outside the sphere of North India .

  • Thus, Sinhalese texts provide us with evidence missing in North India itself. Sinhalese is doubtless a treasure-house of clues to the most archaic stages of Indo-European, often giving us clues to even older stages than Sanskrit (e.g. the word watura for "water", as in Germanic English water and Hittite watar). These clues are not recognized because of the blinkers of the AIT, which treats all "Indo-Aryan" languages (i.e. Indo-European languages native to India) as belonging to one branch which entered India in its earliest form as the Vedic Sanskrit language. Orthodox opponents of the AIT, who also want to accord primacy to the Vedic language, also adopt these blinkers.

In this second stage, therefore, it is clear that there was a vigesimal-effect where only the numbers 11-19 acquired distinctly inflected forms but not the other in-between numbers from 21-99. This second stage of development of the Indo-European number system is not found recorded in any text or document in North India because the older Sanskrit numbers of the first stage had become frozen in form and the Prakrits are recorded from a much later post-Buddhist period in the second half of the 1st millennium BCE, long after the departure of the other branches of Indo-European languages westwards from India, and after the diffusion of the Vedic Sanskrit culture to the Dravidian South, all of which must have taken place at a point of time when the Indo-Aryan languages of the North still had a numeral system at the second stage of development.

The third stage of development of the Indo-European number system, where the number system continued to become more and more subject to inflection and fusion between the tens-forms and the unit-forms, and the inflection in the formation of compound numbers spread to all the numbers from 11-99, is found in its earliest forms in most of the Prakrits and much more so in the modern Indo-Aryan languages of North India. In this stage, all the compound numbers between 10 and 100 acquired distinct forms with fusion and inflection between the tens and units words.

Therefore, the area of North India was home to the first stage of development of the Indo-European number system (as represented by Vedic and Classical Sanskrit, and perhaps colloquial Sinhalese?), as well as to the third stage, both of which are found only in North India, while all the other branches of Indo-European languages outside North India (include literary Sinhalese) represent the second stage. This clearly indicates that the Original Homeland of all these languages was in North India, and they migrated from India during a period when the Indo-Aryan languages of the North were at the second stage, and shared a similar vigesimal-affected decimal system.

As we saw, the first and third stages of development of the Indo-European decimal number system, as shown by what we have called the vigesimal-effect (i.e. where the numbers 11-19 are formed in a distinctly different way from the later sets like 21-29, 31-39, etc.), are found only in North India, and the second stage is found in all the branches outside North India (and therefore must have logically existed in North India in an intervening period, even if not recorded), shows that the Original Homeland of all these Indo-European languages was in North India... All this constitutes one more piece of very strong evidence for the OIT and one more nail in the coffin of the AIT.

Reference: [2] [archive]

Talageri notes that four stages of the decimal system can be defined.[20]

  • 1. Stage 1: 11 words for 1-10 and 100.
  • 2. Stage 2: 19 words for 1-10, 20-90 and 100.
  • 3. Stage 3: 28 words for 1-20, 30-90 and 100.
  • 4. Stage 4: 100 words for 1-100.

Stage 1 is not found in IE, but it may be surmised that this stage may have existed in PIE or pre-PIE. It is found in many languages of the world (typically Chinese or Tibetan).

Stage 2 is Found in the earliest languages Tocharian, spoken Sinhalese, Sanskrit.

Stage 3 is found in all the other 9 living branches of IE outside India as well as in the one Indo-Aryan language outside North India, Literary Sinhalese. And also in the Dravidian languages. Except for stray other languages, no other entire language family in the world is in stage three.

Stage 4 , The modern Indo-Aryan languages of North India are the only languages in the world belonging to Stage 4. • In this stage, not only are there separate words for 1-19, the tens 20-90 and 100, but each and every one of the other numbers in between also have to be separately learnt since the tens and units words are all fused together arbitrarily and irregularly[21]

See also

Notes

  1. ↑ David Eugene Smith and Louis Charles Karpinski, The Hindu–Arabic Numerals [archive], 1911
  2. ↑ William Darrach Halsey, Emanuel Friedman (1983). Collier's Encyclopedia, with bibliography and index [archive]. 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. ↑ Rowlett, Russ (2004-07-04), Roman and "Arabic" Numerals [archive], University of North Carolina at Chapel Hill, retrieved 2009-06-22
  4. ↑ Language may shape human thought [archive], New Scientist, news service, Celeste Biever, 19:00 19 August 2004.
  5. ↑ Flegg (2002), p. 6ff.
  6. ↑ Lam Lay Yong: "The Development of Hindu-Arabic and Traditional Chinese Arithmetic", Chinese Science 13 (1996) p45 diagram i to viii
  7. ↑ Lam Lay Yong, "The Development of Hindu-Arabic and Traditional Chinese Arithmetic", Chinese Science, 1996 p38, Kurt Vogel notation
  8. ↑ Lam Lay Yong, Fleeting Footsteps ISBN 981-02-3696-4
  9. ↑ Lam Lay Yong, An Tian Se, Fleeting Footsteps, p47
  10. ↑ Lam Lay Yong, An Tian Se, Fleeting Footsteps, p42-44
  11. ↑ Lam Lay Yong, An Tian Se, Fleeting Footsteps, p143
  12. ↑ Pearce, Ian (May 2002). "The Bakhshali manuscript" [archive]. 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. ↑ 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 ...
  15. ↑ Martin Levey and Marvin Petruck, Principles of Hindu Reckoning, translation of Kushyar ibn Labban Kitab fi usul hisab al-hind, p3, University of Wisconsin Press, 1965
  16. ↑ Bill Casselman (February 2007). "All for Nought" [archive]. Feature Column. AMS.
  17. ↑ 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.
  18. ↑ 18.0 18.1 Qian, Baocong (1964), Zhongguo Shuxue Shi (The history of Chinese mathematics), Beijing: Kexue Chubanshe
  19. ↑ Wáng, Qīngxiáng (1999), Sangi o koeta otoko (The man who exceeded counting rods), Tokyo: Tōyō Shoten, ISBN 4-88595-226-3
  20. ↑ https://talageri.blogspot.com/2021/09/the-complete-linguistic-case-for-out-of.html [archive]
  21. ↑ https://talageri.blogspot.com/2021/09/the-complete-linguistic-case-for-out-of.html [archive]

References

Bibliography

es:Numeración arábiga ko:아라비아 수 체계 mr:दशमान पध्दत pt:Sistema numérico hindu uk:Арабська система цифр

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