Hindu–Arabic numeral system

The Hindu-Arabic numeral system is a positional decimal numeral system developed by the 9th century (the earliest known description is Al-Khwarizmi's, dating to ca. 825) and spread to the western world through Arabic mathematicians by the High Middle Ages.

The system is based on 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 Indian Brahmi numerals, and have split into various typographical variants since the Middle Ages.

These symbol sets can be divided into three main families: the Indian numerals used in India, the Eastern Arabic numerals used in Egypt and the Middle East and the West Arabic numerals used in the Maghreb and in Europe.

Positional notation

Main articles: positional notation and 0 (number)

The Hindu-Arabic numeral 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 13 symbols (the ten digits, decimal marker, vinculum, and an optional prepended dash to indicate a negative number).

Symbols

Various symbol sets are used to represent numbers in the Hindu-Arabic numeral, all of which evolved 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 below labelled "European", descended from the "West Arabic numerals" which were developed in al-Andalus and the Maghreb (There are two typographic styles for rendering European numerals, known as lining figures and text figures).
• the "Arabic-Indic" or "Eastern Arabic numerals" used with the Arabic alphabet, developed primarily in what is now Iraq. A variant of the Eastern Arabic numerals used in Persian and Urdu.
• the Indian numerals in use with scripts of the Brahmic family in India and Southeast Asia.

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/Japanese numbers and Roman numerals). Theorists believe that this is because it becomes difficult to instantaneously count objects past three.

The following is a list of numeral glyphs in contemporary use. Note: Some symbols may not display correctly if your browser does not support Unicode fonts.

Western Arabic	0	1	2	3	4	5	6	7	8	9
Middle East Arabic	٠	١	٢	٣	٤	٥	٦	٧	٨	٩
Eastern Arabic	۰	۱	۲	۳	۴	۵	۶	۷	۸	۹
Devanagari	०	१	२	३	४	५	६	७	८	९
Gujarati	૦	૧	૨	૩	૪	૫	૬	૭	૮	૯
Gurmukhi	੦	੧	੨	੩	੪	੫	੬	੭	੮	੯
Limbu	᥆	᥇	᥈	᥉	᥊	᥋	᥌	᥍	᥎	᥏
Assamese & Bengali	০	১	২	৩	৪	৫	৬	৭	৮	৯
Oriya	୦	୧	୨	୩	୪	୫	୬	୭	୮	୯
Telugu	౦	౧	౨	౩	౪	౫	౬	౭	౮	౯
Kannada	೦	೧	೨	೩	೪	೫	೬	೭	೮	೯
Malayalam	൦	൧	൨	൩	൪	൫	൬	൭	൮	൯
Tamil (Grantha)	೦	௧	௨	௩	௪	௫	௬	௭	௮	௯
Tibetan	༠	༡	༢	༣	༤	༥	༦	༧	༨	༩
Burmese	၀	၁	၂	၃	၄	၅	၆	၇	၈	၉
Thai numerals	๐	๑	๒	๓	๔	๕	๖	๗	๘	๙
Khmer	០	១	២	៣	៤	៥	៦	៧	៨	៩
Lao	໐	໑	໒	໓	໔	໕	໖	໗	໘	໙
Lepcha	᱀	᱁	᱂	᱃	᱄	᱅	᱆	᱇	᱈	᱉
Balinese	᭐	᭑	᭒	᭓	᭔	᭕	᭖	᭗	᭘	᭙
Sundanese	᮰	᮱	᮲	᮳	᮴	᮵	᮶	᮷	᮸	᮹
Ol Chiki	᱐	᱑	᱒	᱓	᱔	᱕	᱖	᱗	᱘	᱙
Osmanya	𐒠	𐒡	𐒢	𐒣	𐒤	𐒥	𐒦	𐒧	𐒨	𐒩
Saurashtra	꣐	꣑	꣒	꣓	꣔	꣕	꣖	꣗	꣘	꣙

History

Main article: History of the Hindu-Arabic numeral system

Predecessors

The Brahmi numerals at the basis of the system predate the Common Era. They replace the earlier Kharosthi numerals indigenous to India following the conquests of Alexander the Great in 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.

Buddhist inscriptions from around 300 BC use the symbols which became 1, 4 and 6. One century later, their use of the symbols which 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.

Development

The development of the positional decimal system takes its origins in Indian mathematics during the Gupta period. Around 500 CE 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.

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).

The numeral system came to be known to both the Persian mathematician Al-Khwarizmi, whose book On the Calculation with Hindu Numerals written about 825, and the Arab mathematician Al-Kindi, who wrote four volumes, On the Use of the Indian Numerals (كتاب في استعمال العداد الهندي [kitab fi isti'mal al-'adad al-hindi]) about 830, are principally responsible for the diffusion of the Indian system of numeration throughout the Islamic world and ultimately also to Europe.[2]. The first dated and undisputed inscription showing the use of zero at is at Gwalior, dating to 876 AD.

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

Adoption in Europe

The bottom row shows the numeral glyphs as they appear in type in German incunabula (Nicolaus Kesler, Basle, 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 the multiplication and division). After having become 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. Robert Chester translated the Latin into English.

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.

In the Arab World—until modern times—the Hindu-Arabic numeral system was used only by mathematicians. 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 of 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.

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Adoption in East Asia

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

Even though, 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 think it was created from the Chinese text space filler "□".

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

Spread of the Western Arabic variant

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).

References

• Flegg, Graham (2002). Numbers: Their History and Meaning. Courier Dover Publications. ISBN 0486421651.
• The Arabic numeral system