|Hindu–Arabic numeral system|
|Positional systems by base|
|Non-standard positional numeral systems|
|List of numeral systems|
The Hindu–Arabic numeral system or Indo-Arabic numeral system  (also called the Arabic numeral system or Hindu numeral system)[note 1] is a positional decimal numeral system, and 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 in Arabic mathematics (also called Islamic mathematics) by the 9th century. Influential were the books of Al-Khwārizmī (On the Calculation with Hindu Numerals, c. 825) and Al-Kindi (On the Use of the Hindu Numerals, c. 830). The system later spread to medieval Europe by the High Middle Ages.
The system is based upon ten (originally nine) 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 (also called "Indic numerals") used in the Middle East, and the Indian numerals in various scripts used in the Indian subcontinent.
The Hindu-Arabic or Indo-Arabic numerals were invented by mathematicians in India. Persian and 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.
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 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. 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:
|#||Used with alphabets||Numerals|
|0||1||2||3||4||5||6||7||8||9||Latin, Cyrillic, and Greek||Arabic numerals|
|〇/零||一||二||三||四||五||六||七||八||九||East Asia||Chinese, Vietnamese, Japanese, and Korean numerals|
|ο/ō||Αʹ||Βʹ||Γʹ||Δʹ||Εʹ||Ϛʹ||Ζʹ||Ηʹ||Θʹ||Modern Greek||Greek numerals|
|༠||༡||༢||༣||༤||༥||༦||༧||༨||༩||Tibetan||Standard Tibetan § Numerals|
|০||১||২||৩||৪||৫||৬||৭||৮||৯||Bengali / Assamese||Bengali numerals|
|೦||೧||೨||೩||೪||೫||೬||೭||೮||೯||Kannada||Kannada alphabet § Numerals|
|୦||୧||୨||୩||୪||୫||୬||୭||୮||୯||Odia||Odia alphabet § Numerals|
|൦||൧||൨||൩||൪||൫||൬||൭||൮||൯||Malayalam||Malayalam script § Other symbols|
|௦||௧||௨||௩||௪||௫||௬||௭||௮||௯||Tamil||Tamil script § Numerals and symbols|
|౦||౧||౨||౩||౪||౫||౬||౭||౮||౯||Telugu||Telugu script § Numerals|
|໐||໑||໒||໓||໔||໕||໖||໗||໘||໙||Lao||Lao alphabet § 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 (Farsi) / Dari / Pashto|
|۰||۱||۲||۳||۴||۵||۶||۷||۸||۹||Urdu / Shahmukhi|
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.
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 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.
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. 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.
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. These books are principally responsible for the diffusion of the Hindu system of numeration throughout the Islamic world and ultimately also to Europe.
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.
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.
Gregor Reisch, Madame Arithmatica, 1508
A calculation tableRoman numerals, used for arithmetic using
Adam Ries, Rechenung auff der linihen und federn, 1522
Robert Recorde, The ground of artes, 1543
Peter Apian, Kaufmanns Rechnung, 1527
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 "□".
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).
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