The Hindu–Arabic numeral system (also known as the Indo-Arabic numeral system,[1] Hindu numeral system, Arabic numeral system)[2] 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.
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 (On the Calculation with Hindu Numerals,) and Arab mathematician Al-Kindi (On the Use of the Hindu Numerals,). 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.[3]
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.
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.
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; medieval and Renaissance European mathematicians generally recognized them as Indian in origin, however a few influential sources credited them to the Arabs, and they eventually came to be generally known as "Arabic numerals" in Europe.[4] According to some sources, this number system may have originated in Chinese Shang numerals (1300 BC), which was also a decimal positional numeral system.[5]
See main article: article, 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"), 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[6]). 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:
| Symbol | Used with scripts | Numerals | ||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | Arabic numerals | ||
| Arabic: ٠ | Arabic: ١ | Arabic: ٢ | Arabic: ٣ | Arabic: ٤ | Arabic: ٥ | Arabic: ٦ | Arabic: ٧ | Arabic: ٨ | Arabic: ٩ | Eastern Arabic numerals | ||
| Persian: ۰ | Persian: ۱ | Persian: ۲ | Persian: ۳ | Persian: ۴ | Persian: ۵ | Persian: ۶ | Persian: ۷ | Persian: ۸ | Persian: ۹ | |||
| 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) | |
| Brahmi numerals | ||||||||||||
| ० | १ | २ | ३ | ४ | ५ | ६ | ७ | ८ | ९ | Devanagari numerals | ||
| Tamil: ௦ | Tamil: ௧ | Tamil: ௨ | Tamil: ௩ | Tamil: ௪ | Tamil: ௫ | Tamil: ௬ | Tamil: ௭ | Tamil: ௮ | Tamil: ௯ | Tamil numerals | ||
| ০ | ১ | ২ | ৩ | ৪ | ৫ | ৬ | ৭ | ৮ | ৯ | Bengali numerals | ||
| ੦ | ੧ | ੨ | ੩ | ੪ | ੫ | ੬ | ੭ | ੮ | ੯ | Gurmukhi numerals | ||
| ૦ | ૧ | ૨ | ૩ | ૪ | ૫ | ૬ | ૭ | ૮ | ૯ | Gujarati numerals | ||
| Modi numerals | ||||||||||||
| ୦ | ୧ | ୨ | ୩ | ୪ | ୫ | ୬ | ୭ | ୮ | ୯ | Odia numerals | ||
| ᱐ | ᱑ | ᱒ | ᱓ | ᱔ | ᱕ | ᱖ | ᱗ | ᱘ | ᱙ | Santali numerals | ||
| Sharada numerals | ||||||||||||
| ౦ | ౧ | ౨ | ౩ | ౪ | ౫ | ౬ | ౭ | ౮ | ౯ | |||
| ೦ | ೧ | ೨ | ೩ | ೪ | ೫ | ೬ | ೭ | ೮ | ೯ | |||
| ൦ | ൧ | ൨ | ൩ | ൪ | ൫ | ൬ | ൭ | ൮ | ൯ | Malayalam numerals | ||
| Meitei | ||||||||||||
| ෦ | ෧ | ෨ | ෩ | ෪ | ෫ | ෬ | ෭ | ෮ | ෯ | Sinhala numerals | ||
| Maithili numerals | ||||||||||||
| ༠ | ༡ | ༢ | ༣ | ༤ | ༥ | ༦ | ༧ | ༨ | ༩ | Tibetan numerals | ||
| ᠐ | ᠑ | ᠒ | ᠓ | ᠔ | ᠕ | ᠖ | ᠗ | ᠘ | ᠙ | Mongolian numerals | ||
| ᥆ | ᥇ | ᥈ | ᥉ | ᥊ | ᥋ | ᥌ | ᥍ | ᥎ | ᥏ | |||
| ၀ | ၁ | ၂ | ၃ | ၄ | ၅ | ၆ | ၇ | ၈ | ၉ | Burmese numerals | ||
| ႐ | ႑ | ႒ | ႓ | ႔ | ႕ | ႖ | ႗ | ႘ | ႙ | |||
| ០ | ១ | ២ | ៣ | ៤ | ៥ | ៦ | ៧ | ៨ | ៩ | Khmer numerals | ||
| ๐ | ๑ | ๒ | ๓ | ๔ | ๕ | ๖ | ๗ | ๘ | ๙ | Thai numerals | ||
| ໐ | ໑ | ໒ | ໓ | ໔ | ໕ | ໖ | ໗ | ໘ | ໙ | |||
| ᧐ | ᧑/᧚ | ᧒ | ᧓ | ᧔ | ᧕ | ᧖ | ᧗ | ᧘ | ᧙ | |||
| ꩐ | ꩑ | ꩒ | ꩓ | ꩔ | ꩕ | ꩖ | ꩗ | ꩘ | ꩙ | |||
| ꧐ | ꧑ | ꧒ | ꧓ | ꧔ | ꧕ | ꧖ | ꧗ | ꧘ | ꧙ | Javanese numerals | ||
| ᭐ | ᭑ | ᭒ | ᭓ | ᭔ | ᭕ | ᭖ | ᭗ | ᭘ | ᭙ | Balinese numerals | ||
| ᮰ | ᮱ | ᮲ | ᮳ | ᮴ | ᮵ | ᮶ | ᮷ | ᮸ | ᮹ | Sundanese numerals | ||
See main article: article and History of the Hindu–Arabic numeral system.
The Chinese Shang dynasty numerals from the 14th century B.C. predates the Indian Brahmi numerals by over 1000 years and shows substantial similarity to the Brahmi numerals. Similar to the modern Hindu–Arabic numerals, the Shang dynasty numeral system was also decimal based and positional.[7] Zero was represented by an empty space.[8] [9] This conceptual similarity has led some scholars to speculate about the possibility of Chinese influences on Hindu–Arabic numerals.[10] [11]
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.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 place-value system is used in the Bakhshali manuscript; the earliest leaves being radiocarbon dated to the period AD 224–383.[12] The development of the positional decimal system takes its origins in 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 Lokavibhagamay 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]
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 (Arabic: كتاب في استعمال العداد الهندي [''kitāb fī isti'māl al-'adād al-hindī'']) around 830. Persian scientist Kushyar Gilani wrote Kitab fi usul hisab al-hind (Principles of Hindu Reckoning), 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.
See main article: article and Arabic numerals.
In Christian Europe, the first mention and representation of Hindu–Arabic numerals (from one to nine, without zero), is in the Latin: [[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 Latin: 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 Latin: [[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 German: Rechenung auff der linihen und federn (Calculating on the Lines and with a Quill) was targeted at the apprentices of businessmen and craftsmen.
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.
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).