Āryabhaṭa numeration is an

alphasyllabic numeral system Alphasyllabic numeral systems are a type of numeral systems, developed mostly in India starting around 500 AD. Based on various alphasyllabic scripts, in this type of numeral systems glyphs of the numerals are not abstract signs, but syllables of ...

based on Sanskrit phonemes. It was introduced in the early 6th century in India by

Āryabhaṭa Aryabhata (ISO: ) or Aryabhata I (476–550 CE) was an Indian mathematician and astronomer of the classical age of Indian mathematics and Indian astronomy. He flourished in the Gupta Era and produced works such as the '' Aryabhatiya'' (whic ...

, in the first chapter titled ''Gītika Padam'' of his ''

Aryabhatiya ''Aryabhatiya'' (IAST: ') or ''Aryabhatiyam'' ('), a Sanskrit astronomical treatise, is the ''magnum opus'' and only known surviving work of the 5th century Indian mathematician Aryabhata. Philosopher of astronomy Roger Billard estimates that th ...

''. It attributes a numerical value to each syllable of the form consonant+vowel possible in

Sanskrit phonology The grammar of the Sanskrit language has a complex verbal system, rich nominal declension, and extensive use of compound nouns. It was studied and codified by Sanskrit grammarians from the later Vedic period (roughly 8th century BCE), culminating ...

, from ' = 1 up to ' = 10¹⁸.

History

The basis of this number system is mentioned in the second stanza of the first chapter of ''

''. The Varga (Group/Class) letters ''ka'' to ''ma'' are to be placed in the varga (square) places (1st, 100th, 10000th, etc.) and Avarga letters like ''ya'', ''ra'', ''la'' .. have to be placed in Avarga places (10th, 1000th, 100000th, etc.). The Varga letters ''ka'' to ''ma'' have value from 1, 2, 3 .. up to 25 and Avarga letters ''ya'' to ''ha'' have value 30, 40, 50.. up to 100. In the Varga and Avarga letters, beyond the ninth vowel (place), new symbols can be used. The values for vowels are as follows: ''a'' = 1; ''i'' = 100; ''u'' = 10000; ' = 1000000 and so on.

Aryabhata Aryabhata (ISO: ) or Aryabhata I (476–550 CE) was an Indian mathematician and astronomer of the classical age of Indian mathematics and Indian astronomy. He flourished in the Gupta Era and produced works such as the ''Aryabhatiya'' (which ...

used this number system for representing both small and large numbers in his mathematical and astronomical calculations. This system can even be used to represent fractions and mixed fractions. For example, ''nga'' is , ''nja'' is and Jhardam (''jha''=9; its half) = .

Example

The traditional Indian digit order is reversed compared to the modern way. By consequence, Āryabhaṭa began with the ones before the tens; then the hundreds and the thousands; then the

myriad A myriad (from Ancient Greek grc, μυριάς, translit=myrias, label=none) is technically the number 10,000 (ten thousand); in that sense, the term is used in English almost exclusively for literal translations from Greek, Latin or Sinospher ...

and the

lakh A lakh (; abbreviated L; sometimes written lac) is a unit in the Indian numbering system equal to one hundred thousand (100,000; scientific notation: 105). In the Indian 2,2,3 convention of digit grouping, it is written as 1,00,000. For ex ...

(10⁵) and so on. ''(cf.

Indian numbering system The Indian numbering system is used in all South Asian countries (Bangladesh, Bhutan, India, Maldives, Nepal, Pakistan, Sri Lanka and Afghanistan) to express large numbers. The terms ''lakh'' or 1,00,000 (one hundred thousand, written as ''100,00 ...

)'' Another example might be ',Āryabhaṭīya 1(gītikā).3 . Note that in this case, 10⁶(ṛ) and 10⁸(ḷ) parts are swapped, and 10⁶(ṛ) part is ligature.

Numeral table

In citing the values of Āryabhaṭa numbers, the short vowels अ, इ, उ, ऋ, ऌ, ए, and ओ are invariably used. However, the Āryabhaṭa system did not distinguish between long and short vowels. This table only cites the full slate of क-derived (1 x 10^x) values, but these are valid throughout the list of numeric syllables.

References

* Kurt Elfering: ''Die Mathematik des Aryabhata I. Text, Übersetzung aus dem Sanskrit und Kommentar''. Wilhelm Fink Verlag, München, 1975, * Georges Ifrah: ''The Universal History of Numbers. From Prehistory to the Invention of the Computer''. John Wiley & Sons, New York, 2000, . * B. L. van der Waerden: ''Erwachende Wissenschaft. Ägyptische, babylonische und griechische Mathematik''. Birkhäuser-Verlag, Basel Stuttgart, 1966, * * {{DEFAULTSORT:Aryabhata Numeration Numerals Classical ciphers Indian mathematics

History

Example

Numeral table

See also

References