Āryabhaṭa numeration

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Āryabhaṭa numeration is an alphasyllabic numeral system based on Sanskrit phonemes. It was introduced in the early 6th century in India by Āryabhaṭa, in the first chapter titled Gītika Padam of his Aryabhatiya. It attributes a numerical value to each syllable of the form consonant+vowel possible in Sanskrit phonology, from ka = 1 up to hau = 10¹⁸.

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

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 values from 1, 2, 3 .. up to 25 and Avarga letters ya to ha have values 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; ṛ = 1000000 and so on.

Aryabhata 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 1⁄5, nja is 1⁄10 and jhardam (jha=9; its half) = 4+1⁄2.^{[further explanation needed]}

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 and the lakh (10⁵) and so on. (cf. Indian numbering system)

Another example might be ङिशिबुणॢष्खृ ṅiśibuṇḷṣkhṛ,^[1] 1582237500. Note that in this case, 10⁶(ṛ) and 10⁸(ḷ) parts are swapped, and 10⁶(ṛ) part is ligature.^[why?]

Another example from Aryabhatiya is a verse for table of sines.^[2]

makhi bhakhi phakhi dhakhi ṇakhi ñakhi
ṅakhi hasjha skaki kiṣga śghakhi kighva
ghlaki kigra hakya dhaki kica sga jhaśa
ṅva kla pta pha cha kala-ardha-jyāḥ

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.^[3]

• Aksharapalli
• Bhutasamkhya system
• Katapayadi system
• IAST

• Kurt Elfering: Die Mathematik des Aryabhata I. Text, Übersetzung aus dem Sanskrit und Kommentar. Wilhelm Fink Verlag, München, 1975, ISBN 3-7705-1326-6
• Georges Ifrah: The Universal History of Numbers. From Prehistory to the Invention of the Computer. John Wiley & Sons, New York, 2000, ISBN 0-471-39340-1.
• B. L. van der Waerden: Erwachende Wissenschaft. Ägyptische, babylonische und griechische Mathematik. Birkhäuser-Verlag, Basel Stuttgart, 1966, ISBN 3-7643-0399-9
• Fleet, J. F. (January 1911). "Aryabhata's System of Expressing Numbers". Journal of the Royal Asiatic Society of Great Britain and Ireland. 43: 109–126. doi:10.1017/S0035869X00040995. ISSN 0035-869X. JSTOR 25189823. S2CID 163070211.
• Fleet, J. F. (1911). "Aryabhata's System of Expressing Numbers". The Journal of the Royal Asiatic Society of Great Britain and Ireland. 43. Royal Asiatic Society of Great Britain and Ireland: 109–126. doi:10.1017/S0035869X00040995. JSTOR 25189823. S2CID 163070211.