# 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.

## History

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; = 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 15, nja is 110 and jhardam (jha=9; its half) = 4+12.[further explanation needed]

## Example

 Example:  299,792,458 100 101 102 103 104 105 106 107 108 58, 24, 79, 99, 2 .mw-parser-output .script-Cprt{font-size:1.25em;font-family:"Segoe UI Historic","Noto Sans Cypriot",Code2001}.mw-parser-output .script-Hano{font-size:125%;font-family:"Noto Sans Hanunoo",FreeSerif,Quivira}.mw-parser-output .script-Latf,.mw-parser-output .script-de-Latf{font-size:1.25em;font-family:"Breitkopf Fraktur",UnifrakturCook,UniFrakturMaguntia,MarsFraktur,"MarsFraktur OT",KochFraktur,"KochFraktur OT",OffenbacherSchwabOT,"LOB.AlteSchwabacher","LOV.AlteSchwabacher","LOB.AtlantisFraktur","LOV.AtlantisFraktur","LOB.BreitkopfFraktur","LOV.BreitkopfFraktur","LOB.FetteFraktur","LOV.FetteFraktur","LOB.Fraktur3","LOV.Fraktur3","LOB.RochFraktur","LOV.RochFraktur","LOB.PostFraktur","LOV.PostFraktur","LOB.RuelhscheFraktur","LOV.RuelhscheFraktur","LOB.RungholtFraktur","LOV.RungholtFraktur","LOB.TheuerbankFraktur","LOV.TheuerbankFraktur","LOB.VinetaFraktur","LOV.VinetaFraktur","LOB.WalbaumFraktur","LOV.WalbaumFraktur","LOB.WeberMainzerFraktur","LOV.WeberMainzerFraktur","LOB.WieynckFraktur","LOV.WieynckFraktur","LOB.ZentenarFraktur","LOV.ZentenarFraktur"}.mw-parser-output .script-en-Latf{font-size:1.25em;font-family:Cankama,"Old English Text MT","Textura Libera","Textura Libera Tenuis",London}.mw-parser-output .script-it-Latf{font-size:1.25em;font-family:"Rotunda Pommerania",Rotunda,"Typographer Rotunda"}.mw-parser-output .script-Lina{font-size:1.25em;font-family:"Noto Sans Linear A"}.mw-parser-output .script-Linb{font-size:1.25em;font-family:"Noto Sans Linear B"}.mw-parser-output .script-Ugar{font-size:1.25em;font-family:"Segoe UI Historic","Noto Sans Ugaritic",Aegean}.mw-parser-output .script-Xpeo{font-size:1.25em;font-family:"Segoe UI Historic","Noto Sans Old Persian",Artaxerxes,Xerxes,Aegean} जल घिनि झुशु झृसृ खॢ ja-la ghi-ni jhu-śu jhṛ-sṛ khḷ

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

— Aryabhata

## 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 10x) values, but these are valid throughout the list of numeric syllables.[3]

 The   33 × 9  =  297   Sanskrit alphabetic numerical syllables Nine vowels or syllabics -a -i -u -ṛ -ḷ -e -ai -o -au अ इ उ ऋ ऌ ए ऐ ओ औ × 10 0 10 2 10 4 10 6 10 8 1010 1012 1014 1016 Five velar plosives k - क 1 क or का ka कि or की ki कु or कू ku कृ or कॄ kṛ कॢ or कॣ kḷ के or कॆ ke कै kai को or कॊ ko कौ kau kh - ख 2 ख kha खि khi खु khu खृ khṛ खॢ khḷ खे khe खै khai खो kho खौ khau g - ग 3 ग ga गि gi गु gu गृ gṛ गॢ gḷ गे ge गै gai गो go गौ gau gh - घ 4 घ gha घि ghi घु ghu घृ ghṛ घॢ ghḷ घे ghe घै ghai घो gho घौ ghau ṅ - ङ 5 ङ ṅa ङि ṅi ङु ṅu ङृ ṅṛ ङॢ ṅḷ ङे ṅe ङै ṅai ङो ṅo ङौ ṅau Five palatal plosives c - च 6 च ca चि ci चु cu चृ cṛ चॢ cḷ चे ce चै cai चो co चौ cau ch - छ 7 छ cha छि chi छु chu छृ chṛ छॢ chḷ छे che छै chai छो cho छौ chau j - ज 8 ज ja जि ji जु ju जृ jṛ जॢ jḷ जे je जै jai जो jo जौ jau jh - झ 9 झ jha झि jhi झु jhu झृ jhṛ झॢ jhḷ झे jhe झै jhai झो jho झौ jhau ñ - ञ 10 ञ ña ञि ñi ञु ñu ञृ ñṛ ञॢ ñḷ ञे ñe ञै ñai ञो ño ञौ ñau Five retroflex plosives ṭ - ट 11 ट ṭa टि ṭi टु ṭu टृ ṭṛ टॢ ṭḷ टे ṭe टै ṭai टो ṭo टौ ṭau ṭh - ठ 12 ठ ṭha ठि ṭhi ठु ṭhu ठृ ṭhṛ ठॢ ṭhḷ ठे ṭhe ठै ṭhai ठो ṭho ठौ ṭhau ḍ - ड 13 ड ḍa डि ḍi डु ḍu डृ ḍṛ डॢ ḍḷ डे ḍe डै ḍai डो ḍo डौ ḍau ḍh - ढ 14 ढ ḍha ढि ḍhi ढु ḍhu ढृ ḍhṛ ढॢ ḍhḷ ढे ḍhe ढै ḍhai ढो ḍho ढौ ḍhau ṇ - ण 15 ण ṇa णि ṇi णु ṇu णृ ṇṛ णॢ ṇḷ णे ṇe णै ṇai णो ṇo णौ ṇau Five dental plosives t - त 16 त ta ति ti तु tu तृ tṛ तॢ tḷ ते te तै tai तो to तौ tau th - थ 17 थ tha थि thi थु thu थृ thṛ थॢ thḷ थे the थै thai थो tho थौ thau d - द 18 द da दि di दु du दृ dṛ दॢ dḷ दे de दै dai दो do दौ dau dh - ध 19 ध dha धि dhi धु dhu धृ dhṛ धॢ dhḷ धे dhe धै dhai धो dho धौ dhau n - न 20 न na नि ni नु nu नृ nṛ नॢ nḷ ने ne नै nai नो no नौ nau Five labial plosives p - प 21 प pa पि pi पु pu पृ pṛ पॢ pḷ पे pe पै pai पो po पौ pau ph - फ 22 फ pha फि phi फु phu फृ phṛ फॢ phḷ फे phe फै phai फो pho फौ phau b - ब 23 ब ba बि bi बु bu बृ bṛ बॢ bḷ बे be बै bai बो bo बौ bau bh - भ 24 भ bha भि bhi भु bhu भृ bhṛ भॢ bhḷ भे bhe भै bhai भो bho भौ bhau m - म 25 म ma मि mi मु mu मृ mṛ मॢ mḷ मे me मै mai मो mo मौ mau Four approximants or trill y - य 30 य ya यि yi यु yu यृ yṛ यॢ yḷ ये ye यै yai यो yo यौ yau r - र 40 र ra रि ri रु ru रृ rṛ रॢ rḷ रे re रै rai रो ro रौ rau l - ल 50 ल la लि li लु lu लृ lṛ लॢ lḷ ले le लै lai लो lo लौ lau v - व 60 व va वि vi वु vu वृ vṛ वॢ vḷ वे ve वै vai वो vo वौ vau Three coronal fricatives ś - श 70 श śa शि śi शु śu शृ śṛ शॢ śḷ शे śe शै śai शो śo शौ śau ṣ - ष 80 ष ṣa षि ṣi षु ṣu षृ ṣṛ षॢ ṣḷ षे ṣe षै ṣai षो ṣo षौ ṣau s - स 90 स sa सि si सु su सृ sṛ सॢ sḷ से se सै sai सो so सौ sau One glottal fricative h - ह 100 ह ha हि hi हु hu हृ hṛ हॢ hḷ हे he है hai हो ho हौ hau

## References

1. ^ Āryabhaṭīya 1(gītikā).3
2. ^ Roddam, Narasimha (2001). "Sines in terse verse". Nature. 414 (6866). Macmillan Magazines Ltd.: 851. Bibcode:2001Natur.414..851N. doi:10.1038/414851a. PMID 11780041. S2CID 197930.
3. ^ Ifrah, Georges (2000). The Universal History of Numbers. From Prehistory to the Invention of the Computer. New York: John Wiley & Sons. pp. 447–450. ISBN 0-471-39340-1.
• 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.