# 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 = 10$^{18}$.

## 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; 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.

## Example

Example:  299,792,458 :
| 10$^0$ 10$^1$ | 10$^2$ 10$^3$ | 10$^4$ 10$^5$ | 10$^6$ 10$^7$ | 10$^8$ |
| 58, | 24, | 79, | 99, | 2 |
| 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 (10$^5$) and so on. (cf. Indian numbering system)
Another example might be ङिशिबुणॢष्खृ ṅiśibuṇḷṣkhṛ, 1582237500. Note that in this case, 10$^6$(ṛ) and 10$^8$(ḷ) parts are swapped, and 10$^6$(ṛ) part is ligature.
Another example from Aryabhatiya is a verse for table of sines.
   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

## 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.
The   33 × 9  =  297   Sanskrit alphabetic numerical syllables :
| Nine vowels or syllabics | Nine vowels or syllabics | 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$^8$ | 10$^{10}$ | 10$^{12}$ | 10$^{14}$ | 10$^{16}$ |
| Five velar plosives | Five velar plosives | 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 | Five palatal plosives | 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 | Five retroflex plosives | 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 | Five dental plosives | 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 | Five labial plosives | 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 | Four approximants or trill | 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 | Three coronal fricatives | 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 | One glottal fricative | One glottal fricative | | | | | | | | | |
| h - | ह | 100 | ह ha | हि hi | हु hu | हृ hṛ | हॢ hḷ | हे he | है hai | हो ho | हौ hau |