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 = 1 up to = 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; = 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, nja is and jhardam (jha=9; its half) = .
| Example: 299,792,458 | |||||||||
| width=20% | 100 101 | width=20% | 102 103 | width=20% | 104 105 | width=20% | 106 107 | width=20% | 108 |
| 58, | 24, | 79, | 99, | 2 | |||||
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 Sanskrit: ङिशिबुणॢष्खृ ,[1] . Note that in this case, 106(ṛ) and 108(ḷ) parts are swapped, and 106(ṛ) part is ligature.
Another example from Aryabhatiya is a verse for table of sines.[2]
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 | |||||||||||||||||||||||||||
| width=48 | width=48 | width=48 | width=24 | width=48 | width=48 | width=48 | width=48 | width=48 | width=48 | width=48 | width=48 | width=48 | width=24 | ||||||||||||||
| bgcolor=#F0FFFF height=28 colspan=3 | Nine vowels or syllabics | - | - | - | - | - | - | - | - | - | |||||||||||||||||
| bgcolor=#CCCCCC height=48 | अ | इ | उ | ऋ | ऌ | ए | ऐ | ओ | औ | ||||||||||||||||||
| bgcolor=#CCCCCC height=32 | × | 10 0 | 10 2 | 10 4 | 10 6 | 10 8 | 1010 | 1012 | 1014 | 1016 | |||||||||||||||||
| bgcolor=#F0FFFF height=24 colspan=3 | Five velar plosives | ||||||||||||||||||||||||||
| bgcolor=#CCCCCC height=48 | - | bgcolor=#FFFFCC style="line-height:1.6" | क | 1 | क or का | कि or की | कु or कू | कृ or कॄ | कॢ or कॣ | के or कॆ | कै | को or कॊ | कौ | ||||||||||||||
| bgcolor=#CCCCCC height=48 | - | bgcolor=#FFFFCC style="line-height:1.6" | ख | 2 | ख | खि | खु | खृ | खॢ | खे | खै | खो | खौ | ||||||||||||||
| bgcolor=#CCCCCC height=48 | - | bgcolor=#FFFFCC style="line-height:1.6" | ग | 3 | ग | गि | गु | गृ | गॢ | गे | गै | गो | गौ | ||||||||||||||
| bgcolor=#CCCCCC height=48 | - | bgcolor=#FFFFCC style="line-height:1.6" | घ | 4 | घ | घि | घु | घृ | घॢ | घे | घै | घो | घौ | ||||||||||||||
| bgcolor=#CCCCCC height=48 | - | bgcolor=#FFFFCC style="line-height:1.6" | ङ | 5 | ङ | ङि | ङु | ङृ | ङॢ | ङे | ङै | ङो | ङौ | ||||||||||||||
| bgcolor=#F0FFFF height=24 colspan=3 | Five palatal plosives | ||||||||||||||||||||||||||
| bgcolor=#CCCCCC height=48 | - | च | 6 | च | चि | चु | चृ | चॢ | चे | चै | चो | चौ | |||||||||||||||
| bgcolor=#CCCCCC height=48 | - | छ | 7 | छ | छि | छु | छृ | छॢ | छे | छै | छो | छौ | |||||||||||||||
| bgcolor=#CCCCCC height=48 | - | ज | 8 | ज | जि | जु | जृ | जॢ | जे | जै | जो | जौ | |||||||||||||||
| bgcolor=#CCCCCC height=48 | - | झ | 9 | झ | झि | झु | झृ | झॢ | झे | झै | झो | झौ | |||||||||||||||
| bgcolor=#CCCCCC height=48 | - | ञ | 10 | ञ | ञि | ञु | ञृ | ञॢ | ञे | ञै | ञो | ञौ | |||||||||||||||
| bgcolor=#F0FFFF height=24 colspan=3 | Five retroflex plosives | ||||||||||||||||||||||||||
| bgcolor=#CCCCCC height=48 | - | ट | 11 | ट | टि | टु | टृ | टॢ | टे | टै | टो | टौ | |||||||||||||||
| bgcolor=#CCCCCC height=48 | - | ठ | 12 | ठ | ठि | ठु | ठृ | ठॢ | ठे | ठै | ठो | ठौ | |||||||||||||||
| bgcolor=#CCCCCC height=48 | - | ड | 13 | ड | डि | डु | डृ | डॢ | डे | डै | डो | डौ | |||||||||||||||
| bgcolor=#CCCCCC height=48 | - | ढ | 14 | ढ | ढि | ढु | ढृ | ढॢ | ढे | ढै | ढो | ढौ | |||||||||||||||
| bgcolor=#CCCCCC height=48 | - | ण | 15 | ण | णि | णु | णृ | णॢ | णे | णै | णो | णौ | |||||||||||||||
| bgcolor=#F0FFFF height=24 colspan=3 | Five dental plosives | ||||||||||||||||||||||||||
| bgcolor=#CCCCCC height=48 | - | त | 16 | त | ति | तु | तृ | तॢ | ते | तै | तो | तौ | |||||||||||||||
| bgcolor=#CCCCCC height=48 | - | थ | 17 | थ | थि | थु | थृ | थॢ | थे | थै | थो | थौ | |||||||||||||||
| bgcolor=#CCCCCC height=48 | - | द | 18 | द | दि | दु | दृ | दॢ | दे | दै | दो | दौ | |||||||||||||||
| bgcolor=#CCCCCC height=48 | - | ध | 19 | ध | धि | धु | धृ | धॢ | धे | धै | धो | धौ | |||||||||||||||
| bgcolor=#CCCCCC height=48 | - | न | 20 | न | नि | नु | नृ | नॢ | ने | नै | नो | नौ | |||||||||||||||
| bgcolor=#F0FFFF height=24 colspan=3 | Five labial plosives | ||||||||||||||||||||||||||
| bgcolor=#CCCCCC height=48 | - | प | 21 | प | पि | पु | पृ | पॢ | पे | पै | पो | पौ | |||||||||||||||
| bgcolor=#CCCCCC height=48 | - | फ | 22 | फ | फि | फु | फृ | फॢ | फे | फै | फो | फौ | |||||||||||||||
| bgcolor=#CCCCCC height=48 | - | ब | 23 | ब | बि | बु | बृ | बॢ | बे | बै | बो | बौ | |||||||||||||||
| bgcolor=#CCCCCC height=48 | - | भ | 24 | भ | भि | भु | भृ | भॢ | भे | भै | भो | भौ | |||||||||||||||
| bgcolor=#CCCCCC height=48 | - | म | 25 | म | मि | मु | मृ | मॢ | मे | मै | मो | मौ | |||||||||||||||
| bgcolor=#F0FFFF height=24 colspan=3 | Four approximants or trill | ||||||||||||||||||||||||||
| bgcolor=#CCCCCC height=48 | - | य | 30 | य | यि | यु | यृ | यॢ | ये | यै | यो | यौ | |||||||||||||||
| bgcolor=#CCCCCC height=48 | - | र | 40 | र | रि | रु | रृ | रॢ | रे | रै | रो | रौ | |||||||||||||||
| bgcolor=#CCCCCC height=48 | - | ल | 50 | ल | लि | लु | लृ | लॢ | ले | लै | लो | लौ | |||||||||||||||
| bgcolor=#CCCCCC height=48 | - | व | 60 | व | वि | वु | वृ | वॢ | वे | वै | वो | वौ | |||||||||||||||
| bgcolor=#F0FFFF height=24 colspan=3 | Three coronal fricatives | ||||||||||||||||||||||||||
| bgcolor=#CCCCCC height=48 | - | श | 70 | श | शि | शु | शृ | शॢ | शे | शै | शो | शौ | |||||||||||||||
| bgcolor=#CCCCCC height=48 | - | ष | 80 | ष | षि | षु | षृ | षॢ | षे | षै | षो | षौ | |||||||||||||||
| bgcolor=#CCCCCC height=48 | - | स | 90 | स | सि | सु | सृ | सॢ | से | सै | सो | सौ | |||||||||||||||
| bgcolor=#F0FFFF height=24 colspan=3 | One glottal fricative | ||||||||||||||||||||||||||
| bgcolor=#CCCCCC height=48 | - | ह | 100 | ह | हि | हु | हृ | हॢ | हे | है | हो | हौ | |||||||||||||||
| bgcolor=#CCCCCC height=24 |