Āryabhaṭa numeration

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 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; 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
100 101	102 103	104 105	106 107	108
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 (10⁵) and so on. (cf. Indian numbering system)

Another example might be ङिशिबुणॢष्खृ ṅiśibuṇḷṣkhṛ, 1582237500. 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.

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 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

-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

See also

• Aksharapalli
• Bhutasamkhya system
• Katapayadi system
• IAST

References

• 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. 
• Fleet, J. F. (1911). "Aryabhata's System of Expressing Numbers". The Journal of the Royal Asiatic Society of Great Britain and Ireland (Royal Asiatic Society of Great Britain and Ireland) 43: 109–126. doi:10.1017/S0035869X00040995.