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Āryabhaṭa numeration

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Āryabhaṭa numeration


The Āryabhaṭa numeration is a system of numerals 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. places) and Avarga letters like Ya, Ra, La .. have to be placed in Avarga places (10th, 1000th,...etc. places).

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 : i= 100 ; u = 10000, ru = 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) = 4½

Example

Example:  299,792,458
100 101 102 103 104 105 106 107 108
 85,  42,  97,  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, Āryabhaa 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)

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.[1]

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

See also

References

  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
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