Āryabhaṭa numeration

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The Āryabhaṭa numeration is a system of numerals based on Sanskrit phonemes. It was introduced in the early 6th century 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.

Contents

History

The basis of this number system is mentioned in the second stanza of the first chapter of Aryabhatta.

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, 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)

Numeral table

Here is the complete table:

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
ka		 कि
ki		 कु
ku		 कृ
kṛ		 कॢ
kḷ		 के
ke		 कै
kai		 को
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

References