AryabhaTa's Number System

AryabhaTa (in the AryabhaTIyam, and otherwise) dealt with huge numbers in kAlakriyapAda or time measurement and gOlapAda or astronomical (literally Sphere) measurement - like the number of days in a chaturyuga, the diameter of earth, the diameter of the sun... well ginormous numbers, basically. 

Apparently, there was no place value number system in place at the time of AryabhaTa - or if there was, he wanted an alternate number representation system, so those numbers could be woven into shlOka meter. 

What does the smart guy do? He goes and designs a number representation system (quite possibly the precursor of the Indian Numeral Place Value system), so he could represent these huge numbers, incorporate them into shlokas [actual numerals such as 0,1,2,etc even if they were in use at the time, would mess up the meter], and write numbers sensibly and readably. 

He lays out the entire system in one shlOka (#2) in the daSagItikapAda: 

वर्गाक्षराणि वर्गे अवर्गे अवर्गाक्षराणि कात्  ङमौ यः ।
खद्विनवके स्वरा नव वर्गे अवर्गे नवान्त्यवर्गे वा ॥

Walter Eugene Clark (I generally look at western translations of संस्कृतम् with a fistful of salt and a few kilograms of suspicion) translates this thus:

Beginning with the ka the varga letters (are to be used) in the varga places, and the avarga letters (are to be used) in the avarga places. ya is equal to the sum of ~ga and ma. The nine vowels (are to be used) in two nines of places varga and avarga. navAMtyavarge vA

[Note: Clark uses Roman notation with accents, I have used Baraha notation for simplicity.]

The first line and it seems to be accurate enough, but I get the feeling that Clark glosses over the second line. 

Kripa Shankar Shukla does a better job with the second line (although it is a bit overly verbal with inaccurate punctuation, and thus ends up as a somewhat confused explanation; it actually makes sense, though, after reading it a few times and seeing Shukla's explanation) translates this as:

The varga letters (ka to ma) (should be written) in the varga places and the avarga letters (ya to ha) in the avarga places. (The varga letters take numerical values 1,2,3,etc.) from ka onwards; (the numerical value of the initial avarga letter) ya is equal to ~ga plus ma (i.e., 5+25). In the places of the two nines of zeros (which are written to denote the notational places), the nine vowels should be written (one vowel in each pair of the varga and avarga places). In the varga (and avarga) places beyond (the places denoted by) the nine vowels too (assumed vowels or other symbols should be written, if necessary).

[Note: Shukla also uses Roman notation with accents, which I have replaced with Baraha notation. The emphasis is also mine to keep the translation separate from the explanation in parentheses. It might need a reread of just the bold portion for the translation]

Apparently (as explained by Shukla), the real interpretation has its roots in the way numbers were represented in ancient bhAratavarSha, and how the the vargas are in units place and avargas are in 10s place, which is why the vargas increase in units, and the avargas increase in 10s. Also, it is clearly stated that the sequence starts with क, and the value of य is stated as ङमौ = ङ + म = 5 + 25 = 30

Also, the 'two nines of zeros' for the 'nine vowels' indicates a, i, u, Ru, ~lu, e, ai, o, au are in the places of two nines of zeros - in other words, units place (10⁰) to the 10 quadrillion (10¹⁶). The svaras are written 'in the places beyond' the varga/avarga which is taken to mean that they supply a multiplying value to the varga/avargas.

This makes it clear that the idea of Zero was prevalent at the time of AryabhaTa, even if the place value system may or may not have been.

With the 20:20 hindsight of Clark's and Shukla's translations and Shukla's explanation based on treatises of bhAskara, paramESvara, and others and the practices of the time, I would put a word to word translation [I confess that my training in संस्कृतम् is rudimentary and as indicated, I am only doing this with said hindsight] as:

The varga letters in varga places and in the avarga places, the avarga letters starting from क, with य [the sum of] ङ and म.

Of the svaras [placed] on [two rows of] nine zeroes which are in the varga and avarga places, the nine that are in the varga places follow.

I realize that this doesn't make a whole lot of sense as-is, but it is an attempt at a direct translation of AryabhaTa's shlOka which itself leaves some things out for interpretation. The first line is pretty straightforward. The second line where the svara multiplying factor is explained is much more confusing as it is in both Clark's and Shukla's cases. Reading the translation after knowing how the system works makes somewhat better sense than trying to interpret the system from the shlOka translation.

Based on interpretation, and AryabhaTa's own usage and that of the others after him, the salient features of this system work out as: 

1. varigIya vyaMjanas क to म are assigned values according to place 

   - Starts with a 1 for क and increments by 1 for each letter.

   - The last one is म with the value 25.

2. avarigIya vyaMjanas start with ya being denoted as म + ङ

   - य = म(25)+ ङ(5) = 30

   - These increment by 10, and go on up to ह (=100)

3. svaras multiply the vyanjanas in incrementing powers of 10, starting with अ = 10⁰

   - The powers increment by 2 for successive svaras

   - dhIrGas are not used. Ostensibly these can be fit in to match meter, with no change in value

   - Both  ऋ  and  ऌ  are present.

   - By themselves, svaras do not have any numerical value.  

So, the AryabhaTa system resolves itself to:

अ	इ	उ	ऋ	ऌ	ए	ऐ	ओ	औ
100	102	104	106	108	1010	1012	1014	1016

क	ख	ग	घ	ङ
1	2	3	4	5

च	छ	ज	झ	ञ
6	7	8	9	10

ट	ठ	ड	ढ	ण
11	12	13	14	15

त	थ	द	ध	न
16	17	18	19	20

प	फ	ब	भ	म
21	22	23	24	25

य	र	ल	व	श	ष	स	ह
30	40	50	60	70	80	90	100

Example

With this system, 3963 can be represented as: 

3693 = यिचिसग

vyaMjana	Value	svara	Value	Composite	Value
य	30	इ	102	यि	30 * 102 = 3000
च	6	इ	102	चि	6 * 102 = 600
स	90	अ	100	स	90 * 100 = 90
ग	3	अ	100	ग	3 * 100 = 3

Notice that this has the same number of 'digits' (4) as the standard (Indian numeral) notation. The order of 'digits' is unimportant, since चिगसयि or सगयिचि or गसचियि will all result in a value of 3693.

The same number in Roman notation (another old-world system) would be:

MMMDCXCIII

Wow! That took quite some writing.

More examples: 

* 62,842 = चुनिजिरख 

  (Roman for this would be ~75 characters long, and therefore unreadable)

* 57,753,336 = लृछृशुङुयिगियच  

  (It is virtually impossible to write this number in Roman in a way that makes sense)

  According to AryabhaTa, this is the number of moon 'revolutions' in a yuga, and he writes it as  चयगियिङुशुछ्लृ 

  [From this, it appears as if half-letter followed by a letter with a vowel applies that svara (vowel) to  both letters - the half letter and the second letter containing the svara. Another aid in metering shlOkas. In this case the complex letter of half छ and ल with ऋ vowel, छ्लृ, having the value of 50,000,000 + 7,000,000 = 57,000,000]

Advantages of this system: 

• Representation is similar to the Indian numeral system, with a small number of digits for huge numbers
• There is no "place value". The number/word can be written with the digits/letters interchanged, with no change in value - can be altered for meter and rhyme
• Allows for up to values of 10¹⁸ with a single 'digit' (more than sufficient for 'circumference of the sky' which works out as 12.4 trillon yOjanas [12,474,720,576,000] or Lifespan of brahma which works out as 309 trillion years [309,173,760,000,000] which are probably among the largest numbers AryabhaTa dealt with) 
• While it might be possible to write numbers in more than one way, each representation yields exactly one numerical value, leaving no doubt as to the intended value.

One has to contend, however, that AryabhaTa intended this system only as a representation for huge numbers. Computation is not possible (at least, it isn't apparent) with this system as it is with the place value system.

Endnote: Thanks to Varahamihira Gopu for pointing out a mistake in spelling AryabhaTa's name . It isn't bhaTTa, but bhaTa. Wiki explicitly notes: 

While there is a tendency to misspell his name as "Aryabhatta" by analogy with other names having the "bhatta" suffix, his name is properly spelled Aryabhata: every astronomical text spells his name thus, including Brahmagupta's references to him "in more than a hundred places by name". Furthermore, in most instances "Aryabhatta" does not fit the metre either. References provided are K V Sharma, 2001 and Bhau Daji, 1865.