Wednesday, September 10, 2008

Aryabhatta's pi value

The mathematical value pi has always been a curious number throughout history.

Here is Aryabhatta's version of the value of pi:

Lets split the words and understand the meaning:

चतुरधिकम् शतम् - Four more than hundred (=104)

अष्टगुणम् - multiplied by 8 (104 x 8 = 832)

द्वाषष्टि = 62

तथा सहस्राणाम् = of 1000 as such (=62000; totalling 62832)

अयुत द्वय = 10,000 x 2 (=20,000)

विष्कम्भस्य = of the diameter

आसन्न: - approximately

वृत्त परिणाह: - to the circumference.

In effect, 62832/20000 = 3.1416!

It interesting to note the large numbers he has used to arrive at Pi and the remark that pi is only an approximate value.

Tuesday, September 9, 2008

Aryabhatta's numerical encoding

The devanagari alphabets have been used in different fashions to denote mathematical numbers. A very popular such scheme is the katapayAdi sAnkhya notation, where a number is denoted by the formula

कादि नव टादि नव पादि पञ्चक याद्यष्टक
kAdi nava tAdi nava pAdi panchaka yAdyashtaka |

The katapayAdi scheme has been credited to vararuchi - वररुचि (author of chandra vAkya - चन्द्र वाक्य). Literature for this scheme is widely available on the net.

Another popular scheme is using commonly existing materials as representing numbers. For example, moon (चन्द्र / इन्दु) = 1; eyes (नेत्रे) = 2; rishi (ऋषि) = 7 etc. A more comprehensive list can be found here.

The katapayAdi scheme was probably not in vogue during the times of Aryabhatta. In his monumental work AryabhatIya, he introduces a very different scheme, specifically suited for representing astronomical numbers. His scheme was flexible while representing large numbers, which is especially true for astronomical calculations.

In his very innovative scheme, he successfully combines positional system with the devanAgarI-vowels and effortlessly represents large numbers upto 10^17.

Aryabhatta classifies the devanAgarI alphabets as varga (square) and avarga (non-square). This is purely his own classification based on positional system and has nothing to do with the alphabets itself.

Lets look at some definitions:

Defintion #1 - varga (square) consonants - वर्गीय व्यञ्ज्न

The vargIya consonants are represented like this:

1 2 3 4 5 6 7 8 9 10 11 12 13
क् ख् ग् घ् ङ् च् छ् ज झ ञ ट ठ ड
14 15 16 17 18 19 20 21 22 23 24 25
ढ ण त थ द ध न प फ ब भ म

Defintion #2 - avarga (non-square) consonants - अवार्गीय व्यञ्जन

The avargIya consonants are represented like this:

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

Definition #3 - Position of vowels

18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1
औ औ ओ ओ ऐ ऐ ए ए लृ लृ ऋ ऋ उ उ इ इ अ अ
Definition #4 - Position of varga-s

Remember, 10th power := 10 ^ (#position - 1).

Varga positions are at squares of 10, namely 1, 100, 10000 etc (#position = 1,3,5 etc)
Avarga positions are at non-squares of 10, namely 10,1000 etc (#position = 2,4,6 etc)

The usage of vowels for positioning/notational system is a very striking feature of this scheme. The simplicity of denoting large numbers will become evident as we explain the scheme.

With such a system in place, the following can be established:

  • Numbers take their positional life when a consonant is combined with the vowel.
  • The avargIya consonants can go only into positions of 2,4,6 etc.
  • The vargIya letters can go only into positions of 1,3,5 etc.
  • The length of vowel does not influence the number (ka and kA का) are one and the same.

Finally, remember that in Samskrita math tradition, the numbers are read from right to left. This is by the rule अङ्काणाम् वामतो गति: (Numbers go from right)

Before we jump into deciphering numbers, lets recap on what letters should go where.

18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1
यौ यो यै ये य्ल्रृ यृ यु यि
30 30 30 30 30 30 30 30 30

Now lets look at some real world examples:

Simple examples
Well, 100 is pretty much pre-defined (ha - )

Eg: khi (खि)

kh = 2, varga, i () denotes position 3 = 10^2, so khi = 200

Eg: ju (जु)

j = 8, varga, u () denotes position 5 = 10^4, so ju = 80,000

Eg: yRu (यृ)

y = 30, a-varga, Ru () denotes position 8 for avarga = 10^7, so yRu = 30x10^7 = 3x10^8.

So a simple यृ denotes 3x10^8! (think of speed of light in km/s in a vaccuum).

Some more larger numbers:

nu (नु) = 20 x 10^5 = 2,000,000 (2 million)
mau (मौ) = 25 x 10^17! (2.5 billion billion)

Lets look at other numbers:

Number Number in reverse Scheme
125 = 25+100 ma-ha
225 = 25+200 ma-khi (2*10^2)
5335 = 5+30+300+5000 = ङयगि
432,000 = 2000 + 30000 + 400000 = ख्युघृ

Aryabhatta gives several such numerals based on this scheme including number of rotational years of planets and moon, sine tables etc.

But his scheme did not gain popularity. One reason could be the difficulty in pronouncing the words as a result of such an encoding. Another reason could be the katapayAdi scheme, which could blend beautifully with the poetry.

Comparison of katapayAdi and Aryabhatta's scheme:

An easy difference between katapayAdi and the Aryabhatta's scheme is, whereas the former uses 1:M mapping of numbers:letters (one number can be represented by several letters; for eg 1 can be represented by ka-,ta- or pa-), Aryabhatta's is pretty much a 1:1 mapping. The flexibility of the former yields itself to great poetry.

In katapayAdi scheme, vowels do not carry any significance (all vowels are either ignored or equated to 0). Also the half-consonants are ignored. But in Aryabhatta's scheme, vowels play a major part, in fact it "powers" the numbers.

Tuesday, September 2, 2008

Who is equal to Kalidasa?

That mahAkavi kalidAsa is one of the greatest poets is perhaps already a cliche. From Bhavabhuti to Goether several personalities have paid very rich tributes to him. In the history of world literature, from time to time one finds a gem that lauds him to the sky. Here is one such a beauty.

पुरा कवीनां गणनाप्रसङ्गे कनिष्ठिका अधिष्ठित कालिदासा ।

"Long ago when poets met together, they agreed that KAlidAsa is equal to the pinky finger."

Very insulting indeed! How could one equate Kalidasa the greatest of poets to a pinky finger?! Isnt it preposterous? Has this guy even read any of Kalidasa's works?

Well here comes the unassuming punch line as the next part of the subhAshita:

अद्यापि तत्तुल्य कवे: अभावात् अनामिका सा अर्थवती बभूव ॥

"Even today due to the absence of a poet of equal stature, the next finger remains meaningfully to be called as "anAmikA" (Unnamed)."

(In Samskrita, the ring finger is called "anAmikA".)

What a master stroke!