Wednesday, March 18, 2015

Katapayadi system of verses

As I am researching the π, I find interesting ways ancient indian researchers described their findings. One such methodology used is Katapayadi System of verses. It is basically an system of code so that things can be defined in a way so that people can remember. The code is as follows.

1 23456789
क  ख ग घ ङ च  छ  ज  झ 
ट  ठ ड ढ ण त थ द ध 
प   फ  ब  भ  म  
य   र  ल  व  श  ष  स  ह  क्ष  
With the above key in place, Sri Bharathi Krishna Tirtha in his Vedic Mathematics gives following verse.
गोपी भाग्य मधुव्रात  श्रुङ्गिशो दधिसन्धिग  |
खलजीवित खाताव गलहालारसंधार |
If we replace the code from the above table in the above verse, here is what we get.
31 41 5926 535 89793
23846 264 33832792
That gives us \( \frac{π}{10} = 0.31415926535897932384626433832792\)

Apprantly this methodology of remembering digits of π has a name. It is called Piphilology.
Piphilology comprises the creation and use of mnemonic techniques to remember a span of digits of the mathematical constant π. The word is a play on the word "pi" itself and of the linguistic field of philology.

Sunday, March 15, 2015

π Day and India

Yesterday was π day and my friend Haldar Rana posted this quiz on π. I started to look around the Indian contribution with respect to π. Here are some of the stuff that I found.

Aryabhatta talks about π in following way.

  • Add 4 to 100
  • Multiply by 8
  • Add 62000. 
  • The result is approximately the circumference of the circle whose diameter is 20000. 
With above calculation the value of PI comes to 3.1416.

Madhava has done some more serious work with π. He defines π as the ratio of the circumference (2,827,433,388,233) of a circle of diameter \( 9*10^{11} \). Which yields 3.14159265359.

Madhava also came up with Madhava Series.

\(\fracπ4 = 1 - \frac 13 + \frac 15 - \frac17+...\)

Ramanujam later discovered another series that converged much faster.

I will write another post with details of all that.