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	2	3	4	5	6	7	8	9
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.

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.

Loksabha Elections 2014: My Chinese arithmetic

Yesterday all the phases of polls closed in the country and all the television channels were out with their voodoo science that they call exit polls. Since everybody seem to be doing it, I thought I should also do it. So here it goes.

Exit polls have been wrong in pretty much all the elections in past. Here is a snipped of how exit polls have performed in past.

As we can see in the chart above, there has always been a significant difference in the exit poll results and actual results. Barring 1998, BJP has been always over-estimated in exit polls while congress has always been underestimated in exit polls. So, let's look at kind of error that has been seen in exit polls and actual results.

In the chart above, select Errors for Size, Actual for Y Axis and Time for X axis and click on Play and then we see that both BJP and congress seem to have increasingly higher errors over the years with BJP being over-estimated and congress underestimated.


So, I have decided that I am going to take the average underestimation for congress and BJP from 2004 and 2009 and try to predict the result for this year.

• Average BJP overestimation = 27%
• Average Congress underestimation = 21%
• Average seats predicted for BJP in this year's exit polls = 273
• Average seats perdicted for congress in this year's exit polls = 110

So my predictions for BJP and congress are as follows.

India Elections Tidbits #2: Margin of victory

Psephology in Indian elections of always interesting. Television commentators and professors from some mathematics istitute discus on television as if what they discovered is a forgone conclusion. I have heard statements like "Now that congress has lost it, what next for them?".

Winning margin for constituencies with less than 5% margin

Following is the list of constituencies with the winning margin.

Constituency No	Constituency	State	Percentage Margin
5	Zahirabad	Andhra Pradesh	1.71%
6	Medak	Andhra Pradesh	0.57%
10	Chelvella	Andhra Pradesh	1.60%
11	Mahbubnagar	Andhra Pradesh	2.43%
12	Nagarkurnool	Andhra Pradesh	4.66%
23	Kakinada	Andhra Pradesh	3.53%
24	Amalapuram	Andhra Pradesh	3.91%
25	Rajahmundry	Andhra Pradesh	0.21%
27	Eluru	Andhra Pradesh	3.97%
28	Machilipatnam	Andhra Pradesh	1.19%
29	Vijayawada	Andhra Pradesh	1.17%
30	Guntur	Andhra Pradesh	3.77%
31	Narasaraopet	Andhra Pradesh	0.15%
37	Hindupur	Andhra Pradesh	2.22%
40	Tirupati	Andhra Pradesh	1.82%
42	Chittoor	Andhra Pradesh	1.02%
43	Arunachal West	Arunachal Pradesh	0.46%
45	Karimganj	Assam	1.16%
50	Barpeta	Assam	3.38%
51	Gauhati	Assam	1.07%
53	Tezpur	Assam	3.58%
54	Nowgong	Assam	4.54%
57	Dibrugarh	Assam	4.68%
58	Lakhimpur	Assam	4.90%
69	Katihar	Bihar	1.88%
74	Vaishali	Bihar	2.70%
79	Hajipur	Bihar	4.45%
89	Pataliputra	Bihar	3.75%
100	South Goa	Goa	4.60%
102	Banaskantha	Gujarat	0.84%
105	Sabarkantha	Gujarat	2.39%
118	Panchmahal	Gujarat	0.19%
126	Valsad	Gujarat	0.51%
129	Sirsa	Haryana	2.41%
147	Chikkodi	Karnataka	4.42%
149	Bagalkot	Karnataka	2.47%
157	Dharwad	Karnataka	3.70%
163	Dakshina Kannada	Karnataka	3.22%
167	Mysore	Karnataka	1.70%
170	Bangalore North	Karnataka	3.62%
172	Bangalore South	Karnataka	2.90%
180	Malappuram	Kerala	0.00%
183	Alathur	Kerala	2.53%
184	Thrissur	Kerala	3.20%
186	Ernakulam	Kerala	1.58%
196	Bhind	Madhya Pradesh	4.38%
203	Satna	Madhya Pradesh	0.44%
209	Balaghat	Madhya Pradesh	3.82%
216	Ujjain	Madhya Pradesh	4.48%
218	Ratlam	Madhya Pradesh	4.65%
222	Khandwa	Madhya Pradesh	4.28%
224	Nandurbar	Maharashtra	3.97%
228	Buldhana	Maharashtra	3.29%
253	Mumbai South Central	Maharashtra	4.93%
264	Latur	Maharashtra	1.51%
276	Mizoram	Mizoram	0.00%
286	Dhenkanal	Orissa	2.17%
287	Bolangir	Orissa	4.82%
289	Nabarangpur	Orissa	0.00%
295	Bhubaneswar	Orissa	1.46%
303	Hoshiarpur	Punjab	2.93%
308	Ferozpur	Punjab	2.20%
339	Chennai North	Tamil Nadu	2.90%
340	Chennai South	Tamil Nadu	4.52%
342	Sriperumbudur	Tamil Nadu	2.23%
344	Arakkonam	Tamil Nadu	1.41%
350	Viluppuram	Tamil Nadu	0.35%
360	Karur	Tamil Nadu	3.12%
362	Perambalur	Tamil Nadu	0.95%
368	Sivaganga	Tamil Nadu	0.43%
370	Theni	Tamil Nadu	0.41%
380	Kairana	Uttar Pradesh	3.10%
381	Muzaffarnagar	Uttar Pradesh	1.60%
401	Badaun	Uttar Pradesh	1.96%
408	Sitapur	Uttar Pradesh	2.79%
423	Jalaun	Uttar Pradesh	0.91%
433	Ambedkar Nagar	Uttar Pradesh	1.87%
437	Gonda	Uttar Pradesh	1.37%
449	Salempur	Uttar Pradesh	4.64%
480	Jadavpur	West Bengal	4.54%
483	Howrah	West Bengal	3.75%
484	Uluberia	West Bengal	4.64%
509	Mahasamund	Chattisgarh	4.11%
523	Lohardaga	Jharkhand	1.45%

In summary, based on last election, there are 149 constituencies which were won with less than 5% margin.

• Less than 1% margin  -- 16 seats
• Between 1% and 2% margin -- 19 seats
• Between 2 % and 3% margin -- 30 seats
• Between 3% and 4% margin -- 36 seats
• Between 4% and 5% margin -- 48 seats

With such a large number of seats won with so low margin, how can these television commentators be so confident of seats won. Also how can they take the margin of error in votes polled and extrapolate that to number of seats.