The Madhava Infinite Series and Value of Pi (π)
गणित ≠ Mathematics
Image of Madhava
Paper On Madhava by Royal Asiatic
Fast converging fraction with correction
Madhava's Infinite Series that has been transformed into Madhava-Leibnitz Infinite Series after constant proofs provided that Madhava did this many centuries ago before Leibnitz was even born.
Mādhava actually devised two methods to calculate the values of any irrational number, improper fractions among other things.
One of the ways is of a slowly converging function and another a fast converging one.
The Slow Method:
These two lines create a slow converging fraction that gives the value of pi to exactly 13 decimal places.
व्यासे वारिधिनिहते रूपहृते व्याससागराभिहते।
त्रिशरादिविषमसंख्याभक्तमृणं स्वं पृथक् क्रमात् कुर्यात्।।
This method used by Madhava utilizes "bhoota-sankhya".
Tantra-Samgraha, Adhyaya 2, Shlok 271
Bhoota-sankhya uses names of different elements to represent numbers (just like Katapayadi system uses different varna or alphabet combinations).
सागर for example represents 4 (an ocean surrounds land from all 4 sides). All synonyms (पर्यायवाची) of सागर like वारिधि, रत्नाकर etc mean the same.
Meaning of the shloka:
व्यासे (diameter) वारिधि (4) निहते (अर्थ: संबद्ध, संलग्न meaning multiply) रूप (1, everyone has unique form) हृते (divide) व्यास सागरा (4) भिहते।
त्रि (3) शर (5) आदि (etc) विषम (odd) संख्या भक्तम् (dividing) ऋणं (-) स्वं (+) पृथक क्रमात (alternatively) कुर्यात् (do)।
So the meaning of the first line becomes, "Four times the Diameter (in numerator) divided by one."
Mathematically represented as (4D/1)
In the second line it defines again four times the Diameter divided by a series of odd numbers starting with 3 having alternating plus and minus signs.
4D*(1/3-1/5+1/7-1/9...)
The second one is then subtracted from the first giving the value of pi.
This gives Circumference as stated in further shlokas.
Society ¹
Hence,
Circumference(परिधि) = (4D/1)-[4D/(1/3-1/5+1/7-1/9....)]
This comes as,
C/D = 4[1-1/3+1/5-1/7...]
This provides value of pi correct till 13 decimal places (most extensively used in astronomy and many other places).
π = 3.141592535922
Cambridge University Press article on Infinite Series in 15th Century.²
This series calculates value of Sin and Cos tables. It will be dealt in some later blog.³
The Fast Method:
The series however as said converges very slowly with the given method.
To ease the process and make the method fast another shloka was given.
The method is to force the series to converge at a desired point where the method becomes cumbersome.
The odd term where the process is to be stopped (due to boredom), we take the half of next even number as numerator.
That is if we stop at an odd number suppose p then we take (p+1)/2, where p+1 obviously becomes an even number.
The square of that (even number) added to unity is the denominator.
The addition in the end is like a correction term to definitely end the series on a specific point. This correction term is called अंत्य संस्कारः (meaning to end the process).
The sign of addition and subtraction in the correction term depends on the preceding term sign.
The result is quite accurate; in fact, more accurate than the one which may be obtained by continuing the division process (with a large number of terms in the series).
Yet another way to find Pi:
Another way of finding the value of pi given by Madhava is
विबुधनेत्रगजाहिहुताशनत्रिगुणवेदाभवारणबाहवः।
नवनिखर्वमितेवृतिविस्तरे परिधिमानमिदं जगदुर्बुधः।।
विबुध (33) नेत्र (2) गज (8) अहि (8) हुताशन(3) त्रि (3) गुण (3) वेदा(4) भ (नक्षत्र 27)
वारण (8) बाहु (2)।
नव (9) निखर्वम् (10¹¹) इते वृति (circle) विस्तरे परिधिमानम् (Circumference) इदं जगदुर्बुधः (as said by wise men).
The first line so stands as "2827433388233" as numbers are taken in opposite order.
Meaning of the shloka:
9 x 10¹¹ व्यास वाले वृत्त की परिधि 2827433388233 होगी।
That is,
(2827433388233/9x10¹¹) = 3.141592653922 or value of pi.
References:
1. Transactions of the Royal Asiatic Society of Great Britain and Ireland
Vol. 3, No. 3 (1834), pp. 509-523 (15 pages) (https://www.jstor.org/stable/25581775)
3. Special reference to the meanings explained by Sri K. Ramasubramanian.
4. Primary sources are from Tantra-Samgraha of Madhava and Yuktidwipika of Sankara Valiyar from Kerela School of Astronomy.
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