Madhava's correction term is a mathematical expression attributed to Madhava of Sangamagrama (c. 1340 – c. 1425), the founder of the Kerala school of astronomy and mathematics, that can be used to give a better approximation to the value of the mathematical constant π (pi) than the partial sum approximation obtained by truncating the Madhava–Leibniz infinite series for π. The Madhava–Leibniz infinite series for π is
π 4 = 1 − 1 3 + 1 5 − 1 7 + ⋯ {\displaystyle {\frac {\pi }{4}}=1-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+\cdots }
Taking the partial sum of the first n {\displaystyle n} terms we have the following approximation to π:
π 4 ≈ 1 − 1 3 + 1 5 − 1 7 + ⋯ + ( − 1 ) n − 1 1 2 n − 1 {\displaystyle {\frac {\pi }{4}}\approx 1-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+\cdots +(-1)^{n-1}{\frac {1}{2n-1}}}
Denoting the Madhava correction term by F ( n ) {\displaystyle F(n)} , we have the following better approximation to π:
π 4 ≈ 1 − 1 3 + 1 5 − 1 7 + ⋯ + ( − 1 ) n − 1 1 2 n − 1 + ( − 1 ) n F ( n ) {\displaystyle {\frac {\pi }{4}}\approx 1-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+\cdots +(-1)^{n-1}{\frac {1}{2n-1}}+(-1)^{n}F(n)}
Three different expressions have been attributed to Madhava as possible values of F ( n ) {\displaystyle F(n)} , namely,
F 1 ( n ) = 1 4 n {\displaystyle F_{1}(n)={\frac {1}{4n}}}
F 2 ( n ) = n 4 n 2 + 1 {\displaystyle F_{2}(n)={\frac {n}{4n^{2}+1}}}
F 3 ( n ) = n 2 + 1 4 n 3 + 5 n {\displaystyle F_{3}(n)={\frac {n^{2}+1}{4n^{3}+5n}}}
In the extant writings of the mathematicians of the Kerala school there are some indications regarding how the correction terms F 1 ( n ) {\displaystyle F_{1}(n)} and F 2 ( n ) {\displaystyle F_{2}(n)} have been obtained, but there are no indications on how the expression F 3 ( n ) {\displaystyle F_{3}(n)} has been obtained. This has led to a lot of speculative work on how the formulas might have been derived.
Correction terms as given in Kerala texts The expressions for F 2 ( n ) {\displaystyle F_{2}(n)} and F 3 ( n ) {\displaystyle F_{3}(n)} are given explicitly in the Yuktibhasha, a major treatise on mathematics and astronomy authored by the Indian astronomer Jyesthadeva of the Kerala school of mathematics around 1530, but that for F 1 ( n ) {\displaystyle F_{1}(n)} appears there only as a step in the argument leading to the derivation of F 2 ( n ) {\displaystyle F_{2}(n)} . The Yuktidipika–Laghuvivrthi commentary of Tantrasangraha, a treatise written by Nilakantha Somayaji an astronomer/mathematician belonging to the Kerala school of astronomy and mathematics and completed in 1501, presents the second correction term in the following verses (Chapter 2: Verses 271–274):
English translation of the verses:
"To the diameter multiplied by 4 alternately add and subtract in order the diameter multiplied by 4 and divided separately by the odd numbers 3, 5, etc. That odd number at which this process ends, four times the diameter should be multiplied by the next even number, halved and [then] divided by one added to that [even] number squared. The result is to be added or subtracted according as the last term was subtracted or added. This gives the circumference more accurately than would be obtained by going on with that process." In modern notations this can be stated as follows (where d {\displaystyle d} is the diameter of the circle):
Circumference = 4 d − 4 d 3 + 4 d 5 − ⋯ ± 4 d p ∓ 4 d ( p + 1 ) / 2 1 + ( p + 1 ) 2 {\displaystyle =4d-{\frac {4d}{3}}+{\frac {4d}{5}}-\cdots \pm {\frac {4d}{p}}\mp {\frac {4d\left(p+1\right)/2}{1+(p+1)^{2}}}}
If we set p = 2 n − 1 {\displaystyle p=2n-1} , the last term in the right hand side of the above equation reduces to 4 d F 2 ( n ) {\displaystyle 4dF_{2}(n)} . The same commentary also gives the correction term F 3 ( n ) {\displaystyle F_{3}(n)} in the following verses (Chapter 2: Verses 295–296):
English translation of the verses:
"A subtler method, with another correction. [Retain] the first procedure involving division of four times the diameter by the odd numbers, 3, 5, etc. [But] then add or subtract it [four times the diameter] multiplied by one added to the next even number halved and squared, and divided by one added to four times the preceding multiplier [with this] multiplied by the even number halved." In modern notations, this can be stated as follows:
Circumference = 4 d − 4 d 3 + 4 d 5 − ⋯ ± 4 d p ∓ 4 d m ( 1 + 4 m ) ( p + 1 ) / 2 , {\displaystyle {\text{Circumference}}=4d-{\frac {4d}{3}}+{\frac {4d}{5}}-\cdots \pm {\frac {4d}{p}}\mp {\frac {4dm}{\left(1+4m\right)(p+1)/2}},}
where the "multiplier" m = 1 + ( ( p + 1 ) / 2 ) 2 . {\textstyle m=1+\left((p+1)/2\right)^{2}.} If we set p = 2 n − 1 {\displaystyle p=2n-1} , the last term in the right hand side of the above equation reduces to 4 d F 3 ( n ) {\displaystyle 4dF_{3}(n)} .
Accuracy of the correction terms Let
s i = 1 − 1 3 + 1 5 − 1 7 + ⋯ + ( − 1 ) n − 1 1 2 n − 1 + ( − 1 ) n F i ( n ) {\displaystyle s_{i}=1-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+\cdots +(-1)^{n-1}{\frac {1}{2n-1}}+(-1)^{n}F_{i}(n)} . Then, writing p = 2 n + 1 {\displaystyle p=2n+1} , the errors | π 4 − s i ( n ) | {\displaystyle \left|{\frac {\pi }{4}}-s_{i}(n)\right|} have the following bounds:
1 p 3 − p − 1 ( p + 2 ) 3 − ( p + 2 ) < | π 4 − s 1 ( n ) | < 1 p 3 − p , 4 p 5 + 4 p − 4 ( p + 2 ) 5 + 4 ( p + 2 ) < | π 4 − s 2 ( n ) | < 4 p 5 + 4 p , 36 p 7 + 7 p 5 + 28 p 3 − 36 p − 36 ( p + 2 ) 7 + 7 ( p + 2 ) 5 + 28 ( p + 2 ) 3 − 36 ( p + 2 ) ⋯ 4 p 5 + 4 p − 4 ( p + 2 ) 5 + 4 ( p + 2 ) < | π 4 − s 3 ( n ) | < 36 p 7 + 7 p 5 + 28 p 3 − 36 p . {\displaystyle {\begin{aligned}&{\begin{aligned}{\frac {1}{p^{3}-p}}-{\frac {1}{(p+2)^{3}-(p+2)}}&<\left|{\frac {\pi }{4}}-s_{1}(n)\right|<{\frac {1}{p^{3}-p}},\\[10mu]{\frac {4}{p^{5}+4p}}-{\frac {4}{(p+2)^{5}+4(p+2)}}&<\left|{\frac {\pi }{4}}-s_{2}(n)\right|<{\frac {4}{p^{5}+4p}},\end{aligned}}\\[20mu]&{\begin{aligned}&{\frac {36}{p^{7}+7p^{5}+28p^{3}-36p}}-{\frac {36}{(p+2)^{7}+7(p+2)^{5}+28(p+2)^{3}-36(p+2)}}\cdots \\[10mu]&{\phantom {{\frac {4}{p^{5}+4p}}-{\frac {4}{(p+2)^{5}+4(p+2)}}}}<\left|{\frac {\pi }{4}}-s_{3}(n)\right|<{\frac {36}{p^{7}+7p^{5}+28p^{3}-36p}}.\end{aligned}}\end{aligned}}}
Numerical values of the errors in the computation of π The errors in using these approximations in computing the value of π are
E ( n ) = π − 4 ( 1 − 1 3 + 1 5 − 1 7 + ⋯ + ( − 1 ) n − 1 1 2 n − 1 ) {\displaystyle E(n)=\pi -4\left(1-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+\cdots +(-1)^{n-1}{\frac {1}{2n-1}}\right)}
E i ( n ) = E ( n ) − 4 × ( − 1 ) n F i ( n ) {\displaystyle E_{i}(n)=E(n)-4\times (-1)^{n}F_{i}(n)}
The following table gives the values of these errors for a few selected values of n {\displaystyle n} .
Continued fraction expressions for the correction terms It has been noted that the correction terms F 1 ( n ) , F 2 ( n ) , F 3 ( n ) {\displaystyle F_{1}(n),F_{2}(n),F_{3}(n)} are the first three convergents of the following continued fraction expressions:
1 4 n + 1 n + 1 n + ⋯ {\displaystyle {\cfrac {1}{4n+{\cfrac {1}{n+{\cfrac {1}{n+\cdots }}}}}}}
1 4 n + 1 2 n + 2 2 4 n + 3 2 n + ⋯ ⋯ + r 2 n [ 4 − 3 ( r mod 2 ) ] + ⋯ = 1 4 n + 2 2 4 n + 4 2 4 n +