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 Taking the partial sum of the first
terms we have the following approximation to π: Denoting the Madhava correction term by
, we have the following better approximation to π: Three different expressions have been attributed to Madhava as possible values of
, namely, In the extant writings of the mathematicians of the Kerala school there are some indications regarding how the correction terms
have been obtained, but there are no indications on how the expression
This has led to a lot of speculative work on how the formulas might have been derived.
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
appears there only as a step in the argument leading to the derivation of
[1][2] 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):[3][1] English translation of the verses:[3] In modern notations this can be stated as follows (where
is the diameter of the circle): If we set
, the last term in the right hand side of the above equation reduces to
The same commentary also gives the correction term
in the following verses (Chapter 2: Verses 295–296): English translation of the verses:[3] In modern notations, this can be stated as follows: where the "multiplier"
, the last term in the right hand side of the above equation reduces to
have the following bounds:[2][4] The errors in using these approximations in computing the value of π are The following table gives the values of these errors for a few selected values of
It has been noted that the correction terms
are the first three convergents of the following continued fraction expressions:[3] The function
that renders the equation exact can be expressed in the following form:[1] The first three convergents of this infinite continued fraction are precisely the correction terms of Madhava.
has the following property: In a paper published in 1990, a group of three Japanese researchers proposed an ingenious method by which Madhava might have obtained the three correction terms.
Their proposal was based on two assumptions: Madhava used
as the value of π and he used the Euclidean algorithm for division.
[5][6] Writing and taking
express them as a fraction with 1 as numerator, and finally ignore the fractional parts in the denominator to obtain approximations: This suggests the following first approximation to
which is the correction term
The fractions that were ignored can then be expressed with 1 as numerator, with the fractional parts in the denominators ignored to obtain the next approximation.
Two such steps are: This yields the next two approximations to
exactly the same as the correction terms