Around 1400 there lived an Indian astronomer and mathematician by the name of Madhava of Saṅgamagrāma. He was the greatest mathematician of his time and, among other mathematical feats, he and his followers managed to discover a lot of calculus 200 years before Newton and Leibniz did their thing. While preparing a video about this Indian calculus it occurred to me that some of Madhava’s discoveries can be used to give a nice intuitive explanation of Powell’s Pi Paradox, a very counterintuitive property of the famous Leibniz formula
that Martin Powell stumbled upon in 1983. In the end, giving an introduction to Madhava’s discoveries and giving that intuitive explanation is what I ended up doing in this video. ("Leibniz formula" should really be "Madhava formula"!)
00:35 Powell’s Piradox 🙂
04:08 Calculus made in India
15:18 Explanation of the paradox using Madhava’s correction terms
19:37 Calculus: Neither Newton nor Leibniz
24:22 Palm leaf music sequence
Videos in which I prove the Madhava formula:
Euler’s infinite pi formula generator: https://youtu.be/WL_Yzbo1ha4?t=558
Fermat’s Christmas theorem: Visualising the hidden circle in pi/4 = 1-1/3+1/5-1/7+… :https://youtu.be/00w8gu2aL-w
Euler’s real identity NOT e to the i pi = -1: https://youtu.be/yPl64xi_ZZA?t=956
The Wikipedia articles about Madhava, his school and his discoveries are excellent starting points if you are interested in more details:
My explanation of how Madhava may have discovered his correction terms is based on this article by Hayashi, T., T. Kusuba, and M. Yano. "The Correction of the Madhava Series for the Circumference of a Circle." Centaurus 33 (1990): 149-174. This article is sitting behind a paywall. However, the wiki article linked to above is a good summary.
The original article by Powell in which he reports on his observation and asks for an explanation is here: https://www.jstor.org/stable/3616550
Five explanations were subsequently given in this article published in the same math journal: https://www.jstor.org/stable/3617175 (note on JSTOR this collection of articles is broken up into four parts. This link is only to the first part).
The most in-depth article about the Powell’s Pi Paradox is this one here by the Borwein brothers and K. Dilcher on "Pi, Euler Numbers, and Asymptotic Expansions": https://www.maa.org/sites/default/files/pdf/pubs/amm_supplements/Monthly_Reference_4.pdf
In this article they also analyse similar paradoxical behaviours of closely related series like ln(2)=1-1/2+.1/3-1/4+1/5-…
The photo of that palm leaf manuscript page shown at the end of the video was sourced from the slideshow of the 2022 International Congress of Mathematicians invited lecture by K. Ramasubramanian. https://www.youtube.com/watch?v=ctrROj3Tv-E . Also check out his website for LOTS of information about ancient Indian mathematics. https://www.kramasubramanian.com/ I have no idea what it says on this palm leave page, but I trust my colleague to have shown us the right thing here 🙂
The picture of Madhava in the thumbnail is what Google is pushing. However, this image is not a true likeness of the actual person: https://commons.wikimedia.org/wiki/File:Madhava_sangamagrama.jpg
A couple more links to check out:
The Discovery of the "Series Formula for π by Leibniz, Gregory and Nilakantha" by Ranjan Roy: https://www.jstor.org/stable/2690896 Goes into a lot of detail in terms of proofs.
3:36 one of the digit difference towards the end not highlighted
14:39 In the 121 terms sums the correction terms features a minus in the place of a plus.
18:36 In the fourth correction term it should be …N+9/(4N)
Music: Adventure of a lifetime by Campagna