Powell’s Pi Paradox: the genius 14th century Indian solution

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
π/4=1-1/3+1/5-1/7+1/9-...
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"!)
00:00 Intro
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
24:56 Thanks!
Videos in which I prove the Madhava formula:
Euler's infinite pi formula generator: • Euler's infinite pi fo...
Fermat's Christmas theorem: Visualising the hidden circle in pi/4 = 1-1/3+1/5-1/7+... : • Fermat's Christmas the...
Euler's real identity NOT e to the i pi = -1: • Euler's real identity ...
The Wikipedia articles about Madhava, his school and his discoveries are excellent starting points if you are interested in more details:
en.wikipedia.org/wiki/Madhava...
en.wikipedia.org/wiki/Madhava...
en.wikipedia.org/wiki/Kerala_...
en.wikipedia.org/wiki/Madhava...
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: www.jstor.org/stable/3616550
Five explanations were subsequently given in this article published in the same math journal: 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": www.maa.org/sites/default/fil...
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. • Krishnamurthi Ramasubr... . Also check out his website for LOTS of information about ancient Indian mathematics. 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: commons.wikimedia.org/wiki/Fi...
A couple more links to check out:
The Discovery of the "Series Formula for π by Leibniz, Gregory and Nilakantha" by Ranjan Roy: www.jstor.org/stable/2690896 Goes into a lot of detail in terms of proofs.
www.pas.rochester.edu/~rajeev...
Some bugs:
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
Enjoy!
Burkard

[@paradox5854]

As an Indian it is so disappointing that so far I wasn't aware about this Indian gem...... But I am grateful to you for revealing it before us🙏🙏🙏

[@sagarpuri7838]

True

[@liveinpeace1189]

Please read Aryabhattiyam. I am sure also don't know what is sine and from where it comes. Read Aryabhattiyam

[@user-pq4ep3zd3j]

@@liveinpeace1189 also Suryasidhant

[@yj9032]

That because you don’t read books. Please don’t expect spoon fed knowledge by schools and media.

[@douglasdeoliveiracardoso9345]

there are a lot of great indian mathematicians, I'm surprised you don't know that, didn't the schools talk about them?

[@higherbeingX]

Madhava, Neelakanda, Chomathiri, etc, mathematicians belong to the 13th and 14th-century Kerala School of Mathematics. The math at that time was very advanced touching the basics of calculus and infinite series, but most were written as 'Shlokas' in Sanskrit or old malayalam. ie as poems. Kerala was the epicenter of global spice trade and it's possible that many of the original ideas were transferred to Europe which benefitted European mathematicians. Taylor series expansion and various other infinite series were discovered by Kerala mathematicians

[@honourabledoctoredwinmoria3126]

It's possible Fermat knew of the work of the Kerala school. It's even more likely that Cavalieri did. Although there's no direct evidence, his mind seemed to run down very similar lines. But I think there's a very concrete way to show Europeans did not know the results of that work at all. It became fashionable in the 16th Century for Mathematicians to show their prowess at calculation by working out the digits of pi using Archimedes's method and polygons with ridiculous numbers of sides. If any of them had known of Madhava, then we would expect someone to make use of it or attempt to reverse engineer it, and yet, no one does so until Newton, Gregory, and Leibniz.

[@prateemmandal743]

One correction to the above. Yuktibhasha written by Jyeshthadeva from where we know these discoveries and their proofs was written in Malayalam and not Sanskrit and in fact that is one of the reasons why these exploits of Madhava was slow to percolate (lots of Sanskrit scholars were scouring Sanskrit books because well all knowledge was supposed to have been recorded in Sanskrit missing out on other stuff)

[@higherbeingX]

@@prateemmandal743 Yes it's a typo..its written in ancient Malayalam. I don't know if its proto Malayalam or vattezhuthu..its hard to read the palm leaf though

[@user-nx4ti8xs1o]

Now Kerala is epicenter of Jihad terrorism and Hammer+Sickle Communism.

[@caret4812]

it is also possible that two unrelated people from two different places to come up with the same idea

[@kamidhunkumar3021]

When you showed the manuscripts(written in my mother tongue Malayalam, spoken over by only less than 40 million people in a small state known as Kerala in India) towards the end of the video, it gave me chills literally. I am glad that the efforts of these ancient Indian mathematicians are being appreciated by the mathematics fraternity all over the world.

[@Tatvomasi]

Malayali bro 😁👏

[@Mathologer]

For fun I included some auto-translated (from English) subtitles for Malayalam and a couple of other Indian languages. Maybe have a look. I would be interested in finding out how good/bad/useful these sort of subtitles are these days.

[@dejablueguitar]

THAT is a COOL STORY!! - WOW!! (damn... I love the comments on these videos!)

[@aditya95sriram]

​​@@Mathologer I went through a few minutes of the Hindi subtitles (another Indian language), it's not perfect but it's pretty decent in my opinion. Definitely better than no subtitles.

[@warpdrive9229]

40 million is a huge number for the West LOL. Even Afghanistan has a lesser population.

[@alexpotts6520]

Imagine discovering this in the 1980s, then learning that actually some geezers had gotten there not only before the invention of computers, but before the printing press...

[@Mathologer]

Well, it's really like this. Because of the lack of computing power in those ancient times, it would have been very unlikely for mathematicians to observe that paradox. On the other hand, if Powell had travelled back in time and told Madhava about this paradox, Madhava would have been able to come up with the explanation in this video straightaway. Having said that, as far as I know I am the first to make the connection between Madhava's correction terms and the paradox :)

[@bhaiya.jakhwal]

​@@Mathologer A lot of genius that India was has either been destroyed or wrongly attributed passing through the winds of colonialism

[@DebanjanBasu]

Imagine that even though your discovery had been predated by a lot of old geezers, your discovery still gets named after you. Okay enough with the snarkiness!! Indian philosophy and mathematics was atleast as important as the Greeks to the development of the Renaissance and should be acknowledged as such if mathematics claims to be truly a global community. Not sure about the flying chariots and unicorns though. Those might have been made up!!

[@DebanjanBasu]

Oh and thanks to @mathologer for referring to the madhava series. That's how it is done folks!!

[@kumudaroy7607]

In my opinion, You may have researched and applied it, but not discovered it

[@ComboClass]

The other paradox: how every Mathologer video manages to fit in so many interesting facts and beautiful demonstrations!

[@asheep7797]

omg domotro

[@dejablueguitar]

SOOOO AGREE!! This guy makes math MAKE SENSE!

[@simonmultiverse6349]

I noticed the first few seconds of Babooshka by Kate Bush as this video started. We're not ignorant, you know!

[@WippSheridan]

Your channel is super!

[@wyattstevens8574]

Domotro!

[@agrajyadav2951]

Madhava was truly one of the greatest mathematicians of all time

[@Mathologer]

Yes, but hardly anybody seems to know about him, not even in India!

[@LuckyKumar-wb9xm]

​@@Mathologer that's the sad state of affairs today sadly 😔

[@agrajyadav2951]

​@@Mathologer It really is sad, colonialism extends to every aspect of society, even its perception of something as objective as science. You would mostly only hear Indians trying to show India's scientific achievements by misrepresenting them, if not straight up making things up, and its not because they are so enthusiastic about science and its history, but because they are insecure about the perception of India in the world, which ultimately is a result of colonialism. Its really sad how conquest and colonialism affect scientific progess. India went from one of the few bright spots in the scientific world to a mediocre country after centuries of destruction. Not to mention the texts that were literally destroyed during invasions. Its scary to think how much knowledge humanity has lost just because of wars and plundering.

[@shree-wl9bs]

@@Mathologer thanks to you,now more people will know him.

[@aek12]

@@Mathologer I think it is because when we saw the West with all these technological advances and compared to our own country, We just see that we have nohope for the future with ever increasing population. So nobody even bother to look at the history, because we never had any kind of industrial achievement. But we forgot that it is just the brain and knowledge which is a necessary recipe for advancement. Can you describe the mathematics and engineering required to build Ellora caves in India. We just have a feeling of inferiority based on the current condition of India in comparison to the country like U.S.

[@manomishra]

Madhava, Aryabhatta, Brahmagupta, Baraha- the list of such geniuses goes on and on. This talk was so beautiful it brought me immense joy. A flowers fragrance benefits all - no problem. The problem is the same people not recognizing, others walking nearby not recognizing. It is good to see western and other intellectuals finally recognizing. We must recognize where it is due - like the agricultural genius from South America, Math and philosophy from India, material science from China.

[@RudraSingh-pb5ls]

Learnt quite significant stuff from ur comment

[@jamesandrew6188]

Chanakya

[@manomishra]

@@jamesandrew6188 - I am not aware of any significant mathematical work by Chanakya. I have read his most famous work -Arthashastra. It is more a book on political philosophy, some chapters are on taxation. It was not mathematics or even economics in the modern sense.

[@srinumedaram5822]

How we will able to know our magnificent history when our school textbooks are written by Marxist Historians(distorted).

[@jacobzimmermann59]

To be fair, the lack of recognition of the ancient Indian and Chinese mathematicians' work is probably not all because of a Western colonial mentality or sense of superiority, although that undoubtedly plays a role as well, don't get me wrong. However, I think it's also simply the result of a cultural and linguistic barrier. In their time, Western scholars like Leibniz and Gregory had little chance of even looking at a medieval Indian manuscript within their lifetime. But even if they could, hardly anyone in Europe would have been capable of reading it, yet alone realising that it contains a mathematical discovery that predated their own by two hundred years.

[@KarlFredrik]

Always found it insanely impressive when ancient mathematicians used normal language and text instead of our modern formalism to transfer ideas and results. Imagine how hard it must be to do mathematics that way.

[@Nikhil_Sahu_G]

Here is an example how our ancestors knowledge about Maths written in form of Sanskrit Mantras/Shlokas ( verses) चतुराधिकं शतमष्टगुणं द्वाषष्टिस्तथा सहस्राणाम्। अयुतद्वयस्य विष्कम्भस्यासन्नो वृत्तपरिणाहः॥ It represents the equation linking the radius and circumference of a circle. It was developed in India. Later, it came from abroad, in a more brief and decoded version (π = 22/7). When this formula is converted into numbers, will find the following: (100 + 4)* 8 + 62,000/20,000 = 3.1416 However, in Rig Veda, the value of π is mentioned until the 32nd number after the decimal. It is shown as the following: 3.1415926535897932384626433832792…

[@rudraksh111]

But with all due respect wouldn't it make maths more easy , interesting and fun to be introduced and talked about like normal language and easy to exchange ideas. Unlike today, although today's structure increases the efficiency infinite times but this can introduced to a scholar later in studies after formation of concepts. I want to make this a reality soon , wish me luck.

[@trollarasan]

Actually, Indian mathematicians were experts in maths as well as literature. This is the same as a person being an expert in both science and arts. So it is actually easier for them to convey it in normal language. everything got messed up thanks to European mathematicians who are not willing to accept that mathematics is a mess now thanks to their failure to understand Indian mathematics.

[@radhegovinda2]

In fact it was & IS much easier to remember & apply the complex Mathematical Formulae in Language ~ The Values attributed to the Varnas (consonants) are easily remembered & interpreted/deciphered ~!!! That's how Vedic Mathematics is Practiced ~!!!

[@radhegovinda2]

​@@rudraksh111In fact it was & IS much easier to remember & apply the complex Mathematical Formulae in Language ~ The Values attributed to the Varnas (consonants) are easily remembered & interpreted/deciphered ~!!! That's how Vedic Mathematics is Practiced ~!!!

[@a123s1l]

For those who are interested, contributions of Madhava's school are covered in the book titled, "A Passage to Infinity" by George Gheverghese Joseph. Another book, "Mathematics in India", by P.P. Divakaran covers 5000 yrs of history of mathematics in India starting from the Indus valley civilization. P.P. Divakaran also states that Madhava was a pioneer in the methods of calculus much earlier than it development by Newton and Leibniz.

[@sureshadusumilli4960]

Also in the book "THE IMPERISHABLE SEED: How Hindu Mathematics Changed the World and Why this History was Erased" by Bhaskar Kamble.

[@aditichavan3427]

👌👌👍

[@vivekkt89]

As an Indian and a Keralite I'm ashamed of not knowing anything about this great Mathematician. Thanks a lot sir for enlightening us with this information. My country had to deal with lot of violence in the past for centuries. But I feel if we can move past them and try to rebuild the pieces to understand what was lost would bring us more peace. And what you're doing goes along that way. Salute.

[@shesadrisekharbagchi3869]

All because of communists and islamist

[@euthyphr0]

The manuscript in the end was really enlightening. It makes you wonder how much past (and possibly present) mathematicians were (/are) influenced in their thinking by the writing conventions of their time.

[@Mathologer]

For me one of the most significant contributions of Leibniz to mathematics was his calculus notation. Absolutely genius to what extent this notation manages to capture the maths itself :)

[@hueckelaromat]

To see the manuscript passing by was quite the revelation. Just imagine how much facts were already there and had been lost for whatever reason?!

[@NoNTr1v1aL]

When I got 4th rank in Kerala for the Madhava Mathematics competition, I realised that I had the potential to be a mathematician. Thank you for letting more people know about him!

[@Mathologer]

So are you actually from Kerela?

[@johanlindeberg7304]

Is that close to the region where Srinivasa Ramanujan was from ? Was there a tradition of teaching higher mathematics in that part of India ?

[@NoNTr1v1aL]

@@Mathologer Yup.

[@NoNTr1v1aL]

@@johanlindeberg7304 Yes. Ramanujan was born in the neighbouring state of Kerala called Tamil Nadu. Madhava was the founder of the Kerala school of astronomy and mathematics. Ramanujan was a genius who went through a book(i forgot which math book exactly) and produced all those wonderful results!

[@prasoonjha6314]

@@NoNTr1v1aL Probably you are referring to G. S. Carr's "A Synopsis of Pure and Applied Mathematics".

[@tamasdhgebrq5968]

There were/are mathematicians, who can smart formulas not only elaborate, but they really see and feel the underlaying deep relationship. Also Ramanujan was one of them.

[@hujackus]

That image of Madhava seriously looks like you. I could not believe it wasn't an AI generated image of you from 15th century India.

[@haniyasu8236]

I was literally about the comment the same exact thing hahaha

[@Mathologer]

That's what my wife said too :)

[@at7388]

Can it be that Madhava was reborn as Mathologer?

[@debasishraychawdhuri]

I guess great men look alike?

[@DipayanPyne94]

​@@debasishraychawdhuri Socrates waa ugly. I guess you are wrong 😂

[@WasickiG]

Nine years ago I edited the Wikipedia article on Madhava of Sangamagrama to include the continued fraction correction term to the Madhava-Leibniz series. Previously the article stated that it was not clear how Madhava had found the three correction terms in polynomial form he had provided. They happened to be the first three approximants of that continued fraction. I have found the pattern of the continued fraction numerically, using pi to 34 digits rather than the few given by 355/113. (Edited to fix a couple of typos)

[@Mathologer]

Glad to make your acquaintance :)

[@WasickiG]

My pleasure! That was the byproduct of what began as a programming exercise to compute the well known series for log(2) [ Google for “Programming Exercise (HP-15C, 15C LE - and others)” ] I have never had access to the article you mention in the description, but I’ve always curious about its contents, so thank you very much for this glimpse of it. I have come up with a third continued-fraction that works as a correction factor for the Wallis Product, which unlike the previous ones appear to be something really new. In the WolframAlpha notation the complete expression can be written as “ Product((4k^2)/(4k^2 - 1),{k,1,n})*(2 + 4/(8n + 3 + ContinuedFractionK[4k^2 - 1,8n +4,{k,1,n}])) ” This produces 25n/12 correct decimals of pi, however a proof is still missing.

[@Mathologer]

The next Mathologer is about a nice paradox that arises from the alternating harmonic series. Maybe something you'd be interested too.

[@WasickiG]

@@Mathologer I’m looking forward to it. Thank you very much!

[@curtiswfranks]

My goodness! These people were so smart. This is crazy. Wonder is essential to beauty, and I am feeling both wonder and beauty right now. It is a travesty that I was unaware of Madhava's existence.

[@bikrampatra2734]

As an Indian and a student of mathematics it is ridiculous that I came to know about Madhava and his discoverys through you sir, although I am familiar with that series but don't heard Madhava's namae before Thank you sir for this video .🙏🙏

[@denshi_lives29]

Same here . I was just given some names for like 30 secs that's all no more mention as if they were not important . I am frustrated with the colonial system that my parents generation holds so dear .

[@TheMalllu]

well so called "south" come s in the way

[@curious_banda]

There is also Bhaskara 2. Look him up.

[@aniket385]

Coz we Indians instead spend our time arguing about mythological stories

[@curious_banda]

@@aniket385 cope harder. All these mathematicians trace the lineage of their knowledge to Brahma.

[@phyarth8082]

Ramanujan-Sato series is fastest known pi converging algorithm to calculate pi values using computers. The Chudnovsky algorithm is based on Ramanujan’s π formulae.

[@neerajmenon8107]

I belong to Madhavan's home town, present day Irinjalakuda, Kerala. I am really happy to see the world recognizing his works and contributions. Really appreciate the efforts put in by Mathologer😃

[@alokaggarwal6859]

Thanks! Excellent video as always. So interesting to learn that this type of math was being done in the 1400s.

[@Mathologer]

Thank you very much :)

[@leif1075]

@@Mathologer Who first discovered or formulated infinite series representation of functions like cosine x or sine x or e^× or tan x if it wasn't Newton or Leibnitz..does anyone know for sure? Thanks ks for sharing.

[@Mathologer]

@@leif1075 As far as we know it was Madhava. Of course, what he and his colleagues did was not as general as what was developed later in the west. As mathematicians interested in astronomy they had a very narrow focus on things to do with trigonometry. So no e^x for example. I find it really surprising that none of this is widely know among mathematicians. I also only found out about Madhava say 10 years ago.

[@TuxedoMaskMusic]

This is an astoundingly efficient and absolutely priceless piece of knowledge. It may sound cleche but this is precisely the sort of lesson that changes a mans perspective and life. Ty for that. This was an absolute treat!

[@Mathologer]

Glad that this one worked so well for you. :)

[@TuxedoMaskMusic]

@@Mathologer truly blows my mind. The efficiency of a method from 1400's! Just amazing.

[@dejablueguitar]

Hey MAX!! (I remember ya on TV LOL!) and you are SPOT ON! TOTAL TREAT! Did it scramble yer CPU too?? Made the inside top of skull tingle as the lights went ON!

[@gcewing]

Having to write all your formulas as poetry sounds both amazing and horrifying at the same time. It must have made everything so much more difficult!

[@Mathologer]

Yes, really quite unbelievable.

[@BakulShah]

@@Mathologer It is said that Sanskrit was invented to write poetry! Given its free word order, sandhi rules and many synonyms (objects are denoted by their properties), one can pick the right words and rearrange a sentence just right to make it memorable and easy to chant, without altering its meaning!

[@rahulvats95]

They only passed on mathematical formulas using this, the real math was done with notations only.

[@NishanthSalahudeen]

@@BakulShah to be clear, madhava wrote in malayalam. its not sanskrit. Malayalam is largely tamizh family language which may be more related to Pali (Buddhist influence) though there are several words borrowed into malayalam from sanskrit later. As a malayali myself, it was always said that malayalam was never a language of knowledge. As it turns out,... thats not true!

[@BakulShah]

@@NishanthSalahudeen We know about Mādhava's work via other later works such as Tantrasamgraha (1501) which was in Sanskrit and Yuktibhāṣā (~1530) which was in Malayālam. Going back further, Mādhava's student Parameshvaran wrote Drigganita (1430) in Sanskrit (acc. "A history of the Kerala school of HIndu Astronomy" book). So it is more than likely that Mādhava used Sanskrit but we can't be sure! FInally note that Sanskrit was often written in the local script.

[@danieljudah8992]

I am so glad that you are still around!

[@Mathologer]

I am glad that you are glad :)

[@gix8464]

India is very rich in mathematical and metaphysical knowledge as well as philosophy. The mathematics is earlier times were in the form of shlokas . Respected sir , i am an IIT student currently pursuing my degree in aerospace engineering, but i love maths . Here is a request from me to please make a video on baudhyan theorem and Pythagoras theorem , baudhyan theorem came earlier than Pythagoras , but baudhyan stated it in a very different way , leading to the same result as Pythagoras . It would be intresting to know about both of them , their differences and similarities from a great teacher like you .

[@varunb9661]

Thank you for educating us on Madhava...never taught in Indian school books..

[@sskiyer]

Not true.

[@denshi_lives29]

And will never be taught unless..

[@gix8464]

@@sskiyer have you being taught ?

[@alexandrewashington6447]

Yet another Mathologer gem. It's definitively my preferred channel! Thx Mr. Polster, for give us such a wonderful presentation in all senses.

[@jadayus55]

Proud to be born in this Legends place. He used to lie down on a rockbed inside the temple for observing the sky for long hours. The sad fact is that many of the Indians especially people from Kerala are not aware of his existence.

[@sathishkumarkanaparthy1569]

For more than 1000 years India was under foreign rule. You can imagine how much cultural, economic, scientific, literature etc damage had happened. So much so that Indians themselves don't believe such geniuses were their ancestors. Thank you so much for showing the world about Madhava. There are so many other people who require special attention and mention. The world will only benefit by this exploration.

[@AshishKumar-jj7yw]

This video made me fall in love with Mathematics again. To know that people have been doing such amazing things from 1400s is absolutely humbling to me.

[@Deutschebahn]

Thank you for showing the brilliance and true history here! Often non-western mathematicians, scientists, inventors, artists and others are not given due credit so it is wonderful to see the opposite be the case ❤

[@idothing_s]

Thank you for giving the credit that Indian mathematicians really deserve.

[@filspeaks]

holy shit i found the convergance speed ups like a month ago bruh why did an indian have to find it 600 yrs ago, i felt like such a gamer

[@oddlyspecificmath]

That panel at the end...a reminder how easily momentous things can be overlooked, by anyone not understanding what they're holding.

[@anujmishra4412]

Thanks a lot, Mathaloger! I have studied some works of Madhava but didn't know about these beautiful correction terms. Our textbooks in India still reflect a post-colonial mindset, resulting in a significant lack of awareness and ignorance. Unfortunately, many important contributions from Indian scholars are overlooked or not adequately highlighted. For instance, Baudhayana, who authored the world's first geometry textbook, including the Pythagorean theorem long before Pythagoras, is rarely mentioned. Similarly, Pingala's profound insights into permutations and combinations within the context of literature are often disregarded, despite their beauty and significance. Aryabhata's foundational work in trigonometry, which forms the basis of our modern understanding, also receives insufficient attention. Furthermore, there's no mention of Madhava, or any mathematician from the Nila school, in our great textbooks. They only perpetuate a subtle sense of inferiority complex by failing to acknowledge these remarkable achievements. It is high time for a change in our educational curriculum. One aspect I particularly admire is India's significant contributions to philosophy and spirituality. These traditions are not only highly logical but also explore the limitations of logic itself, as evident in the concept of non-dualism. Additionally, architectural marvels like the Kailash temple exemplify India's profound knowledge in the field of architecture. Regrettably, we have lost a great deal of knowledge due to constant invasions, including the destruction of the precious Nalanda University, the world's first residential university. There is so much more to be said about India's marvelous contributions to the world. It is my hope that our generation recognizes and appreciates the immense legacy left by our ancestors. Thank you so much again. Keep up the amazing work that you do!!

[@premshah7685]

This is the reason I love to read and learn about Indian History, but the problem is that, in my achool, classes, college, I was the only one who loved history ... Other people are of the opinion that history is boring and cannot help us in any way, but we say na "bandar kya jaane adrakh ka swaad !"😄 I really think everyone should learn Indian History it is really interesting and never fails to make us proud of what India was before colonization!

[@clp2149]

Very aptly described by you, after Modi era Hindus are trying to reclaim what has been lost to fake history and biased teachings.

[@clp2149]

@@logohub1234 which year?

[@AnujMishra-is5uf]

@@logohub1234 Thanks for pointing this out, but I specifically wrote the world's first "residential" university, according to most historians, where people used to visit and gain knowledge.

[@upwardogdownwardog]

Thank you for the fantastic presentation!! 🙏

[@Mathologer]

Glad you enjoyed it!

[@v.gopalakrishnan350]

I'm 65 years old. I graduated in mathematics with a first class in the year 1977 from Madras University! It's a shame none of my Indian professors had anything to say about these great Indian mathematicians! It's all Newton and Lebnitz and Euler!

[@GowthamV07]

Because in those times they din't have the internet to teach theme real history of india. But now we can learn if you have the will to learn. And also don't forget Congress!

[@mauryanempire7503]

​@@GowthamV07 lol because during that time his contribution wasn't well known it was only recently it came to light due to archeological surveys.

[@mauryanempire7503]

Newton and leibniz did invented important components of Calculus like limits, derivatives and integration but we should not ignore the contribution of south indian mathematicians like madhava.

[@Mrcometo]

3:20 there is a tiny mistake: after the four non coincident digits you have in one case10582 and in the other10581. The "2" and the "1" should be coloured as different digits. Amazing video, many thanks.

[@harrydsouza286]

🎉

[@shawnthomas3802]

I knew somebody else would notice too.

[@Mathologer]

Well spotted :)

[@eroraf8637]

Glad someone else spotted that.

[@imlassuom]

I found it too!

[@Grumpum]

I'm not much into history or race when it comes to math. It's nice to notate who made what discoveries, but I care more about the how they figured it out than the who. Pride gets in the way of many things this way. ESPECIALLY in math as mathematics has nothing but facts to give us. No politics, no pride, no religious...just facts. Noting that the discovery was made earlier gives us factual view of history that says we weren't just mud and straw builders and DID have the knowledge to accomplish great feats of engineering. I enjoyed your video immensely.Thank you so very much.

[@utkarsh_ratnakar_official]

Prof C.K Raju [INDIA], also complied with previous old mathematician work, you can also see that. thank you for the amazing video.

[@bishalsarkar73]

So proud for belonging from India... Sir, Madhava need much attention in our Modern Science today.

[@stefangabor5985]

It is well known in Europe that the India had the best knowledge of Mathematics before anybody else. In fact, the Persians and then through the Arabs some of that knowledge got to Europe.

[@kunjupulla]

Yeah, ancient Indians loved converting everything and anything into Sanskrit verses. If you don't know how to properly decode it (like most people today), there is a high chance that you will mistake it to be a flowery story or poem. I think they did this so that the students could memorize easily. Edit: Oh, and being a Malayali myself I am surprised I couldn't completely read the manuscript. It is so fascinating that the script changed a lot with time!

[@dharmalingeswarparasa1609]

Correct, numbers are hard to remember, where as coded stories are easy. It was an old practice.

[@hueckelaromat]

Such another great piece of art you were able to produce! I like the pure mathematics combined with the archeology. Isn’t is amazing that we just slowly discover how far advanced medical scions was advanced? I am so glad that we are now more open minded and allow to combine the facts of the past. Past scientist would have been happy and proud of us seeing the collaboration and joy.

[@rickostidich]

At 0:18:33 it should be ...N+9/(4N), and at 0:21:16 the two "4n+" should be swapped with the two "n+" (also: I would replace the "n"s with "m"s in this yellow box, since they don't need to be equal to the "n"s above). However: this video made my day; I've been waiting for the new Mathologer video for 2 months! Great work Burkard, most of your videos are in my "Favorites" list, and Mathologer is definitely my favorite channel of all. 🤗🍻

[@Mathologer]

Very well spotted :) Glad you like the videos so much.

[@rickostidich]

Correction to my comment: in the yellow box, only the n² and the (n mod 2) should become m² and (m mod 2); the other "n"s stay as they are. 😜

[@stevewithaq]

3:20 minor formatting bug at the 53rd decimal place - the values are 2 and 1 but they're formatted as equal.

[@Paul71H]

I've read about or seen videos about this extremely-slowly converging series for Pi (or for Pi/4) many times. Why did the idea of an error correction term never occur to me? This is brilliant, and it's amazing that this mathematician figured this out in the 1400s!

[@Rishabh-Dev]

I was aware of this fact but never looked into much deeply. Thank you for addressing it.🙏🏼 Knowledge knows no boundaries.

[@rnoro]

Very beautiful demo and excellent explanation!

[@jixpuzzle]

This is really interesting! Thank you 💕 You are my favorite math youtuber!

[@Mathologer]

That's great. Your favourite KZbinr needs to hear this every once in a while :)

[@NagendraGd]

Just a brilliantly done video, thank you for the amazing explanation and the hidden gems! ❤

[@amigo4lifeUS]

Every time I talked about Newton, I should've started with Madhava. Thanks for sharing this info.

[@johnlinley2702]

I admit, I enjoyed this very much. The most fun with differing rates of convergence, and the distributive property, I've had in years. You have reawakened me. Thank you.

[@6ygfddgghhbvdx]

Mathologer does look like reincarnation of Madhava at least visually!

[@amkamath]

Such a lovely insight! This explains the error term in any base, but certainly easier to notice in base 10. I wonder if Madhava also considered series for Pi based on the series expansion for arctan(1/sqrt(3)) = Pi/6, which converges faster. It is hard to imagine how they discovered these results without using our present day notation.

[@Mathologer]

Thank you very much for that. You are asking " I wonder if Madhava also considered series for Pi based on the series expansion for arctan(1/sqrt(3)) = Pi/6, which converges faster." The answer is "Yes" he did :)

[@amkamath]

@@Mathologer Haha! That's great!😄I would love to learn more about his discoveries. Thank you for opening this new door for exploration!

[@bhanubaisIITM]

I deeply appreciate your remarkable dedication in simplifying the profound teachings of ancient Indian acharyas. Undoubtedly, the task of transforming such extensive knowledge into a more accessible language, as demonstrated in your video, must have been exceptionally challenging. 🙏

[@chandrasekhar4330]

A quotation by David Mumford (a nobel prize recipient in Mathematics and is considered to be one of the founders of Algebraic geometry). Only a fraction of this mathematics has become generally known to mathematicians in the West. Too many people still think that mathematics was born in Greece and more or less slumbered until the Renaissance. It is right time that the full story of Indian mathematics from Vedic times through 1600 became generally known. I am not minimizing the genius of the Greeks and their wonderful invention of pure mathematics, but other people have been doing math in different ways and they have often attained the same goals independently. Rigorous mathematics in the Greek style should not be seen as the only way to gain mathematical knowledge. The muse of mathematics can be wooed in many different ways and her secrets teased out of her. In another instance he says Though Panini is usually described as the great grammarian of Sanskrit, codifying the rules of the language that was then being written down for the first time, his ideas have a much wider significance than that. Amazingly, he introduced abstract symbols to denote various subsets of letters and words that would be treated in some common way in some rules; and he produced rewrite rules that were to be applied recursively in a precise order. One could say without exaggeration that he anticipated the basic ideas of modern computer science. Chandrasekhar

[@JohnLincolnUSA]

Mind blowing & amazing! Thank you sir for sharing this ancient wisdom with us lay folks. Please share other insights that you are aware from the ancient sages. It might not be too far fetched for us to assume that Madhava and his fellow mathematicians at the Kerala School of Mathematics might have discovered many other concepts that we might not be aware yet.

[@airfluxe2095]

One of your best videos, IMHO. I absolutely loved it.

[@Var03]

Thank you so much for this video. For a student in the field of history of Indian astronomy and mathematics, i am very happy to see KZbinr like you talking about the correct chronology and also putting out the works of them.

[@dnsaxena8732]

I am so grateful to you sir, for giving the credit of discovering Calculus back to Medivieal Indian Mathematician from that of Newton ... Kindly search for more Mathematicians like Bhaskaracharya and you'll be amazed to discover most of the discoveries of Maths as modern world know of ... Thanks once Again sir

[@dcterr1]

Another excellent video! Those medieval Indian mathematicians sure were smart, weren't they? At the same time in Europe, theologians were still debating how many angels could fit on the head of a pin!

[@benefactor4309]

Actually Jesuit priest were present in Southern Indian in the 16 th century and they learnet these things from mathematicians of Kerela schools .

[@dcterr1]

@@benefactor4309 Where were the Kerela schools located? In any case, according to this video, Indian mathematicians knew how to estimate pi way before the 16th century.

[@benefactor4309]

@@dcterr1 Madhava lived in a village named "sangamagrama". Today the village is in Kerela province of India .Kerela is in Southern India

[@benefactor4309]

@@dcterr1 yes they did,but Europeans arrived in Kerela in the 16 th century and they started learning it

[@dcterr1]

@@benefactor4309 Thanks for the info! I guess this is just another example of Western knowledge stolen from the Eastern world. Very sad IMO!

[@CrystallineCurrency]

How wonderful that you clearly demonstrated the connection with chemistry and that the answer again is 42! Thank you!

[@rmat007]

Brilliant explanation and kudos for revealing to the world Madhava's contribution to this domain of mathematics 🙏

[@vishatubeful]

it’s amazing to also know that the flavors of derivatives and integrals were already discovered by Indian mathematicians before Newton ❤

[@revivehinduglry5176]

I can bet (cant prove) that the Europeans got all their scientific base from India... no doubt about it

[@sriramlamsal]

K. Ramasubramanian is a professor at Indian Institute of Technology Mumbai. He is my favourite professor. The manuscript is his production.

[@Daniel.Blomberg]

Props to you for bringing awareness to this.

[@kartikeyasharma6558]

Your videos are so well made and so informative! Keep up the good work, sir. Big fan.

[@vickyshrivastava7177]

Thanks for giving his due credits . You are a nice teacher 👍👍👍👍👍

[@christosvoskresye]

"Of course, it would have been very hard to put into action in 1400 without a computer...." That's what graduate students are for!

[@neomooooo]

Continued fractions, and their connection to the Euler-Maclaurin formula would make a nice addendum. Bravo!

[@Mathologer]

Definitely.

[@kaustubhhirve7734]

Absolutely Fantastic!! Grateful for this reckoning!

[@aryanmahajan5558]

Very Great! Indian Mathematicians are getting the attention they deserve

[@harshavandu]

Ah, finally ! Math on KZbin. Subscribed. I was always enamoured by pentile formations. Could we have a video on that please ? 🎉❤

[@7684anitadey]

Thank you sir for sharing Indian Mathematicians like Madhava's contributions.

[@bearcubdaycare]

Thanks for injecting a bit of mathematical joy into my day, and informing me of this Indian mathematician.

[@Mathologer]

Thanks for that. Glad you enjoyed the video :)

[@kanishqprasad9378]

If we somehow were able to get back the burnt books of Nalanda University, then I am pretty sure that more than 90% of today's "modern" discoveries in maths and science would be found in them.

[@Wildevis]

Wow astounding how they could have done all this so long ago!

[@yashbakshi3725]

Huge Respect for your sir for giving an Genius Mathematician Sir Madhava, a credit and fame he deserves !!. Being Indian, I felt so proud about how brilliant our minds are, specially in Mathematics.

[@draxelchang5411]

Exceptional work. Thanks for sharing your passion for math.

[@johnchessant3012]

Great video! Another way to speed up the convergence of the Madhava series is to use the Shanks transformation; this makes the assumption that the difference between the nth partial sum and pi is roughly geometric, or at least that the ratio between successive errors changes very gradually. Repeatedly applying Shanks can give 7 digit accuracy from just the first 10 or so terms! I like the correction terms in the video better, though; they're more specific to this series and they help to explain the paradox.

[@Mathologer]

That's a very good point. Just in case anybody else is reading this, this wiki article has some details about what exactly happens when you apply the Shanks transformation to the Madhava series en.wikipedia.org/wiki/Shanks_transformation

[@BikeArea]

Thanks from anybody! 😊

[@geemcd]

"Without a computer" underrated words x

[@shubhamrawat_69]

Is this some gift for viewers from india? One of many now. Keep em coming.

[@abhirama]

Astounding video. Thank you so much for the service to humanity.

[@debasishraychawdhuri]

It is impossible that they did Tailor's series (well, we have to use the same term for clarity) with verses. I think they did do some form of symbolic math, but it was the tradition to use only verses in the final copy. One reason for that may be that they thought it would be harder for future mathematicians not familiar with their symbolism to understand the symbolic math compared to reading verses.

[@Nikhil_Sahu_G]

Ha ha ha 😂😂 Thats may be true. Imagine finding a ancient mathematics books with weird symbols

[@NishanthSalahudeen]

May be thats their handicap... having no symbols. yet they did this much! remember romans also managed to do some calculations even though they didnt have the decimal system that Indians used. This severely limited what romans could do perhaps. In a similar way, the whole tradition of shruthi and smrithi in south asian culture should have been a framework on which later knowledge was fashioned. Inorder to follow that, everything need to be verses. practically every book in every subject was written in verses due to the traditional habit of "memorizing verbatim". it sure helps to memorize, but at the same time, tough to manage beyond a certain complexity. it is what it is. they did well with what they had. I also always wondered why the chinese language writing never evolved beyond "one symbol per word" creating practically infinite written characters to learn in old times. I think cultural momentum is real... once you get stuck with some thing that works for a while in some context, it refuses to give way. As more and more stuff snowballs on it, it becomes impossible to change. QWERTY keyboard is a modern similar thing. Then a disruptive concept would arise which shatters this cultural framework (much like the disruptive technologies). In this case, it may be printing. the chinese adapted their language to a more manageable set of symbols after the arrival of computer keyboard (disruption).

[@mayurrathan]

The final theorems were written in verse as shlokas or poems. The shlokas had rhyming terms set in a meter. It's like learning the periodic table set to the tune of a song rather than rote learning it.

[@dynamitebsb4520]

Glad u recognised the Kerala mathamatcian. He has a great importance in the qorld of mathematics

[@a123s1l]

@Mathologer thanks for your great content again. My love for math has been kindled again watching your excellent videos.

[@a_maxed_out_handle_of_30_chars]

this popped in my feed and watching it made me happy :) thank you Mathologer

[@vick229]

Truly a polite and precise explanation of Pi easy to understand youtube video of the day.. Great work Mathologer 👏

[@last3239]

I'm from Kerala and I am feeling proud after seeing Malayalam.

[@Mathologer]

I even put some Malayalam auto-generated subtitles :)

[@miyamoto900]

@@Mathologer just use CHSTGPT 3, PUT IN your entire transcription and ask it to translate to any language you want... 👏

[@Kapritchosa]

This is an absolutely wonderful video. Thank you for your effort.

[@SaturnCanuck]

Thanks again. I love to watch these on Sunday afternoon with my coffee.

[@QuantumHistorian]

If we really want to go back to the beginning of calculus, there's strong evidence that Archimedes had something very very close to it (including rigorous definitions of limits), if not bona fide integral calculus itself. Which is some 1600 years before Madhava. Of course, I would only be half survived if some even older Indian, Chinese, Babylonian, etc... mathematician had invented something equivalent to calculus even before that. Scholars typically remember the start of still-living lineages, rather than the original discoveries that faded for centuries before being rediscovered.

[@David-bh7hs]

Sorta, but I wouldn’t say calculus was invented. It was discovered.

[@kodfkdleepd2876]

All mathematics comes from language. all languages have infinite descend. Calculus is just a manifestation of what people did all the time implicitly. Everyone applies calculus. The formalization of it is what many mathematicians did(including non-mathematician mathematicians and the lesser/un knowns). Math, nor anything, is created in a vacuum nor is it independent of everything else.

[@user-zc1vi8tx7b]

Your name tho 😂 Phy+ History or what

[@QuantumHistorian]

@@David-bh7hs A long standing debate. What was discovered was the logical consequence of a set or axioms that were chosen _ab initio_ by people. Axioms are most certainly not discovered, but invented, hence everything that follows from them is just discovering the consequence of an invention. A good analogy is engineering: if you build a car, and then test it on a track to find out it's top speed, did you discover or invent the top speed? A bit of both really, you're discovering a property of your invention.

[@SocratesAlexander]

I almost invented calculus when I was in 9th grade. I thought that the volume of a sphere was the sum of its thin slices, but couldn't find a way to realize the calculations in an acceptable way.

[@1AdityaSingh]

*A few of his contributions:* Something of peculiar interest regarding this mathematician includes the fact that he discovered the series equivalent to the Maclaurin expansions (sin x, cos x and arctan x), more than two hundred years ahead of Europe. He determined the explanations of a few ‘transcendental’ equations by a process of repetition. A procedure was unfolded by him to determine the siting of the Moon every 36 minutes, and techniques to evaluate the motions of the planets. His contributions to infinite series and infinitesimal quantities attest his writings’ existence; a lot of his work has been transmitted through other mathematicians after him. To suggest his popularity, there is a formula named Madhava-Leibniz which many students are taught during their college years.

[@Sameer_S_Kulkarni]

Why do you just copy paste comments from other sites bro? You can just express whatever you think and/or feel...

[@1AdityaSingh]

@@Sameer_S_Kulkarni I don't understand what is problem in that brother. If someone is reading it that would be simply beneficial. And I believe there is nothing wrong with it.

[@1AdityaSingh]

@@Sameer_S_Kulkarni I am a jee aspirant of class 11th and I find these things fascinating so I do read articles for different sites and simply sharing it with those who find it interesting. There is nothing to argue with it.

[@1AdityaSingh]

@@Sameer_S_Kulkarni Typing or copying and pasting some interesting radicals or facts from other sites is not an issue instead of writing nice video or good content. I am not done PhD in math I am just learning and yet there are many things to understand. I myself got confused in f(x)2 it. So please forgive me brother 🙏

[@mostlysunny4u]

Thank you for bringing this up in a video 👏 😃'Madhava Series'

[@mrsvsingh4690]

Thank you u r doing a great service to KNOWLEDGE and TRUTH...the pillars of humanity

[@vv6533]

Thanks for shining the light on Indian mathematics contribution. Kerala school of mathematics invented many things before the westerners did.

[@benstallone6784]

Western mathematics invented many things before indians did

[@uday9574]

@@benstallone6784 you racist

[@abi3751]

​@@benstallone6784 they did but we too did

[@striker44]

Alliens did everything before anyone here on earth did. 😂

[@RaiAdarsh-bi8bh]

got to know about him today. Its such a same, even I have studied PCM in my class 11th and 12th and in my drop year. What a legend he was. When I learn about these people I love mathematics more. My fellow Indian please don't only know about them we have to get to the top in every field to regain our legacy of Golden Bird. Let's do our best to make India great again

[@GowthamV07]

Thanks to congress and western appeasement policies that won't happen and we will be thought of how good british were and how barbaric india was in our books in school.

[@hsinghal1]

Really amazing video. Always feel humbled after knowing about such legends.

[@MATHSWISDOM]

🙏 Thankyou sir. Lots of respect from an Indian student

[@rakheshsekhar6087]

The brilliance is that they all were "oral" recitations - imagine a full paper worth of mathematical notations in 4 lines of Sanskrit stanzas! YES - it "has" to be exact 4 lines of Sanskrit stanza - otherwise they will flout the Sanskrit grammar rules. So imagine the plight!!! It has to be the mathematical the full proof; needs to be Sanskrit grammatically correct- and YES -easy to learn and record in minds as well - no paper. Very small order indeed! This exact proof it like so - it is written like this in Madhava's mother tongue Malayalam script but in Sanskrit : വ്യാസേ വാരിധിനിഹതേ രൂപ ഹൃതെ വ്യാസ സാഗരാഭിഹതേ ത്രിശരാദി വിഷമസംഖ്യാഭക്തം ഋണം സ്വം പൃഥക് ക്രമാത് കുര്യാത് Those 4 lines - summarize the entire summation series like so : വ്യാസേ വാരിധി ->4 x Diameter ത്രിശരാദി -> 3, 5 etc വിഷമസംഖ്യാഭക്തം -> Div by odd nos ഋണം സ്വം -> plus & minus Which goes like this - Do the 4 times of the diameter of a circle and then …

[@krishnanunnimadathil8142]

Thank you for this. Enlightened.