Magic of Mathematics | Vinay Nair | TEDxGSMC

Summary

In his TEDx talk, Vinay Nair shares the intriguing history and marvels of Indian mathematics, delving into ancient concepts that have shaped the mathematical world. Beginning with the inception of 'zero', he explores how early Indian mathematicians conceptualized nothingness. He discusses various mathematical advancements, such as geometric explorations linked to the construction of fire altars, and fascinating algorithms found in the realm of poetry and dance. Highlighting polymaths like Pingala and Brahmagupta, Nair illustrates how these ancient scholars, through their interdisciplinary learning, made groundbreaking discoveries like the Fibonacci sequence and Pythagorean triplets centuries before their rediscovery in Europe. This talk encourages us to reflect on the origins of mathematical thought and the benefits of integrating diverse disciplines for educational and intellectual growth.

Highlights

• The concept of zero originated in ancient India and was formalized with nine other symbols, revolutionizing mathematics. 🎇
• Both Hanna and Pingala delved into mathematics through non-mathematical pursuits like poetry, music, and rituals. 🎵
• Brahmagupta explored number theory and geometry, laying foundations for algebraic understanding long before European scholars. 🔢
• Pingala's work revealed early knowledge of binary numbers and the concept of zero as integral in computation. 🚀
• The discovery of the Fibonacci sequence in two different cultures highlights the universality and timelessness of mathematical truths. 🌍
• The integration of diverse disciplines in ancient India led to a comprehensive understanding and advancement in mathematical ideas. 📚

Key Takeaways

• Discover the fascinating origin of zero in Indian mathematics, an idea that revolutionized math across civilizations. 🌌
• Explore how ancient Indian scholars like Pingala used poetry to delve into mathematical algorithms. 📝
• Learn about Brahmagupta's contributions, including operations with negative numbers centuries before Europe embraced them. ➕➖
• Understand the intertwining of different cultures leading to parallel mathematical discoveries like the Fibonacci sequence. 🔄
• Reflect on the benefits of non-compartmentalized learning, which fueled groundbreaking innovations in ancient times. 🔍

Overview

Vinay Nair takes us on a whimsical journey through the annals of mathematics, tracing its magical evolution from the realms of poetry, music, and rituals in ancient India. He begins with the story of zero, an abstract symbol that sparked a mathematical revolution and transformed numerical systems across the world. By showcasing the rich tapestry of ideas and discoveries that emerged from India, Nair sheds light on how these ancient insights continue to influence present-day mathematics.

We dive deep into the ingenious minds of historical polymaths like Pingala and Brahmagupta, whose explorations in poetry and astronomy led to groundbreaking mathematical concepts. From the geometric necessities of religious fire altars to the crafting of poetic meters, mathematical thinking was an intrinsic part of cultural practices, sparking innovation. Nair presents a vivid portrayal of how these creative pursuits crafted a melding of disciplines that advanced mathematics remarkably.

The talk culminates in a thoughtful reflection on how these ancient Indian practices could enliven and enrich modern education. By highlighting the interconnection of diverse fields, Nair suggests adopting a similar openness in contemporary learning environments. He underscores the significance of teaching mathematics in a way that resonates with real-life applications and cultivates multidimensional thinkers, urging us to embrace the past's wisdom to inspire future innovators.

Chapters

• 00:00 - 01:00: Introduction to Zero In the introduction to the concept of zero, the speaker uses a humorous and engaging approach by introducing an imaginary friend named Leo. This analogy is used to relate how people might have reacted when the idea of a number representing nothing was first proposed. The chapter sets up the context for exploring this groundbreaking mathematical concept.
• 01:00 - 02:30: Geometry and the Discovery of Square Root of 2 The chapter explores the historical development of mathematical concepts, particularly focusing on the idea of the square root of 2. It discusses how ancient civilizations like the Greeks, Mayans, and Babylonians grappled with the concept but couldn't fully accept it. The breakthrough came in India, where mathematicians developed a new number system with a symbol for zero, revolutionizing mathematics. The talk delves into these transformative ideas and their impact.
• 02:30 - 04:30: Pingala and Binary System The chapter discusses the ancient contributions to mathematics by figures such as Pingala, focusing on the origins of binary system in ritualistic contexts. It highlights how poetry, music, dance, and religious rituals indirectly facilitated mathematical thinking. The text goes back 3,000 years to describe practices involving fire altars, which led to the exploration of geometrical shapes like squares and right triangles.
• 04:30 - 06:30: Brahma Gupta's Contributions In this chapter, the focus is on Brahma Gupta's contributions to mathematics, particularly his observations about the diagonal of a square. Ancient mathematicians noticed that the ratio of the diagonal to the side of a square was a constant, although they did not know its exact value. This chapter explores how Brahma Gupta derived an interesting formula to calculate this ratio, known today as the square root of 2, which is an irrational number.
• 06:30 - 09:30: Pel's Equation and its Historical Significance The chapter titled "Pel's Equation and its Historical Significance" delves into the intriguing mathematical formula related to Pel's Equation. It begins by discussing a complex operation involving fractions, highlighting a specific calculation: 8 of one-third of 134, subtracted from 1, which equates to an intriguing fraction, 77 over 4. Interestingly, this result approximates the square root of 2, and the text encourages readers to verify this using a calculator, due to its accuracy up to five decimal places. The text teases the reader's curiosity by hinting at the intriguing nature of how this specific formula was derived.
• 09:30 - 13:00: Nature of Indian Mathematical Research The chapter "Nature of Indian Mathematical Research" discusses the use of the term 'ceviche shahe' by an Indian mathematician, indicating an acknowledgment of approximation rather than exact values. The significance of this term lies in its indication of the mathematician's awareness of the limitations of his calculations, especially in relation to the number root two and its irrationality. Historians analyze this acknowledgement to understand the depth of mathematical understanding during the period.
• 13:00 - 16:30: Interdisciplinary Learning and Modern-Day Applications The chapter titled 'Interdisciplinary Learning and Modern-Day Applications' explores historical and cultural intersections in the development of mathematical concepts. It opens with a reference to ancient practices, specifically mentioning the construction of fire altars and the irrationality of the square root of two as motivations for the advancement of geometry. The narrative then travels across time to a region that corresponds to modern-day Pakistan, introducing the reader to Pingala. Pingala was known for his work on the treatise Sandesara, which is a study focusing on the structural rules governing poetic compositions. This indicates an integration of mathematical thought processes into art and literature, underscoring the interdisciplinary nature of knowledge and its applicability in diverse creative fields.

Magic of Mathematics | Vinay Nair | TEDxGSMC Transcription

• 00:00 - 00:30 hello everyone I would first like you to meet my friend who has come up on stage with me and his name is Leo and as you can see he does not exist right this is how probably people felt when somebody proposed the idea of having a number that would represent nothing and this
• 00:30 - 01:00 idea was toyed by civilizations like the Greeks the Mayans and the Babylonians but they couldn't accept it completely so they rejected it until some great minds in the Indian subcontinent came up with this idea of having a symbol that would represent nothing and along with nine other symbols they came out with a very new number system that would change the face of mathematics in the years to come in this talk I will be sharing with you some incredible ideas that developed in India in the past thousands of years
• 01:00 - 01:30 which was going to be you know from very unassuming areas like poetry music dance rituals and so on so to get started let's go back about 3,000 years where we we come across both Hana he was a person who lived during the time when people were worshiping the nature by making fire altars and when they made these fire altars they had to use geometrical shapes like squares right triangles and so on and when they explored the shape
• 01:30 - 02:00 of squares they observed that the ratio of the diagonal of the square to its side was some constant but they weren't able to get the exact value of that particular constant and today we know that that constant is square root of 2 and I'm sure you all would remember your high school days when your teacher told you that it's an irrational number which sounded very irrational to us because you know it goes unending for ever and ever and ever and both Hara gives a very interesting formula to calculate the value of to about 3,000 years ago he says add one third to one and a
• 02:00 - 02:30 quarter of one third x 134 subtracted from one and you will get an interesting fraction why 77 over 4 not 8 and that will be equal to or approximately equal to the value of square root of 2 and I would urge you to take out your calculator and do this operation and you will see that the values eight up to five decimal places I know what's running in your mind how on earth did he get that particular formula right
• 02:30 - 03:00 but that's not all in fact he uses a word ceviche shahe that appears at the end of the words where he makes a big claim he says that this value that I'm giving you is not accurate but only close to or approximate now how does he know that this value is not accurate and approximate so there is a lot of discussion that happens on this particular word by historians where they believe that the value of root or the irrationality of root two was known in some sense to both Hana and that is why
• 03:00 - 03:30 he specifically uses the word ceviche shahe which means close to remember what is the motivation for discussion and irrationality of root two it was construction of fire altars and geometry we move on for about a few centuries and we are in some place in probably modern-day Pakistan where we meet Pingala and Pingala was the author of sandesara and chanda's is a branch of study that deals with the rules for writing poetry and as many of us know
• 03:30 - 04:00 that in those days indian texts were written in political form which means it had to follow a particular structure so that it could be sung in melodious tunes and while making these structures a poet had to keep in mind the syllables in sanskrit which were of two types one is logo and the other is guru now what does this mean so if I say the word Sri Rama so Sri Andhra take more time to pronounce and more takes lesser time so Sri Andhra our gurus and myrrh is Allah
• 04:00 - 04:30 who the question that Pingala was trying to explore is if I want to construct a four syllable meter how many combinations can I have using Lagos and gurus I can have a verse with all four gurus or maybe I can have with some combination of logos and gurus and then he continued his thought he said can I have an algorithm in which I can list all these things down and if I have an algorithm is there a particular way in which I can find the raw number of a particular pattern without doing the entire listing and if I am given a
• 04:30 - 05:00 particular row number can I find the actual appear of the actual pattern of the string without doing this listing and today students of computer science would say that hey these are nothing but our listing and sorting algorithms which was were which was which he was discussing and we find the motivation for these discussions for Pingala was not computer science but poetry so Pingala continues his exploration and if you see if you replace all these G's with 0 and L with
• 05:00 - 05:30 1 you will see that it gives you the binary listing from 0 till 15 now the Christmas did Pingala no 0 or did he talk about 0 in fact Pingala text is the oldest text that we see the appearance of 0 as a symbol that we are using today and he uses the word SUNY 'm which means 0 in sanskrit and he is using this particular words to give an algorithm to find the powers of 2 now we might think what's the big deal about powers of 2 if let's say we want to calculate 2 to the
• 05:30 - 06:00 power 100 1 ways we can multiply to 99 times but that's not a very efficient way of doing that particular calculation so what Pingala did was he came up with an algorithm with which we can do 2 to the power hundred in much lesser number of steps in fact this algorithm is still used in computer science as a very optimum algorithm for finding powers of 2 he continued his exploration and then he came up with this particular triangle and as I can see most of you are from science background so you might have seen this triangle back in 11 standard
• 06:00 - 06:30 or 12 standard and we use this for finding the binomial coefficients and he called this triangle as Varna Meru and of course we know that Blaise Pascal discovered this particular triangle about 2000 years later when we hear about all these things about Pingala we might think that oh such a great genius all these things came out of one head but that is not true because we also see works before Pingala that talk about permutations combinations binary mathematics and all that and interestingly those texts were not
• 06:30 - 07:00 mathematical texts they were texts on medicine and in fact we also see a lot of people who followed Pingala and we cannot call a single one of them as mathematicians because they were writing texts on different fields and they were using all these areas of mathematics in some cents or the other for example we see musicians we see astronomers in this list in fact we see Haymitch Andra who lived in the 12th century who was a contemporary to Fibonacci who gave the famous Fibonacci sequence and Fibonacci
• 07:00 - 07:30 discovers the sequence while he observes the breeding pattern in rabbits and humans and Roy discovers the same sequence when he is observing the math behind what do you call that percussion instruments so it is interesting to see that at two different civilizations during the same time two different people talk about the same mathematical concept and the motivation is completely different we move ahead now 2,000 years and we go to France where we meet firma
• 07:30 - 08:00 firma was a great mathematician and a lawyer and he was he was considered as the father of modern day number theory and firma poses a problem to his contemporary in fact for my his famous for throwing open a lot of problems that a lot of mathematicians worked on in the years ahead so firm ass is it possible to find the smallest integer solution for this equation X square minus 61 y square equals 1 now for those of you who took a vow that you'll not touch mathematics beyond 12 standard
• 08:00 - 08:30 let me just try to explain what that means that is can you find two whole numbers X and Y such that if you square Y and multiply it with 61 and subtract it from X square the result should be one so what is the smallest such pair of two whole numbers and we will see an interesting connect to this particular question that was posed by firma but let's go back thousand years to India where we made Brahma Gupta who gave some rules for multiplication with negative integers and 0 and what is interesting is that by the by the time of Brahma
• 08:30 - 09:00 Gupta in 7th century the decimal place value system was already in place and people were comfortable using zero and a bit of negative integers so Brahma Gupta is the first person who gives rules for operations with negative integers and interestingly we see that negative integers came to Europe much later in fact they were accepted as something sensible only in the 18th or 19th century in Europe brahma-bhutah continues his exploration in one of my favorite topics which is geometry and he says
• 09:00 - 09:30 if you take two triplets you can do a lot of things with that and I'm sure you all recognize when we learned Pythagorean triplets that is if we have three whole numbers a B and C a square plus B Square should be equal to C square right so if you take any two triplets like that let's say 5 12 35 3 4 5 & 5 12 13 and you multiply them in an interesting way you will see some really amazing numbers that emerge we might
• 09:30 - 10:00 think what's so amazing about those four numbers what is interesting is that if you add a 39 Square and 52 square you will get the answer the same as 25 square plus 60 square now for those of you who are not interested in Mathematica I do what what's the big deal about it but this thing is today we call it tetrads and it is used in this area of geometry called cyclic order littles and that is what brahma-bhutah was trying to explore when he started with triplets and then he continues his
• 10:00 - 10:30 exploration in fact he's unstoppable he just goes ahead using tetra as he goes into number theory and an area under number theory called the Pels equation so what is the Pels equation this example that is X square minus 2y square equals one is one form of pulse equation instead of that two you can put anything over there and it will still be a part of pulse equation and he is looking for integer solutions for these equations what does that mean again if you substitute in this case three in the place of X and two in the place of Y we
• 10:30 - 11:00 can see that three square minus two times four gives you one so 3 comma 2 is a valid solution for this particular equation and brahma-bhutah said you can have more such solutions and you know you you will get infinitely many solutions but then one might think was brahma-bhutah really jobless why was he doing all these things that does it didn't have any other things because he was an astronomic chilling so he is talking about well situation because he wants to do something with you know
• 11:00 - 11:30 these numbers x and y and if you remember the last solution 577 for not 8 just a while ago we had discussed that Bhavana gives the same two numbers as a fraction using you know to find the approximate square root of 2 and that is precisely why Brahma loopty is also using this particular equation so he gets into the value of Euro square root of 2 using algebra and geometry so the motivation is to find square root of numbers and then brahma-bhutah says instead of 2 if you put any integer D
• 11:30 - 12:00 which is not a square number if you are able to solve this equation and get values for x and y you can get better approximation of square root of d remember this is happening in seventh century back in India but Brahma is not able to solve this for any value of D so he poses the question and then probably leaves it to his successors to work on it and the equation that we had seen the question that was posed by firma thousand years later is the same equation where the value of D is 61 and there is an interesting connect to this
• 12:00 - 12:30 which we will come back just in a while what Firma didn't know that the problem that he is posing as a very challenging problem was already solved by some mathematicians back in India five or six centuries earlier and we see that the problem that he poses as a very challenging problem the smallest integer solution for this one we see in the work of viscacha ray the second in the 12th century and as you can see the numbers are pretty small it is something like 1.7 billion odd and 226 million odd
• 12:30 - 13:00 numbers and this was worked on with hand and head mind you back in 12th century or 11th century so what we see here we see that a question that was posed by Brahma Gupta in the 7th century was continued by mathematicians in the coming centuries and over a period of 500 years they had developed a perfect algorithm to solve this particular problem which later was discovered in Europe as the Pels equation this is a funny remark that French mathematician made so that what might have been firm
• 13:00 - 13:30 us you know surprised if somebody would have told him that hey the problem that you are posing as a very challenging problem is something that has been already solved five six centuries back by some native Indians just to put everything into context whatever we have seen so far the first thing is we see a beautiful line of research that happened in India because we see works that have been cited I mean works which were written centuries or even millennia later and that has in continued by people who came later on
• 13:30 - 14:00 and this is precisely how research is done even today so it is not one independent person's work it is a lineage that is continued ahead what made Indians come up with these abstract ideas like zero infinity and all that was it some philosophically ideas that was present in India because we see philosophies that supported nothingness exist and on the other hand we had philosophies that believed in a supreme infinite being so this kind of diversity
• 14:00 - 14:30 did it help Indians come up with these abstract ideas because Greeks were also very good in geometry in mathematics as such but they couldn't come up with these kind of things they rejected the idea in fact we see similar diversity in Europe especially during the Renaissance period and we see a lot of new ideas that were coming up because of you know the diversity probably so is diversity in India that has always existed a very good thing that supports come up coming
• 14:30 - 15:00 up with all these good ideas imagine a computer science teacher coming to the class and teaching permutations and combinations through music poetry and dance I am sure all of us would want to sit in that particular lecture right so I think there are a lot of ideas that we can borrow from history of especially Indian mathematics where we see that every idea in mathematics is backed by some kind of real life application and the motivation came from real-life instances so those things I think we can
• 15:00 - 15:30 still use in modern day classrooms the last point but not the least is the kind of people that we have seen so far all these names we have seen on the screen we cannot call them as only mathematicians because they were good in many areas so they were polymaths we cannot say that this person only worked in mathematics so what made these kind of people experts in various disciplines was it the D compartmentalize learning that they had in those times when subjects were not
• 15:30 - 16:00 put into different buckets as against how it is today because we we learn algebra in a different bucket geometry is in a different bucket and people who learn Vilas you don't even look at and vice-versa right so is this the compartmentalization of subjects really hampering our intelligence and brilliance to come out and this is something that I think we need to think and keeping all these things in mind I think it will be a good idea to take a route back to the roads and start thinking how we can use these ideas in
• 16:00 - 16:30 modern-day classrooms and I think that's an idea worth sharing thank you [Applause]