[Music] [Music] circle a very common geometric shape circle and by extension sphere we find it everywhere around us in the objects even within our own body and that ratio of circumference to diameter which we very commonly call as pi is a very crucial element of a circle to understand its properties we all learned about pi in our high school while we're learning geometry of course the value is pretty simple 22x7 or 3.14 all of us know the value of pi but only few of us know the history of pi in this short documentary let's try and understand the evolution of pi in ancient india just in case if you're wondering why only the contributions from ancient india why not from the world over we'll get to that in a minute so let's get started as i said we all know the value of pi but beyond just the value what is the purpose of pi why was that mathematicians were obsessed with the value of pi for well over thousands of years why do we need pi in first place so let's try and understand and for the maths geeks out there please bear with me for a minute mathematics is as old as human civilization and measurement of things has become a very intrinsic part of that civilization and one of the basic scenarios for measurement is measurement of length say for instance if we have a line of length a and you want to measure that length what is the value of a or how long is that line now there are different methods or standards of units in different civilizations that people could measure the length of a line using a hand span using a length of a rope using a yardstick so there are a lot of other ways to measure the length of this line a right now let's take this to the next level think of this line of length a as our long wooden log and we want to construct a platform with this wooden logs so i need to place five wooden logs side by side to construct a platform so now what is the length of all these five wooden locks put together then we encounter a new subject which is called as area it's no longer a length of one dimension but it is an area of two dimensions then you have to add each of these wooden logs a plus a plus a plus a five times or you simply multiply a by five because multiplication is nothing but addition done in a faster way so the area of this wooden platform is five times a but now imagine i would like to construct a wooden platform which is of equal size on all sides then i have to extend this platform to its base is equal to height that is a times a so that is how if you add up all these a's vertically then you will have a times a or a square that is the area of a square so this is how organically the formula of the area of a square is derived so now you understood how organically the area of a square is derived right it is not just about the formulas we all know the formulas what's important here is to understand the thought process that went behind derivation of these formulas that is very important now imagine that we would like to construct a circular wooden platform instead of a square one so now we need to place this wooden logs like spokes of a bicycle wheel across the circular shape so that all these wooden logs perfectly cover up to create a circular platform they should not be overlapped there should not be any gaps left between the logs so imagine that kind of a wooden platform now if we need to answer that question as to how many wooden logs we need to construct such kind of a wooden platform that is the question and this question in the case of a square is very easy because we know that the height of the square should be equal to the base of the square so you have defined boundaries i need a by a but in circle it is not the case we don't have the definitive boundaries as to how much we need to spin this wooden log or place other wooden logs on top of it to sweep an area that is completely covering the area of a circle you could say 360. but back then it was not evolved to that level where we can already measure the degrees in a circle all that the ancient civilizations could measure is length in linear terms a straight length so long story short we know that we need to multiply the side of a square with its height which is a times a to get the area of that square but in case of circle we definitely know that we have to add this a or multiply it an unknown number of times but what is that unknown quantity that was the question that was encountered by the ancient civilizations back then so what was the unknown quantity that we need to multiply to the diameter of a circle to derive the area formed by that circle this question led to the thought of discovering a constant a value that is always constant no matter how big or small the circle is that very thought gave rise to this concept of pi and many civilizations egyptians chinese greeks babylonians indians many contributed to the value of pi they perfected it over a period of time not everyone called it as pi they had their own terminologies but nonetheless the contributions that came in from every civilization is just remarkable and the reason i'm doing this documentary only on ancient indian contributions is a vast majority of us are of the understanding that greeks invented pi and by extinction geometry and we have been learning the same but that's fundamentally incorrect pi has evolved in ancient india over thousands of years having said that there are a lot of whatsapp university myths about this pie that go hyper jingoistic about that ancient indians were great with pai like no other civilization even that is not right completely ignoring the contributions of india or going hyper jingoistic about the contributions of india both these stances are very stupid so let us try and understand what is actually the story of the evolution of pi in ancient india to understand the evolution of pi in ancient india we need to take these four important milestones into account because something very significant happened during these periods and in each of these periods we'll take three things into account purpose proof and precession purpose of pipe why did people needed pi back then for what purpose during that milestone and the second one is proof what is the ancient scripture that codes this specific value of pi and the third one is precession how precise was the value of pi back then so comprehensively will try to cover how pi has evolved in india over thousands of years in a most objective manner without resorting to any kind of myths or extrapolations and all that sorts of things it is just pure facts undisputable mathematical facts let's get started so let's start with the oldest reference of pi in ancient indian history that is in yes or whether the approximations of pi as per azuram if we take the entire landscape of the vedic knowledge system there are four vedas that we all know and there are six vedangas and there are four upavedas especially there is one branch of vedanga called as kalpa or kalpasutras which details out how the rituals need to be organized that were mentioned in years or with them especially yesterday and besides the theological aspects that are detailed out about how a yagyun needs to be organized there are a lot of geometrical principles also detailed out as part of the kalpasutras that is where our story starts so as part of these rituals which are mentioned in yeshua them how to practice them in day-to-day life so there are a lot of yognas that needs to be organized and here what you're seeing in the picture is one such ritual which holds three fire altars or yagna kunda with the names aha agni in the square shape dakshin agni in a semi circle and gravath yagni in a circular shape and these are not some kind of random constructions they are to be aligned perfectly as per the principles of geometry defined in kalpa sutras and that specific part of kalpa sutras which deals with this geometrical principles are called as sulba sutrani and these are the set of geometrical principles or rather the world's oldest applied geometry that is still in practice today and coming back to these three fire altars which are in the shapes of square semicircle and circle a very important constraint to be followed was all three of them should be of exactly the same area this is one of the earliest scenarios where the ancient vedic civilization people they encountered dealing with the areas of a circle and back then the area of a circle square or whichever geometric shape that we take their only equipment back then was nails and rope to take measurements of the shapes to execute this vedic geometry and how to construct these three firealters there were detailed instructions rather algorithms that were given in kalpasutras how each and every geometric shape has to be calculated on this topic i already made a separate doc film where i explained this algorithm step by step as to how the squaring of a circle and vice versa the problem that pestered mathematicians for thousands of years was solved in india back then and very much in use as part of conducting this yetness even today so i will not repeat that again in this doc film rather i'll give a link to this one if you're interested to watch it so for now let's stick only to pi as to how the value of pi was approximated so as part of these algorithms which were given to transform a circular firealter to a square shape or a square shaped fieralter to a circular one here is a relationship that was quoted a rope when stretched across the circle also goes three times around the circle which essentially means that the ratio of a circumference to the diameter of a given circle is three divided by one which gives the value of pi as 3. not very accurate as opposed to the actual value of pi of 3.1415 but considering their purpose back then and the fact that it was given thousands of years ago but that's how things evolve so that is one of the earliest valleys of pi that was found in india and of course the sulba sutrani gives few other values of pi as well extending to one or two other decimal places but essentially around three is the value that was proposed as part of silva sutrani not specifically calling it out as pi but just by quoting a relationship between the diameter and the circumference of a circle and here i am quoting the references from katya and sul sutras and there are many other sulba sutras also which talk about this relationship which we call today as pi there is a lot of academic research also that has been done on yazoo veda to understand the mathematical and the geometrical concepts that were detailed out and if you're interested you can do your own research so like i said what's very interesting is these mathematical principles that were given thousands of years ago are still being practiced in constructing this firealters or yagna kundum which are geometrically aligned with each other in terms of area so to sum it up all the approximation of pi in vedas the purpose was to construct these circular shaped firealters the proof was in sulba sutras the bow dhaina katya and aposta there are many other sulba sutras which detail out these mathematical concepts in construction of these fire altars and specially in bowdhy and kachina sulbasutras the approximations of pi were given as i just quoted and coming to the precession the value of pi in some instances it is given as exactly 3 and in some instances it is given as 3.1 so that is a quick view about the vedic approximations of pi and one very important thing here an indicative date that i've given here as 3700 bce but theologically if you see the vedas have a very long history so there cannot be a proper dating that could be done for the vedas so a simpler and a more conservative and correct answer would be how old was yesterdam that old was these approximations of pi as part of user rhythm so now moving to the next milestone that happened in 6th century aryabhatta he gave the approximations of pi so let's try to understand that aryabhatta is part of aryabhattiya the astronomical treaties that he wrote he gave a numerical approximation for pi directly this is what he quoted hundred added four multiplied by eight and added sixty two thousand this is the circumference of a circle whose diameter is twenty thousand so that gives us a fraction of sixty two thousand eight 832 divided by 20 000 and that gives a value of pi as 3.1416 as opposed to the accurate value of pi as 3.14159 and so on and he says that this value is an approximate figure and also like i said pi is the name given by the greeks to call this constant with a name but in ancient indian scriptures if you see it is always given as a relationship between diameter and the circumference of a given circle which of course translates to the pi what we use today and very interestingly aryabhatta also gave a formula to calculate the area of a circle without using the value of pi and here it is half of the circumference multiplied by the radius of a given circle essentially gives the area of that circle and like i said in ancient india geometry was executed using ropes and nails so figuring out the circumference of a circle with a rope and its radius it's pretty doable and here is where the geometry that originated in europe which is heavily based on compass and scale it had certain limitations the flexibility that a rope offers in geometrical calculations it's far better than the flexibility that a compass offers but anyways that's just a minor observation so let's get back to this one so the purpose of pi given by aryabhatta is to employ it in the astronomical and geometrical calculations which belongs to the larger corpus of jyoti shastra especially out of ujjain ujjain was a powerhouse of mathematics and time competition and astronomy geometry and a lot more back then and coming to the proof the scripture in which this value was detailed out coming to the precession it is 3.1416 as opposed to the actual value of 3.14159 moving next to the 16th century milestone this is the game-changing invention that came in from india an approximation of pi given by madhava one of the earliest methods of calculating the value of pi was given by archimedes through the method of exhaustion so basically what it means is so what is pi it is the ratio of circumference to the diameter of a given circle right now what if we calculate the value of pi for a square instead of a circle that is the circumference of a square divided by its diameter but square does not have a diameter so what is it is it a side or a diagonal or an average of both so with this thought process archimedes embarked on a method of exhaustion which is to calculate the ratios of circumference to the diagonals of polygons extended one side at a time because archimedes had a very interesting perspective that a circle is nothing but a polygon which has n number of sides where n could be a very large number if you try to analyze the thought process of archimedes it exactly matches with the thought process of madhubacharya who in the year 1300s came up with a similar kind of an approximation of pi but only the difference is archimedes was doing it geometrically while madhava tried to do it in an arithmetic manner so let's try and understand what is that madhava gave the value of pi as 4 multiplied with an alternating series of 1 minus 1 by 3 plus 1 by 5 minus 1 by 7 plus 1 by 9 and so on and this formula for pi is given in this small samskrutam slocum which elaborates to the infinite series of the value of pi the infinite series what it means is you can calculate the value of pi depending on the number of digits that you incorporate into your equation longer the number of terms that we considered in the series accurate the value of pi would be and this was revolutionary because till then for thousands of years civilizations have been adding one digit after another perfecting the value of pi but there was no generalized formula for pi which can give you a number of digits as per your need and madhwa invented just that and not just that this is also the first instance in the history of mankind where a finite quantity of pi could be expressed as a sum of an infinite series laying the stepping stone for calculus integration by parts and this was 350 years before newton conceived his version of calculus madhava and his inventions in the field of calculus or be the infinite series of pi and many other contributions that came in from him they are very well popular in the academic circles but not so much in the general public in fact the homi baba center for science education and tata institute of fundamental research they organize a competition called as madhava's mathematics competition in the honor of this great mathematician who invented the infinite series of pile and many more including the foundations of calculus which was first invented in bharata the purpose for which madhava invented the infinite series of pai was to devise calculus as an instrument for astronomical calculations as part of jyotishastra he's an astronomer and mathematician based out of kerala and especially in astronomical calculations it is very important to have more precise value of pi as per your need in fact madhwa was one of the first astronomers to accurately calculate the position of moon for every 36 minutes so there are a lot of great things about madhava but coming back here so that is the purpose and coming to the proof he detailed out this infinite series of pi in mahadjana in a prakarana but treaties on calculations of science co-science tangents etc with infinite series coming to the procession of pi given by madhava as i said it's an infinite series but nonetheless it takes incredible amount of effort to calculate using this infinite series but yes the precession can be obtained depending on your patience but it precisely gives the value of pi following the formula given by madhava we can't really call the 20th century as ancient india but if we don't talk about this contribution that came in from srinivasa ramanujan about the value of pi this video will be grossly incomplete ramanujan was the one who proposed the fastest calculation for the value of pi while madhava's infinite series for the value of pi is very doable but it takes incredible amount of effort to refine the value of pi to more and more decimal places you have to incorporate thousands of terms into that infinite series to get the value of pi to few tens of decimal places srinivasa ramanujan came with his own formula for pi what you're seeing here sounds to be very complex and weird but this formula of pi is the fastest one of those times to calculate the pi in matter of minutes and there are tons of other examples that he scribbled in his notebook as to how to reach to the value of pi quickly and of course precisely is one of the very few mathematicians who gave the fastest convergence for the value of pi and even today these formulas given by surinva sara manager form the bedrock for calculations of the values of pi and many other concepts around it so that is a complete outlook about how pi has evolved over thousands of years in india there are many other mathematicians and astronomers who worked on this transcendental number pi but we just took these four as they are relatively more significant than other contributions that came in from bharat refining the value of pi so the earliest approximations for the value of pi from yazoo vedam which is given around 3 is well known for its antiquity one of the oldest references of pi not just in bharat but around the world but there are also other civilizations who calculated pi more accurately than what was given in yazuru and that's very important to note nonetheless the antiquity of this particular approximation really stands out moving next to aryabhatta his value of 3.1416 is very handy but at the same time and during the same period in china a better approximation was proposed and moving next to madhava this one definitely is a groundbreaking revolution that came from india where there was a full stop to adding one digit after other to the decimal places rather calculating the value of pi to an infinite series so it was an unprecedented accuracy and then coming to srinivasaramanagin his formula gave an unprecedented speed to the calculations of pi the contributions that came in from india in refining the value of pi are just incredible and these historical milestones with their purpose proof and precession the history speaks for itself and this is the story of the evolution of pi in india which is wide deep and rich than a simple 22x7 and as always thanks for watching [Music]