A Brief History of Pi

This video is brought to you by Squarespace whether you need a domain website or online store make it with Squarespace take any circle measure its circumference and its diameter the ratio of these two numbers is a mathematical constant we call pi while this definition is simple pi has been studied for thousands of years and History of our understanding not just of the value of pi But also what it means forms a history of all of mathematics it takes us from the Middle East to Europe to China to India and even America It's a history, which involves revolutions murder and the infinite Maths is as old as civilization older even There's evidence of counting going back thirty thousand years and two of the very earliest civilizations the ancient Egyptians and Babylonians both investigated pi around 4000 years ago the Babylonians estimated PI to be 3 and 1/8 or 3.125 now that's the first of a few estimates you're gonna hear in this video so for reference remember that the first few digits of pi are 3.1415926 There are more that means that the Babylonian estimate of Pi is accurate to 1% of its true value Which is kind of astonishing when you remember that this is a time in human history when? iron was first being used and the last mammoths went extinct the ancient Egyptians on the other hand estimated PI slightly less accurately as 3.16 but how do you even estimate the value of pi You have to count it by definition measure a curved surface, which is super tricky to do accurately Well one way of doing it is to cheat and actually use a square compare a square and a circle well It's quite a little bit like a circle But that was much like a circle as a Pentagon which has one more side than a square and a Pentagon doesn't look quite as much like a circle as a hexagon which has one more side again and a hexagon doesn't look quite as much like a circle as a heptagon and So on you can think of a circle as a regular polygon just want with an extremely large number of sides So many sides in fact that each individual one is Infinitesimally small meaning that the circle looks round this was exactly the thinking that legendary ancient Greek mathematician Archimedes Used when estimating pi around 220 BC in fact It was probably the very last thing he ever did to approximate pi He reasoned why not measure the perimeter of a square adding up the lengths of all of its edges And then dividing that number by the squares diameter But what is the diameter of a square is it the length of its diagonal or the length of one of its edges? Why not both said Archimedes draw one square with its corners just touching the perimeter of a circle Another square with its faces just touching the perimeter of that same circle add up the lengths of the sides of each square divided by their effective diameters And you have two estimates for the value of pi the true value of which lie somewhere between those two numbers But here's the really clever part Because the difference between those two values is pretty big if you're using squares because a square isn't much like a circle But replace those squares with Pentagon's and you shrink the difference between those two numbers Meaning that there's a smaller range of values that PI could be your estimation just got more accurate And if you replace those Pentagon's with hexagons you'll get an even more accurate estimates keep increasing the number of faces on the shape that you're drawing inside and outside the circle and your estimate will get more and More accurate as long as you have the time and patience to draw said shapes There is a reason why this thing was called the method of exhaustion Archimedes got up to a 96 sided shape which incidentally is called an a neocon Turki hexagon I really hope I said that right giving an estimate of Pi between three point one four zero eight and three point one four To nine so accurate to two decimal places as I mentioned earlier this was likely his final contribution to science because into 1/2 BC he was killed by Roman soldiers who invaded his hometown Zaira Q's he was apparently performing this Calculation at the time allegedly his final words were don't disturb my circles European progress in the study of pi died with Archimedes for well over a thousand years fortunately however there was plenty of the world which was not in Europe a mathematicians here were also interested in PI in particular three mathematical superpowers of the first millennium ad Were China India and Persia? Ideas when these three nations were soon to change the world first off Chinese mathematicians used a method of exhaustion similar to our comedians But instead of considering the parameters of shapes they considered their areas and this dude no I'm not going to try and pronounce his name because I'll only get it wrong used to polygon with 3072 sides to obtain pi to five decimal places 200 years later a father-and-son team used a polygon with over 12,000 sides to extend that record to six decimal places and that was a world record which stood for 800 years the problem was it was just difficult to do these calculations. They weren't especially hard to understand it was just awkward to write down what you were doing to physically do the Calculation and this was something that would only be resolved by the introduction of two world changing ideas From India and Persia say that you want to do a calculation You know that you and your friends together weigh a hundred and twenty-five kilos And you also know that you weigh 70 kilos the question is how much does your friend weigh? Mathematically, we'd write. This as X plus 70 equals 125 where X is your friends weight in kilos Subtract 17 from both sides and you get the answer 55 kilos now in that simple example. I just used two ideas which were Revolutionary to the classical world firstly I wrote large numbers like 125 and 70 using a simple notation we take it for granted these days But the ability to write any number using just ten symbols and a place value notation Where the position of a symbol in a number determines its size? Massively simplifies arithmetic to see what I mean try and do that calculation Only using Roman numerals our modern decimal notation was first developed in India some time before 400 AD and then rapidly spread to Persia where the second key idea came from the second key idea was Representing your friend's weight using some symbol X and then manipulating both sides of the equation this of course is algebra originally developed by Babylonian and ancient mathematicians But truly established by Persian mathematician and all-round very influential dude Mohammed eben Musa al-khwarizmi using decimal notation and algebra allowed for much easier calculations across all of maths and mathematicians working on calculating PI Used it to turbocharge their work after the Renaissance and a renewed interest in mathematics along with Crucially new tools from the east Europe was back in the game and in 1630 the most accurate estimate of Pi using the polygon method was achieved by Austrian astronomer Christiaan grind Berger who used a shape with 10 to the 40 sides yes really to calculate pi to 38 decimal places and then because mathematicians are sensible people with lives to lead they decided that was accurate enough And they'd leave it there oh wait The adoption of algebra by European mathematicians triggered a whole new way of looking at the world a change in thinking generally grouped under the title the Scientific Revolution Which itself went on to inspire the Age of Enlightenment with thinkers like Rene Descartes and John Locke? Amongst other ideas the Enlightenment movement emphasized the value of Reason over Tradition and new mathematical ideas were held up as Paragons of this they were pure reason the change in how 17th century European mathematicians calculated PI is arguably a Perfect example of the shift from following what the ancients did to new rational? Theoretical approaches because while the ancients like Archimedes may have measured the perimeters of shapes increasingly similar to circles Now European mathematicians were using a method based entirely on reason a method based on Infinite series an infinite series is just an expression Made up of things added together one after the other after the other after the other and so on until Forever if those contributions keep getting smaller as you go on then the series converges to a particular value Sometimes you can work out what that value will be using logical arguments? but sometimes you just have to keep calculating term after term after term until you reach an accuracy that you're happy with the Method of using infinite series to calculate pi was first used not in Europe But again in India you could kind of argue that what Archimedes did was an infinite series? But the first person to write a mathematical function as an infinite series was Indian mathematician math hava of Sangamo grammar in the 14th century He wrote down expressions for the sine cosine and tangent of an angle as well as the inverse tangent quick refresher if you write the expression y equals tan of X the Expansion for the tangent would tell you what y equals? If you already know what X is while the expansion of the inverse tangent would tell you what X is? If you already know what Y is by its definition the function tan of X precisely equals 1 when x equals 1/4 pi That means that if you have an expression for the inverse tangent Then if you plug 1 into that expression and keep calculating terms You'll end up with an increasingly accurate estimate of 1/4 pi madhava did this and calculated PI to 11 digits, but then his method seems to have been forgotten only to be apparently independently rediscovered in 17th century Europe by Scott James Gregory and German, Gottfried Wilhelm lightness and at this point everything kicked off the new decimal notation and algebraic technique allowed for record calculations of Pi in 1699 it was calculated to 271 digits by abraham sharp who was beaten in 1706 when John machen reached a hundred digits who was in turn beaten by thomas von tete de l'année I hope that's how you say his name in 1719 with 112 digits it wasn't just the case that each of those mathematicians had more spare time than the previous one They were competing with each other using different infinite series, which converged on PI faster instead of just using the inverse tangent infinite series They might use a combination of different inverse tangent values or something completely different the competition then became less about which mathematician had done the most calculations and instead which mathematician had the fastest converging infinite series Development of increasingly efficient infinite series continued well into the 20th century With the technique kind of coming full circle as the current infinite series of choice was developed by Indian prodigy mathematician Srinivasa Ramanujan of course by the 20th century Mechanical computers had been invented making it much easier to calculate pi You basically just used one until he got bored in 1949 Americans D. F, Ferguson and John wrench calculated PI to 1120 digits But they were bringing a knife to a gunfight because that very same year the first calculation of Pi by an electronic computer was done nearly doubling their record with two thousand and thirty seven digits from here the history of Pi is basically a list of increasingly powerful computers running for a long time and spitting out Increasingly absurd numbers of digits at the time of recording the world record for digits of pi Calculated is held by peter trib with a shade under twenty two and a half trillion digits Calculated the question of course is if we know that pi is going to keep going on forever. It's a transcendental number Why should anybody bother calculating anymore dishes? Well for one thing calculating pi is actually a really good way of making sure that your brand new shiny computer is working properly Calculating pi uses up a lot of mental brainpower for the computer you have an answer that you can check yours Against and also if you keep going just a little bit longer than the previous person you can have a casual world record Secondly pi is actually a really good random number generator If you look at the first two hundred billion digits of pi. You'll find the number zero occurs almost precisely 20 billion times and the same goes for the other digits 1 through 9 That means that if you were to pick a random digit in those 200 billion There's an almost exactly 10 percent chance of it being one under almost exactly 10% chance being to and so on This makes calculating PI to a large number of digits very Valuable to people that want to generate random numbers people working in cryptography for example But lastly and arguably most importantly People keep calculating More digits of pi for the same reason that why people memorize tens of thousands of digits of pi And the same reason why people climb mountains and swim oceans and invent the double luge because they can Humans are weird. We like to understand the world around us and as our civilization has developed We've built increasingly complex tools to help us understand the world It wasn't essential for our survival that we did that we just did it because of the way we're wired because we could pie is a thread that's gone through all of human history because it's a microcosm of how we interact with the natural world from the ancients to present day through Revolutions in Thor and across the world as long as there are people. There's always going to be somebody who just wanders What's the next digit long may that continue? I'd like to close out this video with two quick announcements announcement the first I Finally launched a website Go to the very shiny and new Simon ox fist calm for a hub for everything I do online including this YouTube channel Sponge in electric and the wiki cast there's a page there Detailing the gear that I used to make these videos and coming soon there's also going to be a page detailing all the advice that I give out to people who are Interested in applying to the universities of Oxford and Cambridge and then announcement number two if you'd like a look at my website And why wouldn't you I built it with Squarespace who very kindly sponsored this video before making this website I didn't have any experience with web development at all But it was super easy guys so easy so quick to use one of their templates customize it to look the way I want and then just fill in my stuff and then bang. It's done. I never have to patch or upgrade or install anything It's all taken care of Brilliant also it was super easy to set up for selling merchandise which um May be happening soon you you'd like to build your own website for your next project then definitely use Squarespace for it go to Squarespace comm for a free trial and when you're ready to hit the big red button and build the Website go to Squarespace com forward slash, Simon Clark to get 10% off your first purchase Thanks to Squarespace for sponsoring this video. Thank you for watching it if you enjoyed it please do give it a share pop it alike maybe leave a comment and Thank you again. I'll see you next time

Channel: Simon Clark

Views: 662,606

Rating: 4.8883657 out of 5

Keywords: pi, math, maths, mathematics, history, history of math, history of maths, history of mathematics, archimedes, transcendental, liu hui, madhava, ramanujan, al khwarizmi, leibniz, infinite series, algebra, place value notation, ancient greeks, babylonians, ancient greek maths, ancient greek math, indian math, indian mathematics, approximation, method of exhaustion, Pi day, why pi is 3.14, pi day, secrets of pi, why pi is constant, what is pi, simon clark, how pi was discovered

Id: 1-JAx3nUwms

Length: 16min 47sec (1007 seconds)

Published: Wed Mar 14 2018

Reddit Comments

[u/lrrpkd]

Would it not have been easier and more accurate for the ancients to instead draw a circle with a form of a compass then lay a string down exactly as they could, cut the string then lay the string flat and measure that? Sure it wouldn’t have been proven by a formula but you would get a fairly accurate number.