How pi was almost 6.283185…

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Happy pi day! Did you know that in some of his notes, Euler used the symbol pi to represent 6.28..., before the more familiar 3.14... took off as a standard?
Plushie creatures now available: http://3b1b.co/store
Why? Well, people asked, and what better way to celebrate pi day?
The idea for this video, as well as the live shots, came from Ben Hambrecht, with the writing and animating done by Grant Sanderson.
Special thanks to:
- University Library Basel, for letting us rummage through their historical collection
- Martin MattmΓΌller from the Bernoulli-Euler center for helpful discussion
- Michael Hartl, author of the Tau Manifesto, for pointing us to obscure references
- Library of the Institut de France
Cinematographer: Eugen Heller
Music by Vincent Rubinetti:
https://vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown
https://soundcloud.com/vincerubinetti/
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Tags:  3b1b  euler  mathematics  pi  how  pi  was  almost  6.283185... 
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Category: Education
Caption: [music]. i'm sure that you're already familiar . with the whole pie versus tao debate a . lot of people say that the fundamental . circle constant we hold up should be the . ratio of a circles circumference to its . radius which is around six point two . eight not the ratio to its diameter the . more familiar 3. 14 these days we often. call that larger constant tao . popularized by michael hurdles tau . manifesto although personally i'm quite . partial to robert police's proposed . notation of a pie with three legs in . either of these manifestos and on many . many other places of the internet you . can read to no end . about how many formulas look a lot . cleaner using tao largely because the . number of radians describing a given. fraction of a circle is actually that . fraction of tau that deadhorse is beat . i'm not here to make that case further . instead i'd like to talk about the . seminal moment in history when pi as we . know it became the standard for this one . fruitful place to look is that the old . notes and letters by one of history's . most influential mathematicians leonardo . euler luckily we now have an official . three blue one brown switzerland . correspondent ben hum brecht who was . able to go to the library in euler's . hometown and get his hands on some of . the original documents and in looking . through some of those it might surprise . you to see oiler write let pi be the . circumference of a circle whose radius. is one that is the six point two eight . constant that we would now call tau and . it's likely he was using the greek . letter pi as ap for perimeter so was it . the case that euler genius of the day . was more notationally enlightened than. the rest of the world fighting the good . fight for 6. 28 . and if so who's the villain of our story . pushing the 3. 1415 constant shoved in . front of most students today well the . work that really established pi as we . now know it as the commonly recognized . circle constant was an early calculus. book from 1748 at the start of chapter 8 . in describing the semi circumference of. a circle with radius one and after . expanding out a full 128 digits of this . number one of them wrong by the way the . author adds which for the sake of . brevity i may write pi now there were . other texts and letters here and there . with varying conventions for the . notation of various circle. since but this book and this section in . particular was really the one to spread . the notation throughout europe and . eventually the world so what monster . wrote this book with such an . unprincipled take towards circle. constance . well euler again in fact if you look . further you can find instances of euler . using the symbol pi to represent a. quarter turn of the circle what we would . call today pi halves or tau fourths . in fact euler's use of the letter pi . seems to be much more analogous to our . use of the greek letter theta it's . typical for us to let it represent an . angle but no one angle in particular . sometimes it's 30 degrees maybe other . times it's 135 and most times it's just . a variable for a general statement it . depends on the problem and the context . before us likewise euler let pi . represent whatever circle constant best . suited the problem before him though . it's worth pointing out that he . typically framed things in terms of unit . circles with radius 1 so the 3. 1415 . constant would almost always have been . thought of as the ratio of a circles . semi circumference to its radius . none of this circumference to its. diameter nonsense and i think oilers use. of this symbol carries with it a general. lesson about how we should approach math . the thing you have to understand about . euler is that this man solved problems a . lot of problems i mean day in day out . breakfast lunch and dinner he was just . turning out puzzles and formulas and . having insights and creating entire new. fields left and right over the course of . his life he wrote over 500 books and . papers which amounted to a rate of 800 . pages per year and these are dense math . pages and then after his death another. 400 publication surfaced it's often . joked that formulas and math have to be. named after the second person to prove . them because the first is always gonna . be euler his mind was not focused on . what circle constant we should take his . fundamental he was on solving the task . sitting in front of him in a particular . moment than writing a letter to the . bernoulliΓ­s to boast about doing so . afterwards for some problems the . quarter-circle constant was most natural . to think about for others the full . circle constant and for others still say. at the start of chapter 8 of his famous . calculus book maybe the half circle . constant was most natural to think about . too often in math education the focus is . on which of multiple competing views . about a topic is right . is it correct to say that the sum of all . positive integers is negative 1/12 or is . it correct to say that it diverges to . infinity can the infinitesimal values of . calculus be taken literally or is it . only correct to speak in terms of limits . are you allowed to divide a number by . zero these questions in isolation just . don't matter. our focus should be on specific problems . and puzzles both those of practical . application and those of idle pondering . for knowledge as own sake . then when questions of standards arise. you can answer them with respect to a . given context and inevitably different . contexts will lend themselves to . different answers of what seems most . natural but that's ok . outputting 800 pages a year of dense . transformative insights seems to be more . correlated with a flexibility towards . conventions than it does with focusing . on which standards are objectively right. so when this pi day the next time . someone tells you that you know we . should really be celebrating math on . june 28th see how quickly you can change . the topic to one where you're actually . talking about a piece of math . [music]. you. .
 

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