How pi was almost 6.283185…


I’m sure you’re well-familiar with the
whole pi vs. tau debate. Many people say that the fundamental circle
constant we hold up should be the ratio of a circle’s circumference to its radius,
around 6.28, not the ratio to its diameter, the more familiar 3.14. These days we call this larger constant “tau”,
popularized by Michael Hartl’s “Tau manifesto”, although personally, I’m quite partial to
Robert Palais’ proposed notation of a pi with three legs. In this manifesto and many other places on
the internet, you can read to no end about how many formulas look at lot cleaner using
tau, largely because the number of radians describing a given fraction of a circle is
actually that fraction of tau. That dead horse is beat, I’m not here to
make the 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 at
the old notes and letters by one of history’s most influential mathematicians, Leonhard
Euler. Luckily, we now have an official 3blue1brown
Switzerland correspondent Ben Hambrecht who was able to go to the library in Euler’s
hometown and get his hands on some of these original documents. It might surprise you see Euler write (in
French), “Let pi be the circumference of a circle whose radius=1”. That is, the 6.28 constant which we call tau
today, where he was likely using pi as the greek letter “p” for “perimeter”. So was it the case that Euler was more notationally
enlightened than the rest of the world? Fighting the good fight for 6.28? If so, who’s the villain of our story pushing
the 3.14 constant that students are shown today? The work that really established 3.14… 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 1, and after expanding out 128 digits of the
this number, the author writes “which for the sake of brevity I may write π”. There were other texts and letters here and
there with varying conventions for the notation of various circle constants, but this book,
and this section in particular, was really the one to spread the notation through Europe,
and eventually the world. So who wrote this text with such an unprincipled
take towards circle constants? Well…Euler again. In fact, we can also find instances of Euler
using the symbol pi to represent a quarter turn of a circle, what we would today call
“pi/2”. 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 particular angle. Sometimes it’s 30o, other times 135o, most
times just a variable for a general statement. It depends on the problem and context before
us. Likewise, Euler just let pi represent whatever
circle constant best suited the problem before him. Though it’s worth pointing out he typically
framed things in terms of unit circles, with radius 1, so the 3.14 constant would have
been thought of as the ratio of a circle’s semi-circumference to its radius, none of
this circumference to its diameter nonsense. And I think Euler’s use of this symbol carries
with it a general lesson about how we should approach to math. What you have to understand about Euler is
that he solved problems. A lot of problems. I mean, day in day out breakfast lunch and
dinner this man was thinking about puzzles, formulas, having insights and creating entire
new fields left and right. He wrote over 500 books and papers during
his lifetime, what amounted to about 800 pages per year, with another 400 publications appearing
posthumously. It’s often joked that formulas in math have
to be named after the second person to prove them, since the first person will always be
Euler. His mind was not focused which circle constant
should be taken as fundamental; it was how do I solve the task sitting in front of him
and writing a letter to the Bernoulli’s boasting about doing so. For some problems, the quarter-circle-constant
was most natural to think about. For others, the full circle, and for others
still, the half circle. Too often in math education the focus is on
which of multiple competing views of a topic is “right”. Is it correct to say the sum of all positive
integers -1/12, or is it correct to say it diverges to infinity? Can the infinitesimals 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 0? 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’s
own sake. When questions of standards arise, then, you
can answer them with respect to a given context. Inevitably, different contexts will lend themselves
to different answers of what seems most natural, but that’s okay. Outputting 800 pages a year of transformative
insights seems to be more correlated with a flexibility towards conventions than it
does with focusing on which standards are objectively right. So on this pi day, the next time someone tells
you we should really be celebrating math on June 28th, see how quickly you can change
the topic to one where you’re talking about an actual piece of math.

Comments 100

  • How is 1+2+3…
    -1/12?

  • 우으..

  • I was thinking about thit two days ago and today it was in my recommended.

  • But pie is math

  • 한국인 손들어

  • This is so good. And its applicable to a lot more than just math. Conventions are overrated

  • tau > pi
    prove me wrong

  • You misspelled περίμετρος (you wrote περιμετρος)

  • im quite partial to pie with mashed spud and cheese on top

  • Stop trying to change the value of an established constant, and use the other constant!
    C = τr
    A = πr²

    If Euler used the symbol ambiguously, why should we adhere to any one single value over another? Semantics! Nitpicky semantics!

  • Please support this channel
    ….The best and and the most intelligent channel on Youtube….

  • I like eta (half pi) more. It's a right angle in radians.

  • 이것은 공대가 높게 평가

  • pi represents a plane in a 3 dimensional room right? Obviously this is the only correct usage XD

  • 1:48 Double check your blurred sections.

  • This some idiots

  • Basically, calculus is the alphabet.

  • I never understood this "pi-tau-discussion".

    The first time I've heard abou it was 2011. And I've not cared at all on this. Why I should?

    For me as a mathematician I ask for "motivation" of a defintion.

    All circles are similar. This is was Pi is saying. If one takes radius or diameter… it's not important.

    So Pi has a meaningful sense.
    Of course Tau has some intuitive attitudes. But maths is not about intuition.

    It's about meaningful axioms. This is what makes mathematics so beautiful.

    Yeah I know there are people who say: "Mathematics should be useful."

    Define "useful".

    x years ago even mathematician laughed about number theory mathematicians. "What do u want with primes?"

    Good Mathematicians are good for one reason.

    They never ask "what is it good for?" That's why they are good.

    Consider this scenario.
    One is developing a new theory from some definition.

    How can one know it's not useful?

    Think about Banarch-Tarski.
    It's totally counter-intiuitive.
    Would u ever dare to doubt the "axiom of choice" in intuitive way?

    No. No one would.
    Only in philosophical way.

    Oh now I remember.
    I understand the pi-tau-discussion.

    The first person, who starts this silly discussion in 2011 was not a mathematician. He is a physician… this explains a lot.

  • 0:19 its 3.1415 not 3.1416

  • I was born on 14 March 1999

  • Does anyone know where i can learn more about the sum at 5:25?

  • Tau many people hate Tau. They shall be liked equally. This is a NumberPhile , or not that, but an infinite-load of numbers. Goodbye, I said my opinion, now go eat my baked number Pi .

  • i like how he used it to represent whichever constant was the most efficient, but i think that concept would be hard to teach to kids in highschool and middle school, so if anything, i think it would be good to teach that concept in advanced math in college and universities.

  • So nobody is gonna make a joke about the pi with 3 legs huh? Guess I have a dumber sense of humor than your average viewer…

  • 6.28 would make more sense imho. Though the floating constant is cool idea too.

  • 3:27 I solve problems. Practical problems

  • The narrator reminds me of the PBS vacation guy, Rick Stevens, I think

  • The Isaac Asimov of the 18th century, in terms of publication

  • Tau looks like a beyblade so I like it better. τ⚡τ

  • when you realise that it's just tau

  • That's not how names work. Pi might not have been called "pi", but its value cannot change.

  • Are those books hand written?!
    I mean seriously…!

  • 0:17 It's not "…" if you rounded up!

  • 1 + 2 + 3 + … = -1/12 still pisses me off
    Yes I have seen the fancy demostration, and yes it makes sense when you put it like that
    Problem is, a FUNDAMENTAL rule of the + operation is that the sum of two positive numbers will always yield a positive number. No matter how many times you do it, no matter how many numbers are involved the result MUST be positive. God it's like they got a lawyer to do the math

  • So Euler was the villain!

    Henceforth, I shall write “let tau be the ratio of a circle’s perimeter to its radius…”, and then proceed with proper math, discarding this pi nonsense…

  • Can you explain the Bailey–Borwein–Plouffe formula? https://en.wikipedia.org/wiki/Bailey%E2%80%93Borwein%E2%80%93Plouffe_formula

  • Tau day is my b day!!!

  • thank you oiler, very cool!

  • If the old system of teaching and learning mathematical processes by equations and formulae continues, then the estimation/calculations of Pi from geometrical ideas of numbered dimensions driven forward, should also be analyzed in reverse/reciprocal, and the conception of the real numbers from e-Pi-i interference positioning processes be contextualized (Holographicly..), along the lines of Analytical Chemistry and Physics. (?)

    Euler apparently didn't need to think about how to find a particular solution, he just did what he somehow knew to do by intuitive experience?, or what is "genius" of any kind/type if it's not just an obsessively focused skill, well practiced? (And the appropriate aptitude)

  • "2pi or not 2pi?"

  • Your mesuring the whole circle!!! Not half!!! What kinda moron would invent a constant where you had to multiply it times two all the time??

  • I still believe using the constant Tau = 6.28… would make trigonometry much easier for students that are previously unfamiliar with angles and functions acting on angles. Because the idea that pi/2 is not half a turn or pi/3 is not a third of a turn around the circle seems to lead students to make several mistakes just because of this arbitrary notation they likely JUST learned about. After that, idgaf what people use, though it would logically follow that you continue using Tau for consistency. Otherwise, I believe engineers should use Pi (=3.14…) because diameter and circumference are direct measurements while radius is an abstract concept relative to diameter, however, I believe mathematicians should use Tau because the concept of a perfect circle in some space comes about from a given center and radius, hence the length of the border line should be taken relative to its radius. But given how many people start hating mathematics around trig level, I think making calculations simpler by using Tau would help maintain a student's interest in math, and that's what I really care about in this argument.

  • wait so, if (pi)r² then, its = (tau)r?

  • 209 viharts dislike this video

  • Wouldn't changing pi mean that you would not be able to derive trigs with as much ease?

  • Well lucky for us the pi symbol visually makes sense as the half circle constant, and the tau symbol visually makes sense for the whole circle constant, so it all worked out for the better :). I think of it this way, the tau sign is a sine wave over 1, pi is a sine wave over 2(11)

  • pi π no es la relación perímetro diámetro pues una circunferencia no es un polígono regular es una curva cerrada una línea curva no es una recta ni está compuesta de rectas

  • el radian está a 56.25 grados

  • You know that TV show called "Ed", with Tom Cavanagh? The narrator sounds like him.

  • 3,14 is not euclidean, it is a surface or american wild geometry

  • 3.14159265358979323846264338327950288419716939937510582097445

  • DarkFrame?

  • 3.1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679 8214808651 3282306647 0938446095 5058223172 5359408128 4811174502 8410270193 8521105559 6446229489 5493038196 4428810975 6659334461 2847564823 3786783165 2712019091 4564856692 3460348610 4543266482 1339360726 0249141273 7245870066 0631558817 4881520920 9628292540 9171536436 7892590360 0113305305 4882046652 1384146951 9415116094 3305727036 5759591953 0921861173 8193261179 3105118548 0744623799 6274956735 1885752724 8912279381 8301194912 9833673362 4406566430 8602139494 6395224737 1907021798 6094370277 0539217176 2931767523 8467481846 7669405132 0005681271 4526356082 7785771342 7577896091 7363717872 1468440901 2249534301 4654958537 1050792279 6892589235 4201995611 2129021960 8640344181 5981362977 4771309960 5187072113 4999999837 2978049951 0597317328 1609631859 5024459455 3469083026 4252230825 3344685035 2619311881 7101000313 7838752886 5875332083 8142061717 7669147303 5982534904 2875546873 1159562863 8823537875 9375195778 1857780532 1712268066 1300192787 6611195909 2164201989 3809525720 1065485863 2788659361 5338182796 8230301952 0353018529 6899577362 2599413891 2497217752 8347913151 5574857242 4541506959 5082953311 6861727855 8890750983 8175463746 4939319255 0604009277 0167113900 9848824012 8583616035 6370766010 4710181942 9555961989 4676783744 9448255379 7747268471 0404753464 6208046684 2590694912 9331367702 8989152104 7521620569 6602405803 8150193511 2533824300 3558764024 7496473263 9141992726 0426992279 6782354781 6360093417 2164121992 4586315030 2861829745 5570674983 8505494588 5869269956 9092721079 7509302955 3211653449 8720275596 0236480665 4991198818 3479775356 6369807426 5425278625 5181841757 4672890977 7727938000 8164706001 6145249192 1732172147 7235014144 1973568548 1613611573 5255213347 5741849468 4385233239 0739414333 4547762416 8625189835 6948556209 9219222184 2725502542 5688767179 0494601653 4668049886 2723279178 6085784383 8279679766 8145410095 3883786360 9506800642 2512520511 7392984896 0841284886 2694560424 1965285022 2106611863 0674427862 2039194945 0471237137 8696095636 4371917287 4677646575 7396241389 0865832645 9958133904 7802759009 9465764078 9512694683 9835259570 9825822620 5224894077 2671947826 8482601476 9909026401 3639443745 5305068203 4962524517 4939965143 1429809190 6592509372 2169646151 5709858387 4105978859 5977297549 8930161753 9284681382 6868386894 2774155991 8559252459 5395943104 9972524680 8459872736 4469584865 3836736222 6260991246 0805124388 4390451244 1365497627 8079771569 1435997700 1296160894 4169486855 5848406353 4220722258 2848864815 8456028506 0168427394 5226746767 8895252138 5225499546 6672782398 6456596116 3548862305 7745649803 5593634568 1743241125 1507606947 9451096596 0940252288 7971089314 5669136867 2287489405 6010150330 8617928680 9208747609 1782493858 9009714909 6759852613 6554978189 3129784821 6829989487 2265880485 7564014270 4775551323 7964145152 3746234364 5428584447 9526586782 1051141354 7357395231 1342716610 2135969536 2314429524 8493718711 0145765403 5902799344 0374200731 0578539062 1983874478 0847848968 3321445713 8687519435 0643021845 3191048481 0053706146 8067491927 8191197939 9520614196 6342875444 0643745123 7181921799 9839101591 9561814675 1426912397 4894090718 6494231961 5679452080 9514655022 5231603881 9301420937 6213785595 6638937787 0830390697 9207734672 2182562599 6615014215 0306803844 7734549202 6054146659 2520149744 2850732518 6660021324 3408819071 0486331734 6496514539 0579626856 1005508106 6587969981 6357473638 4052571459 1028970641 4011097120 6280439039 7595156771 5770042033 7869936007 2305587631 7635942187 3125147120 5329281918 2618612586 7321579198 4148488291 6447060957 5270695722 0917567116 7229109816 9091528017 3506712748 5832228718 3520935396 5725121083 5791513698 8209144421 0067510334 6711031412 6711136990 8658516398 3150197016 5151168517 1437657618 3515565088 4909989859 9823873455 2833163550 7647918535 8932261854 8963213293 3089857064 2046752590 7091548141 6549859461 6371802709 8199430992 4488957571 2828905923 2332609729 9712084433

  • 3.14159… es la mitad de la sumatoria de los lados de un polígono regular de radio igual a 1

  • el echo que un polígono regular tenga muchos lados que su apotema tienda a 1 sus ángulos a 0 no lo convierte en una circunferencia sique siendo un polígono regular

  • si tú tienes una cuerda de 50 centímetros la tencionas toma forma de linea recta cambia su forma no su medida
    una línea curva no es una linea recta ni está compuesta de rectas

  • un fractal se expande manteniendo su característica y su factor numérico al infinitamente grande y el infinitamente pequeño

  • π no es la relación perímetro diámetro de la circunferencia

  • algoritmos como √(2-√4-l²) sirven para encontrar los valores de los lados de un polígono

  • Tau vs Pi

  • Tau makes way more sense to be frank here.

  • If we define a system of units where Pi=1, can we then prove Eulers number is rational (in said system)? Or are all other numbers in this system inherently irrational?

  • Tau is best! The battle is a joke, but I like tau for more things

  • Me interesaste …. en el tema quisiera saber mas …. utilizo /6.28 en la formula para mis confecciones

  • Pi…

  • how get tau do pi*2

  • 3.1415… es la mitad de la sumatoria de los lados de un polígono regular de radio igual a 1 que supuestamente tiene lados infinitos que cada vez se hacen más pequeños son rectas que se convierte en una curva es lo que se supone que es una circunferencia una circunferencia no es un polígono regular es una curva cerrada una línea puede tener diferentes formas puede ser recta o curva

  • "3.1416…" Noype. The ellipsis should be used after the actual digits.

  • Pi=Wrong
    Tau=2Pi
    Tau=2Wrong

  • 1:58 *3,1415…

  • la relación perímetro diámetro de la circunferencia es 3.2

  • 0:49 "The history of our people" lol

  • Pi is useless. Every equation involving pi is multiplying it by 2.

  • How Pi was almost Tau.

  • 하앟 흥분된다 존나 신비로워

  • More like pi is the arc length.

  • All this because we use a base ten system. Use a different base and argue what you must.

  • I feel like this channel just may be too smart for me….

  • Amazing video!

  • So what, then, do all your blue and brown characters represent?

  • Why is Euler's name blurred out at 1:52?

  • Every time I watch one of your videos it gives me a fresh perspective on mathematics

  • I'm going to have to go with Euler on this one

  • 6.28318530717958 is better than 3.20000000000000

  • I, for one, support the quarter circle constant.
    long live 1.57079632679

  • In my country, we don`t read pie, we read pee

  • Great video! I love pi so much! The beginning of pi is the ray.

  • And in a few short years we progressed to this,.,.

  • In 100 years tau day is going to be celebrated for Elon Musk day (his birthday)

  • It's not pronounced "tau" it's "taf" trust me I'm greek

  • I actually started hating Math due to the very lengthy exams in ny university but thanks to 3Blue1Brown love for it has resurfaced again!

  • So pi was almost 2 pi. SCIENCE

  • Well don't irrational numbers represent all number because they are infinite?

  • is it really a problem tho?

  • Actually I feel that student should be taught both pi and tau, and the conversion between them.

  • So this guy Euler goes to Könnigsberg to cross a bridge…

  • That's stupid.

  • Mmmmm pi.

  • π / 2 is the best: the ration between the semicircle and the diameter.
    The reason?
    The diameter of the circle is the hypotenuse of an infinity of inscribed-right-triangles.
    Therefore its "squared" equivalent is the semicircle.

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