Understanding e to the i pi in 3.14 minutes | DE5


One way to think about the function e^t is
to ask what properties it has. Probably the most important one, from some points of view
the defining property, is that it is its own derivative. Together with the added condition
that inputting zero returns 1, it’s the only function with this property. You can
illustrate what that means with a physical model: If e^t describes your position on the
number line as a function of time, then you start at 1. What this equation says is that
your velocity, the derivative of position, is always equal your position. The farther
away from 0 you are, the faster you move. So even before knowing how to compute e^t
exactly, going from a specific time to a specific position, this ability to associate each position
with the velocity you must have at that position paints a very strong intuitive picture of
how the function must grow. You know you’ll be accelerating, at an accelerating rate,
with an all-around feeling of things getting out of hand quickly. If we add a constant to this exponent, like
e^{2t}, the chain rule tells us the derivative is now 2 times itself. So at every point on
the number line, rather than attaching a vector corresponding to the number itself, first
double the magnitude, then attach it. Moving so that your position is always e^{2t} is
the same thing as moving in such a way that your velocity is always twice your position.
The implication of that 2 is that our runaway growth feels all the more out of control. If that constant was negative, say -0.5, then
your velocity vector is always -0.5 times your position vector, meaning you flip it
around 180-degrees, and scale its length by a half. Moving in such a way that your velocity
always matches this flipped and squished copy of the position vector, you’d go the other
direction, slowing down in exponential decay towards 0. What about if the constant was i? If your
position was always e^{i * t}, how would you move as that time t ticks forward? The derivative
of your position would now always be i times itself. Multiplying by i has the effect of
rotating numbers 90-degrees, and as you might expect, things only make sense here if we
start thinking beyond the number line and in the complex plane. So even before you know how to compute e^{it},
you know that for any position this might give for some value of t, the velocity at
that time will be a 90-degree rotation of that position. Drawing this for all possible
positions you might come across, we get a vector field, whereas usual with vector field
we shrink things down to avoid clutter. At time t=0, e^{it} will be 1. There’s only
one trajectory starting from that position where your velocity is always matching the
vector it’s passing through, a 90-degree rotation of position. It’s when you go around
the unit circle at a speed of 1 unit per second. So after pi seconds, you’ve traced a distance
of pi around; e^{i * pi}=-1. After tau seconds, you’ve gone full circle; e^{i * tau}=1.
And more generally, e^{i * t} equals a number t radians around this circle. Nevertheless, something might still feel immoral
about putting an imaginary number up in that exponent. And you’d be right to question
that! What we write as e^t is a bit of a notational disaster, giving the number e and the idea
of repeated multiplication much more of an emphasis than they deserve. But my time is
up, so I’ll spare you my rant until the next video.

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100 thoughts on “Understanding e to the i pi in 3.14 minutes | DE5”

  1. 3Blue1Brown says:

    Complex exponents are very important for differential equations, so I wanted to be sure to have a quick reference for anyone uncomfortable with the idea. Plus, as an added benefit, this gives an exercise in what it feels like to reason about a differential equation using a phase space, even if none of those words are technically used.

    As some of you may know, Euler's formula is already covered on this channel, but from a very different perspective whose main motive was to give an excuse to introduce group theory. Hope you enjoy both!

  2. Lp Xiv says:

    omg I hope you can elaborate more on the convex optimization !!~~ It would be really helpful!

  3. Manuel M. says:

    If it's worth it, would you make a video about affine transformation? (I am interested in 3D computer graphics.)

  4. CO8izm says:

    A nice visualization of some combinatorial optimization problems would be really cool. One fascinating and fun topic that is not covered on youtube at all.

  5. Andreas Maier says:

    e^(i*x) = (cos x -sin x, sin x cos x) = . It is just a shortcut to write a the 2D Rotation matrix. So there you have it. The mystery explained in 3.14 seconds. 😉

    Just to elaborate: e^(i*Pi) = (-1 0, 0 -1), so it rotates a vector (1, 0) to (-1, 0).

  6. Kumar The Cowboy says:

    eye pie reminds me of Elon Musk

  7. Garrett says:

    Can someone explain why he says “velocity, the derivative of position, is always equal to that position” and then places the velocity ahead of the position by the length of the position at 0:30?

  8. Manideep Karthik says:

    Math is sexy.

  9. Học như Thủ khoa says:

    Why not in -1 second

  10. Irfan Darmawan says:

    … i don't understand at all, please explain it… iam a bit dumb when its come to math

  11. tom Jansen says:

    Im too stupid for this

  12. Jason Cheng says:

    You just taught me vector fields better than an entire semester of my second year ODE course.

  13. Σταμάτης Καλλιόστρας says:

    What is that τ in 3:13 ? Is this true τ = 2π ?

  14. Patricia McGeorge says:

    First video to ever get me to understand what the hell this means/why anyone cares. Thank you so much, you are the very reason I like maths

  15. Variety of Everything says:

    But there are 60 seconds in a minute, not 100, so it's not pi minutes, even according to the description :/

  16. Ian song says:

    I feel like I just got scammed.

  17. swetha narayanan says:

    The further away from 0 you are, the faster you move! Brilliant!

  18. AndreOnline says:

    make a series on python programming!

  19. Venkat Babu says:

    9 12 15 20 25 gives rational outcome of e.

  20. Subhajyoti Sarkar says:

    What software do u use to make such good animation

  21. Summer Johnson says:

    Holy crap, I've always wondered why this was, and no one could explain it. Thanks.

  22. Qiwei Shi says:

    I think it's beautiful.

  23. Rudy Pages Jr says:

    Bruh how the hell do i cook a digiorno pizza

  24. Sasmit Vaidya says:


  25. Danilo Arcidiacono says:

    I think this way of teaching "visually" should be introduced in every school in the world… it's amazing how you manage to find the right animation for whatever concept, bravo!

  26. Atirix says:

    So, is this the most beautiful video on the internet?

  27. sherbaj thind says:


  28. Anushasan poudel says:

    tell me if you use some kind of drugs to understand anything so easily… Damn !! everything now makes so much sense .

  29. luther schultz says:

    my mind just exploded, omg

  30. The PineApple says:

    This took my math lecturer about 30 minutes to explain, but you've done it in just 4 minutes!

  31. René Sánchez says:

    This was just a beautiful explanation of a beautiful concept. thanks and keep on making your great vids!

  32. Ma7moud Ali says:

    Can yiu translate your videos to arabic please 😢💕

  33. Taylor Olson says:

    3b1b: Whats that miles down there below us? Looks like a head…

  34. MJBx says:

    wonderful explanation with wonderful animation

  35. Kaung Gyi says:

    It's ridiculously amazing how mathematicians discovered how they couldn't solve certain problems by moving along one dimension with positive and negative numbers and came up with adding another dimension to the system.. This is a truly awesome visualization! Grant, what would the notation be like if we want three dimensions (moving into and out of the plane)?

  36. Cyrus Post says:

    This has never made sense to me until now and I laughed out loud into my hands when I got it

  37. Avik Das says:

    Expectation: Determined to fully understand a 3b1b video
    Reality: Facepalm

  38. LGEnt says:

    Mind blowing

  39. Ornithocowian King says:

    come back grant

  40. lixing shen says:



    there is no explanation : WHY e , not pi or something else ??????

  42. Orocimaro Say says:

    Amazing such a good explication

  43. shubham says:

    Please do Laplace transform

  44. Will R says:

    Cool, I've never understood anything less.

  45. Secnyt Secnyt says:

    This video has such a good explanation!!! Thank you!

  46. jesterthecosmonaut says:

    3B1B can you please be my math teacher for my school career?

  47. Na Cl says:

    Why is it a rotation when you put a real number to the power of imaginary number?

  48. Kiên P.S. says:

    Sometimes it's really hard for me to believe that e^(pi*i)=-1
    Since e=2.718281828…, pi =3.1415925… and i=sqrt(-1), it's really crazy to see pi*i is equal to some kind of special number which when e raise to, it's equal to -1.
    Your video really gives me a lot of sense for the euler's formula.
    While school just shows me of formulas that you have to remember, ask students to solve a problem; your videos gives me ways to approach solutions to the problems, the beauty of math, the ways to look at mathematics,…
    And honestly, I love math a lot since I was grade 6, but then I started to be bored as school just teach me formulas, makes me memorize those formulas and , do homework, but not even showing me at least a interesting fact about Math. That gives me a feeling that Math isn't interesting at all. Without knowing your channel, I think I would stop loving math.
    Thanks a lot, 3Blue1Brown, you are the best mathematics YouTube channel. I really appreciate all your hard work to make these videos.
    I hope that in your future videos, you'll show us much more ways to approach the solutions of any kind of problems, more different beautiful look to math, and more beautiful things in math.
    Also hope you 'll discover more mysterious things in math that no one founded yet :))
    I'll definitely try my best to be participate in IMO.
    Hope you'll see this comment too.
    I still feel crazy to believe the Euler's Formula :)). Please do more videos about it :))

  49. Alive Afterall says:

    e^i*pi = -1
    e^2*i*pi = 1
    ln(e^2*i*pi) = ln(1)
    2*i*pi = 0
    i=0 or pi=0? And may be 2=0? :))

  50. Christopher Weiss says:

    How does one decompose e^(i*pi) into a real component plus an imaginary component in the form a + ib?

  51. mora tony says:

    this is so beautiful

  52. Ibrahim Tuama says:

    E.T. remake comfirmed

  53. Lele Tan at says:

    i finally understand your videos after learning calculus in kumon

  54. R JS says:

    Soooo intuitive

  55. Dan says:

    Mathologer also made a great video on this.

  56. Convantol says:

    My head exploded after 18 seconds :p

  57. Roberto Carlos Rodal Salgado says:

    Hello! 3blue1brown, I am a Mexican mathematician and I am completely in love with his videos, my studies make me understand a lot of mathematical theory, however, this practical and intuitive perspective is what we all need, because thanks to it we can make interesting Mathematics to more people.

    I just discovered you watching a video related to the Fourier Transform and I fell completely in love. In truth this is the math disclosure that is needed.

    I have the idea of ​​creating a YouTube channel where I explain in a semi-formal way the theoretical mathematics, although I would love to complement these explanations by sharing your videos when they are related.

    I really appreciate your work and have won a fan of this channel, I am not someone who comments a lot but I will be watching all your videos to complement my ideas.

    You are amazing. Thank you for developing and sharing these videos.

    You have become a reference for the typical question of And what are these math topics for?

    Excellent day!

  58. syd sings says:

    I’ve watched so many videos explaining this and I’ve never understood but this one is so clear and I understand it so well for my little background in math as it is.

  59. David Hunt says:

    Readers. If you think this is cool try the math department at MIT courses. Then you will have something you can use instead of pretty graphics and technogibber. What he says is true but not useful. For example: the sun rose this morning

  60. _ NO says:

    i respect yooooou. thank you for information.

  61. Frederick Hofmann says:

    I want your babies

  62. Shubham Choure says:

    Plzz sir make a video about transandental numbers in an intuitive way

  63. matqal mazareeb JR says:

    Wow I Understood nothing. But imma take it that he is an A+ guy

  64. Vikash Jangir says:

    Knowlegeble channel, thanks👏👏👍👍👍👌👌

  65. nollix says:

    Genius, genius, genius brilliant explanation. I am moved by your ability to convey intuition so perfectly. This is the greatest

  66. Sakarias Hansen Andersen says:


    Didn’t understand the least bit.

  67. Holc Tomaž says:


  68. Alex Cerqueira says:

    This Chanel deserves all respect as possible

  69. Bambsi says:

    but the video is 4:08 minutes long..

  70. Bennett Austin says:

    Crazy how this equation can be represented in this way and in a Taylor series

  71. Dylan de Jonge says:

    In this example you start from the point e^0 = 1. This is true but the problem is that a^0 = 1 if a is not equal to zero. With this knowledge every number to the power of (pi multiplied By i) should be equal to -1. The real thing that is going on is that e^xi = cos(x) + isin(x). Since cos(pi) = -1 and isin(pi) = 0 you get the result of e^xi = cos(x) + isin(x) with x = pi. You get e^pi multiplied By i = cos(pi) + isin(pi) = -1 + 0 = -1
    So e^pi multiplied By i = -1

  72. harry pooper says:

    i may be dumb here but isnt d/dt of velocity accleration not position? i thought the integral of velocith was position

  73. Fernando Merancia says:

    soy mexicano y me encanta esta página, explica tan bien

  74. jan nedorostek says:

    Nice video, but why you have to use e? I didnt catch it

  75. No Hackers says:

    I got lost at the very end

    Why if you travel pi times you land at -1?

  76. icalbo72 says:

    I'm so lost hahaha

  77. Taky C says:

    Bruh what is e

  78. Rodrigo Antonio Jimenez says:

    That sublime vector animation, OMG I got an erection….

  79. ThatCuteDinosaur says:

    See, this is why I fucking love math

  80. やつむごい says:

    oh my god what

  81. Mitch says:

    That was so beautiful.

  82. Runstar Homer says:

    I am constantly amazed at your ability to so elegantly show the beauty in mathematics. This was simply incredible.

  83. lapidations says:

    Frick this is awesome

  84. Jackson Wald says:

    I must’ve seen like 20 videos explaining e to the pi i and this is the first one where I kind of got it.

  85. Jackson Wald says:

    I must’ve seen like 20 videos explaining e to the pi i and this is the first one where I kind of got it.

  86. Andrew says:

    Most incredible.

  87. Vanshaj Rai says:

    You lost me at 0:00

  88. pratik prajapati says:

    Which software you are using for that ?

  89. Jerry Gundecker says:

    Wouldn't this get us to other planets quicker?

  90. Reuben Adams says:

    Such an intuitive explanation of a formula that has always seemed so abstract to me. I will now always have this concrete image in my head when I think of it. Thank you 🙂

  91. Unni krishnan says:

    If you want to show off in front of someone just play any of 3Blue1Brown videos. They'll suddenly think you're genius.

  92. No Hackers says:

    our velocity is dy/dx of e. At e^t our velocity is e^t

    at e^2t our velocity will be twice our posiotion

    at e^-0.5t our velocity is negative and half of our position

    at e^0.5t our velocity is half of our position

    but i tends to rotate things by 90°

    taking "it" on it's own t=1

    i(1) = i

    e^it = e^-0.5t (in terms of rules)

    e^it ,t =0

    e^i(0) = 1 (postion)

    dy/dx (velocity)

    i(e^it) = i(1)= i

    now if we travel half circumference which is pi

    (r = e^i(0) = 1, 2pi(1) = 2pi, 2pi = circumference, pi = circum/2)

    if we travel a quarter of circle we get to i
    if we travel another quarter we get to -1

    e^ipi = -1

  93. Dadda Purple says:

    Wait, but i*e^it=/=e^it?

  94. KLASA 2GE says:

    (SI (Q) PA)

  95. geraldillo says:

    You are a genius

  96. Pedro Damian Sanchez Jr. says:

    2π = 6.28; try messing with 38 to get 6.66. 🦻

  97. Archimedes says:

    I don't always insist on the pi, but I want to ask one reason why the angle is divisible by 360 °

  98. Kellem Negasi says:

    Reading the comments is even more helpful than the video itself sometimes. I was lost in the multiplying by i part but thanks to the comment section I think I got it now.

  99. Gifford Metz says:


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