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.

Comments 100

  • 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!

  • This is how many people are amazed by the explanation


  • This without a doubt is the best explanation I've seen of this yet.

  • As a chap who dropped Maths as soon as possible, I thought I’d watch to see if even I could understand it in about 3 minutes.
    I am actually pleasantly surprised. Seeming I never did differentiation I don’t understand the significance of what I’ve learnt, but it did make me smirk that i gets its own entire spatial dimension.

  • Why is it adding Position and Velocity?

  • Can u make a video for your thoughts about Ramanujam's infinite serieses. I think everybody wants to enjoy your explanation.
    Thank you.

  • I think i might be having language barrier issues here, but why does it make sense to add the "velocity" vector to the "position" vector

  • This explanation is amazing; so clear and concise. Keep it up!

  • Sir please can you tell me which book do you refer for such basic intuitive maths🙏🙏🙏

  • Look to my nickname there are two π

  • I was comfortable with this identiy before, but I had a great “AHA!” moment. This is probably the most concise and surprising way to look at it I’ve ever seen. Thanks a bunch

  • That is a very good piloting situation

  • I really liked this video, but I disagree with your conclusion at the end. I get where you're coming from, and I agree that the exponential function isn't best thought of as multiply e by itself (though by extension that means the constant e is actually meaningless, it's only really the exponential function that matters and it's just coincidence that it equals e^x), but I've actually came up with away to extend iteration to the complex numbers that is consistent with complex exponentiation as well as the intuitive idea of iteration. Yeah, it's no longer based on recursion, but neither is real iteration (or really any iteration beyond the natural numbers), so I see no problem with extending in such a manner. Basically, it's done by equating iteration with movement along a path rather than as discrete applications of functions, with magnitude determining speed and argument determining direction. It's basically the same vector space shown here, just conceptualized differently.

  • Awesome videos as always! Thank you for your amazing hard work👍

    But I'd like to ask, why did you call your channel 3Blue1Brown what was the purpose?

    And how do you do these perfect animations? What app do you use?

    Thank you, @3Blue1Brown

  • Wow damn it actually makes sense now.

  • By showing animation graph, this channel made me to misunderstand that I am following the principle. But when it finished, I found myself who understand nothing.

  • Wow. Just wow.

  • I did not understand why the yellow line (velocity ) was moving faster then the blow line (position ) even though u said the dervative is just equal to to the position at that time,.

    Can u explain

  • You left out a detail that I think might be slightly unintuitive for some people, So allow me help to help clear things up a bit, it take 2Pi to traverse the circle because the circle has a radius of 1 and the circumference is 2*(Pi)*Radius, that said this is a great video and I learned a lot from it, Thank you

  • the only thing about this video that feels immoral to me is to say that velocity equals position.

  • Hey grant
    Big fan
    Which software do you use to animate this?
    It will be helpful for me to understanding higher math.

  • Please japanese sub😭💃💃💃

  • Why the name 3blue1brown

  • so, that's mean e^pi/2=i??

  • Did anyone else cringe when he said Tau instead of 2pi?

  • If possible, can you please make a series on statistics and probability

  • Grant, love your series. Just a little advice: if you want to distinguish simplified Chinese and traditional Chinese in your video description, I think the 2 types of Chinese are preferred to be described just like ‘simplified and traditional’ rather than China and Taiwan because it’s neutral and avoids political arguments. But if you must mention the regions, it’s preferred as mainland China and Taiwan. Note recent videos don’t involve traditional Chinese subtitles, but still, I really think you should drop “China” in the brackets because it may send wrong message.

  • More on the topic of laplace transforms please.

  • oooooOOOOOOhhhhhhhhhhhh so that that's why "i" is used when transforming sinusoidal signals from the time to phase domain.

  • i like it

  • This is a god tier math video lol, S++ tier

  • what are those..?

  • Amazing video!
    By the way, I bet that youtube doesn't like it for being too short.


  • Hey Grant do code or animate these videos

  • I can demonstrate more easily just search Eulers Formula on google and input Pi=180degrees (i dont have symbol of pi sorry) and you will understand easy and fast

  • Eit? Watching this makes me realize im an id-e-it.. ehh? Ehh?

  • Hey guys upvote if u want an angry looking plush Pi

  • Next video on millennium question Yang Mill Field and Mass Gap…

  • Please upload visual theory and applications of laurant and machlaren series.

  • The only property is phone
    ET phone home

  • Overall thank you for all your videos, I achieved to understand those things by writing my own program in '90, Is because your work is tailored to perfection that I'm writing the following comment: At minute 1:48 the animation about exp(-0.5 t) is for a super slow down animation (t changes constantly from 0.00 to 4.00 in 8 seconds AND the factor 0.5 slows the "exponential" of another further half). You would have the same vector animation for exp(-0.25 t) where t grow in seconds unit coherent to the video. However is more didactic to show exp(-t) which is more representative for the exponential function of a negative exponent, which yes slows down its approach to zero but also because it went toward it a lot (however there is nothing wrong in your video)

  • Make a video on significnace on numbers and derive whole maths from that

  • As math lovers, let's create a group for discussion which I already did at discord. Join mates


  • May i ask how you create these great accurate animations? (Which Software?)

  • Yeah! Maths! It would be really cool if you could demonstrate some equations of fields in physics

  • When you explain something it makes sense

  • SIR, it is a request to make a video about laplace trsnsformation which will be much helpful for us all.

  • Hey Grant! I had a request to make from you.
    I want you to make a video intuitively explaining what exactly are waves, different types of it and how to intuitively visualise all of them.
    Waves are very basic to most of physics and getting a very clear idea about them may help millions of students get better at physics and in general science.

  • You're almost at 2^21 subs! Congrats!

  • e^-iwt used to make some strange signals

  • Y’all ever watch something that you’re too dumb for, but watch any way to feel smarter despite not understanding any of it? Like asking a professional about something and getting an answer you don’t understand, so you just say “oh ok, that makes sense now. Thanks”

  • What happens at t=/=0?

    For example t=1 -> e^(i*1)

    The position now can´t be drawn on to the regular axis.

  • Make videos on complex analysis

  • My brain crashed when you began explaining the "i"

  • What about George Green identity-function?

  • Well, I really appriciate the huge work You do for Maths and for all of the people interested in it and also help people to even be interested in it.. Such amazing explanations, animations and beautiful thoughts.
    But I would really love to see You making similar Physics videos, from the very bases of Physics to some complex and huge physics thoughs or even unsolved mysteries..
    Because I haven't seen any better educational channel with such a good explanations / animations, which helps to improve persons view on the world of maths/physics. It'd be nice to see this even in physics problems / theorems. 🙂

  • C/D = 3.17157.

  • Sir, please make a video on Residue Theorem of complex analysis.

  • Another Sunday comes and goes without the next video.

  • Hey 3b1b thanks for your content! You're videos never fail to leave me satisfied! I was looking for some explanations (intuitive explanations) on topics in Numerical Analysis, anyone know where I can find some of that kinda stuff?

  • I love and respect this channel

  • Very nicely explained and I always used to wonder why is this beautiful equation always left unexplained.
    Thanks for explaining using animations.

  • Excellent! Could you make a visualization on the recent proof to the Sensitivity Theorem by Hao Huang?

  • I can't even put in words how brilliant that explanation was!

  • Thank you alot for your work! I really enjoy spending my time on your channel and discover "real" math and learn why the things we learn are as they are.
    It would be cool if you do another/more videos on music theory. I think alot of musicians who aren't watching you would enjoy it too.
    Have a nice day!


  • Where is the video that explains what e^(matrix) means?

  • Amazing video. I've never had it explained this way to me. I've always just understood E^jX is equivalent to the unit circle and accepted it as so.

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

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

  • 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.

  • e^(i*x) = (cos x -sin x, sin x cos x) = https://en.wikipedia.org/wiki/Rotation_matrix . 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).

  • eye pie reminds me of Elon Musk

  • 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?

  • Math is sexy.

  • Why not in -1 second

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

  • Im too stupid for this

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

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

  • 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

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

  • I feel like I just got scammed.

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

  • make a series on python programming!

  • 9 12 15 20 25 gives rational outcome of e.

  • What software do u use to make such good animation

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

  • I think it's beautiful.

  • Bruh how the hell do i cook a digiorno pizza

  • F

  • 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!

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

  • ❤️

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

  • my mind just exploded, omg

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