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.

Understanding e to the i pi in 3.14 minutes | DE5
Tagged on:                                             

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

  • July 7, 2019 at 2:26 pm
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    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!

    Reply
  • July 16, 2019 at 11:04 am
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    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

    Reply
  • July 16, 2019 at 9:05 pm
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    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

    Reply
  • July 17, 2019 at 3:42 am
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    the only thing about this video that feels immoral to me is to say that velocity equals position.

    Reply
  • July 17, 2019 at 8:23 am
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    Hey grant
    Big fan
    Which software do you use to animate this?
    It will be helpful for me to understanding higher math.

    Reply
  • July 17, 2019 at 3:07 pm
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    Please japanese sub😭💃💃💃

    Reply
  • July 17, 2019 at 3:56 pm
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    Why the name 3blue1brown

    Reply
  • July 17, 2019 at 6:45 pm
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    so, that's mean e^pi/2=i??

    Reply
  • July 18, 2019 at 4:41 am
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    Did anyone else cringe when he said Tau instead of 2pi?

    Reply
  • July 18, 2019 at 4:53 am
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    If possible, can you please make a series on statistics and probability

    Reply
  • July 18, 2019 at 9:09 am
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    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.

    Reply
  • July 18, 2019 at 10:18 am
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    More on the topic of laplace transforms please.

    Reply
  • July 18, 2019 at 10:27 am
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    oooooOOOOOOhhhhhhhhhhhh so that that's why "i" is used when transforming sinusoidal signals from the time to phase domain.

    Reply
  • July 18, 2019 at 10:53 am
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    i like it

    Reply
  • July 18, 2019 at 6:36 pm
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    This is a god tier math video lol, S++ tier

    Reply
  • July 19, 2019 at 9:21 am
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    what are those..?

    Reply
  • July 19, 2019 at 3:52 pm
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    Amazing video!
    By the way, I bet that youtube doesn't like it for being too short.

    Reply
  • July 19, 2019 at 4:38 pm
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    SMART MAN WELL DONE EXPECTING MORE FROM YOU

    Reply
  • July 20, 2019 at 6:27 am
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    Hey Grant do code or animate these videos

    Reply
  • July 20, 2019 at 3:22 pm
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    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

    Reply
  • July 21, 2019 at 2:41 am
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    Eit? Watching this makes me realize im an id-e-it.. ehh? Ehh?

    Reply
  • July 21, 2019 at 12:54 pm
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    Hey guys upvote if u want an angry looking plush Pi

    Reply
  • July 22, 2019 at 9:54 am
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    Next video on millennium question Yang Mill Field and Mass Gap…

    Reply
  • July 22, 2019 at 4:03 pm
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    Please upload visual theory and applications of laurant and machlaren series.

    Reply
  • July 22, 2019 at 11:30 pm
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    The only property is phone
    ET phone home

    Reply
  • July 23, 2019 at 2:30 am
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    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)

    Reply
  • July 23, 2019 at 12:50 pm
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    Make a video on significnace on numbers and derive whole maths from that

    Reply
  • July 23, 2019 at 2:58 pm
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    As math lovers, let's create a group for discussion which I already did at discord. Join mates

    https://discord.gg/Wngy7g

    Reply
  • July 24, 2019 at 4:58 pm
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    May i ask how you create these great accurate animations? (Which Software?)

    Reply
  • July 24, 2019 at 6:20 pm
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    Yeah! Maths! It would be really cool if you could demonstrate some equations of fields in physics

    Reply
  • July 24, 2019 at 7:13 pm
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    When you explain something it makes sense

    Reply
  • July 24, 2019 at 10:48 pm
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    SIR, it is a request to make a video about laplace trsnsformation which will be much helpful for us all.

    Reply
  • July 26, 2019 at 5:33 pm
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    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.

    Reply
  • July 26, 2019 at 8:55 pm
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    You're almost at 2^21 subs! Congrats!

    Reply
  • July 27, 2019 at 12:25 am
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    e^-iwt used to make some strange signals
    https://www.youtube.com/watch?v=QaOUqX4GzXA

    Reply
  • July 27, 2019 at 5:30 am
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    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”

    Reply
  • July 27, 2019 at 11:14 am
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    What happens at t=/=0?

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

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

    Reply
  • July 28, 2019 at 8:46 pm
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    Make videos on complex analysis

    Reply
  • July 28, 2019 at 10:11 pm
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    My brain crashed when you began explaining the "i"

    Reply
  • July 29, 2019 at 3:56 pm
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    What about George Green identity-function?

    Reply
  • July 29, 2019 at 8:59 pm
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    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. 🙂

    Reply
  • July 29, 2019 at 10:34 pm
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    C/D = 3.17157.

    Reply
  • July 30, 2019 at 3:12 am
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    Sir, please make a video on Residue Theorem of complex analysis.

    Reply
  • July 30, 2019 at 3:38 am
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    Another Sunday comes and goes without the next video.

    Reply
  • July 30, 2019 at 4:20 pm
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    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?

    Reply
  • July 30, 2019 at 4:49 pm
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    I love and respect this channel

    Reply
  • July 31, 2019 at 1:39 am
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    Very nicely explained and I always used to wonder why is this beautiful equation always left unexplained.
    Thanks for explaining using animations.

    Reply
  • July 31, 2019 at 9:46 pm
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    Excellent! Could you make a visualization on the recent proof to the Sensitivity Theorem by Hao Huang?

    Reply
  • August 1, 2019 at 2:24 pm
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    I can't even put in words how brilliant that explanation was!

    Reply
  • August 1, 2019 at 2:52 pm
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    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!

    Reply
  • August 2, 2019 at 6:45 am
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    VERY GOOD

    Reply
  • August 3, 2019 at 12:53 am
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    Where is the video that explains what e^(matrix) means?

    Reply
  • August 3, 2019 at 4:29 am
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    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.

    Reply
  • August 3, 2019 at 8:29 am
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    omg I hope you can elaborate more on the convex optimization !!~~ It would be really helpful!

    Reply
  • August 3, 2019 at 11:52 pm
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    If it's worth it, would you make a video about affine transformation? (I am interested in 3D computer graphics.)

    Reply
  • August 4, 2019 at 8:19 am
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    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.

    Reply
  • August 4, 2019 at 11:22 am
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    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).

    Reply
  • August 4, 2019 at 9:48 pm
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    eye pie reminds me of Elon Musk

    Reply
  • August 4, 2019 at 11:52 pm
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    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?

    Reply
  • August 5, 2019 at 5:17 am
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    Math is sexy.

    Reply
  • August 5, 2019 at 1:48 pm
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    Why not in -1 second

    Reply
  • August 6, 2019 at 5:42 am
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    … i don't understand at all, please explain it… iam a bit dumb when its come to math

    Reply
  • August 6, 2019 at 3:23 pm
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    Im too stupid for this

    Reply
  • August 8, 2019 at 5:52 am
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    You just taught me vector fields better than an entire semester of my second year ODE course.

    Reply
  • August 8, 2019 at 3:20 pm
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    What is that τ in 3:13 ? Is this true τ = 2π ?

    Reply
  • August 9, 2019 at 7:13 am
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    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

    Reply
  • August 9, 2019 at 11:24 pm
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    But there are 60 seconds in a minute, not 100, so it's not pi minutes, even according to the description :/

    Reply
  • August 10, 2019 at 12:04 am
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    I feel like I just got scammed.

    Reply
  • August 10, 2019 at 8:07 am
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    The further away from 0 you are, the faster you move! Brilliant!

    Reply
  • August 10, 2019 at 2:17 pm
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    make a series on python programming!

    Reply
  • August 11, 2019 at 1:57 pm
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    9 12 15 20 25 gives rational outcome of e.

    Reply
  • August 11, 2019 at 8:27 pm
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    What software do u use to make such good animation

    Reply
  • August 12, 2019 at 11:23 pm
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    Holy crap, I've always wondered why this was, and no one could explain it. Thanks.

    Reply
  • August 14, 2019 at 1:25 pm
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    I think it's beautiful.

    Reply
  • August 14, 2019 at 6:48 pm
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    Bruh how the hell do i cook a digiorno pizza

    Reply
  • August 16, 2019 at 6:01 am
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    F

    Reply
  • August 17, 2019 at 12:58 pm
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    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!

    Reply
  • August 20, 2019 at 7:27 am
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    So, is this the most beautiful video on the internet?

    Reply
  • August 20, 2019 at 1:56 pm
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    ❤️

    Reply
  • August 21, 2019 at 7:25 am
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    tell me if you use some kind of drugs to understand anything so easily… Damn !! everything now makes so much sense .

    Reply
  • August 21, 2019 at 7:36 am
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    my mind just exploded, omg

    Reply
  • August 23, 2019 at 10:50 am
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    This took my math lecturer about 30 minutes to explain, but you've done it in just 4 minutes!

    Reply
  • August 24, 2019 at 4:51 am
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    This was just a beautiful explanation of a beautiful concept. thanks and keep on making your great vids!

    Reply
  • August 24, 2019 at 10:37 am
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    Can yiu translate your videos to arabic please 😢💕

    Reply
  • August 29, 2019 at 7:18 pm
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    3b1b: Whats that miles down there below us? Looks like a head…

    Reply
  • September 3, 2019 at 2:59 pm
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    wonderful explanation with wonderful animation

    Reply
  • September 4, 2019 at 4:51 am
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    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)?

    Reply
  • September 5, 2019 at 3:19 pm
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    This has never made sense to me until now and I laughed out loud into my hands when I got it

    Reply
  • September 9, 2019 at 8:58 am
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    Expectation: Determined to fully understand a 3b1b video
    Reality: Facepalm

    Reply
  • September 9, 2019 at 7:16 pm
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    Mind blowing

    Reply
  • September 9, 2019 at 9:09 pm
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    come back grant

    Reply
  • September 12, 2019 at 1:13 pm
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    VERY GOOD

    Reply
  • September 13, 2019 at 5:29 am
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    there is no explanation : WHY e , not pi or something else ??????

    Reply
  • September 13, 2019 at 9:16 am
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    Amazing such a good explication

    Reply
  • September 13, 2019 at 11:18 am
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    Please do Laplace transform

    Reply
  • September 16, 2019 at 11:07 pm
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    Cool, I've never understood anything less.

    Reply
  • September 17, 2019 at 11:58 am
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    This video has such a good explanation!!! Thank you!

    Reply

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