Book Store Online

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.

copyright 2019 Blog WordPress Theme By Blog WordPress Theme

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!

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

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

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

Can yiu translate your videos to arabic please 😢💕

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

wonderful explanation with wonderful animation

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)?

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

Expectation: Determined to fully understand a 3b1b video

Reality: Facepalm

Mind blowing

come back grant

VERY GOOD

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

Amazing such a good explication

Please do Laplace transform

Cool, I've never understood anything less.

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

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

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

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 :))

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? :))

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

this is so beautiful

E.T. remake comfirmed

i finally understand your videos after learning calculus in kumon

Soooo intuitive

Mathologer also made a great video on this.

My head exploded after 18 seconds :p

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!

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.

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

i respect yooooou. thank you for information.

I want your babies

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

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

Knowlegeble channel, thanks👏👏👍👍👍👌👌

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

Clickbait

Didn’t understand the least bit.

THIS IS GENIUS

This Chanel deserves all respect as possible

but the video is 4:08 minutes long..

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

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

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

soy mexicano y me encanta esta página, explica tan bien

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

I got lost at the very end

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

I'm so lost hahaha

Bruh what is e

:O

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

See, this is why I fucking love math

oh my god what

That was so beautiful.

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

Frick this is awesome

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.

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.

Most incredible.

You lost me at 0:00

Which software you are using for that ?

Wouldn't this get us to other planets quicker?

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 🙂

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

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

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

(SI (Q) PA)

You are a genius

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

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

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.

Wouhhhhhh