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.

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 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.