(Early view) Understanding e to the i pi in 3.14 minutes

I just thought it was worth knowing that there is a way to place e^t and it's derivative on a diagram. I have a follow-up video discussing how the sum of cosh(t) and sinh(t) is actually e^t, as in it follows all the properties of an exponential function, with a base of e. I haven't uploaded it yet, and I'm not sure if I'll ever get around to it, but this might be a good resource for explaining e^t to new students.

Alright, I need to show this to you. You know how way back in your series on the Essence of Calculus, during your video on e^t: You asked and said you would love it if someone could point to both e^t and it's derivative in one graph. I found a way to do it using the solutions to two differential equations: Here is a video showing what I've found: https://www.youtube.com/watch?v=uEulSQiBquo. The video/audio quality isn't great as this is my first attempt at an actual math video, and it was meant for an audience of people who are less acquainted with the hyperbolic functions used to build e^t, but hopefully, this gives a good enough explanation.

In the next video, I want to talk about the series in a different way, not as a Taylor series, per se.

3blue1brown

Should you mumble the words "Taylor Series" around 3:43 to give just a hint about what that infinite series is?

Ron Jensen

Ayy! Very very cool. I've been working on an interesting intuition about number theory with Euler's formula at its center, which seems to be unique. Your vidyas were behind the genesis of this intuition, and also holding my hand on the journey going forward. So it gives me intense pleasure to see you putting out -- now that's what I call a pun! -- another vidya on the subject. Thank you!

I'm glad you liked it, and your spot on about e being a sort of fingerprint for derivatives, the same way pi is a fingerprint for circles.

3blue1brown

This is precisely what the next couple videos will be focussed on :)

3blue1brown

Or watch this: https://youtu.be/bcPTiiiYDs8

Tau?

Burt Humburg

Grant, Seen as you mentioned exponentiating matrices earlier, can I suggest a short series on the Essence of Exponentiation/Hyperoperations? I think this is something that often glossed over because its so fundamental, but it is surprising how many people don't have a good understanding of what exponentiation really means. I will suggest this on reddit, but it seemed appropriate to at least put a mention in here, given some of the comments.

Okay so think of it this way, you can write "i" as e^(I*pi/2), since when you expand that using rulers formula you get cos(pi/2) + i*sin(pi/2) and the cos(pi/2) is 0 and the sin(pi/2) is 1 leaving you with just i. When you write it like that, i*e^(t*i) (where e^t*i is some complex number) can also be written as e^(i*Pi/2) * e^(t*i), using exponent laws this can be rewritten as e^((pi/2 + t)*i), which is the same as adding pi/2 to t, and since when you have a complex number written in the form of a*e^(t*i) the coefficient of i in the exponent is the angle, and your adding pi/2 to that angle, it is the same as adding pi/2 to the angle. Q.E.D. The only issue with that is that I am already using Euler's formula which this video was deriving so let me write another proof that does not use Euler's formula and then come back to the comments.

The only thing I felt a little in the dark about was the WHY i rotates by 90º. I don't know if it was the animation of seeing the vector rotate instead of seeing the vector in the context or an argand diagram. It's not something I would change the video for, but maybe something in the description box, or have the little i label in the corner come up with a suggestion to one of the Essence of Linear Algebra videos, that shows what it means to multiply one basis vector with the other? I don't think it takes away from the explanation, but if you don't have a good understanding of vector/matrix multiplication, handedness, or complex numbers, it may seem like an arbitrary choice? Great work as always though!

Great video! When you see "e" showing up it usually means that there has to be a derivitive somewhere but I never really thought about it in the context of complex exponentiation. Since multiplying by i rotates 90 degrees and the tangent direction is the same as the derivative vector direction which is just the displacement from the origin multiplied by i, it makes sense that the tangent is always perpendicular to the displacement which means it must be a circle. And the scaler would change the rate that it rotates since it changes the magnitude but not the direction of the derivitive vector meaning the tangent is still 90 degrees and still traces a circle, just at a different rate. I wish they covered stuff like this in high school...

Yes, but you can also exponentiate matrices, and even weirder things like the derivative operator, so letting the notation stay tied to repeated multiplication seems almost misleading at some point.

3blue1brown

I think i is defined as just the positive root. -sqrt(-1) would be -i.

Magnasium

To be fair definition of exponent just need to be extended beyond "multiplying n times" is the gist of why you can do complex powers (or in fact irrational powers also require serious redefinition)

Timur Sultanov

Awesome but don't use ( i.e.) in the text or my head will explode 😁

Please don't introduce i = sqrt(-1) as you are missing half the solutions...

Martin Embeh