> analyse > analyse-complexe > comprendre-visuellement-le-théorème-fondamental-de-l-analyse-algo-zéros-fonctions-complexes-3b1b

Solving 2D equations using color, a story of winding numbers and composition

3Blue1Brown - 2018-03-24

An algorithm for numerically solving certain 2d equations.
Brought to you by...you!  https://patreon.com/3blue1brown

Special thanks to these supporters: http://3b1b.co/winding-thanks

Even though we described how winding numbers can be used to solve 2d equations at a high level, it's worth pointing out that there are a few details missing for if you wanted to actually implement this.  For example, in order to determine how often to sample points, you'd want to have some bounds on the rate at which the direction of the output changes.  We will perhaps discuss this more in a follow-on video!

Music by Vincent Rubinetti: 


3blue1brown is a channel about animating math, in all senses of the word animate.  And you know the drill with YouTube, if you want to stay posted on new videos, subscribe, and click the bell to receive notifications (if you're into that).

If you are new to this channel and want to see more, a good place to start is this playlist: http://3b1b.co/recommended

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Website: https://www.3blue1brown.com
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Electric2Shock - 2018-03-24

To be continued.... Now

And was my loudest YEAAHHH BOI ever.

Flerovium - 2018-03-25

THIS ^^ :D

Anton Wilzewski - 2018-03-31

I was really annoyed, because i hate to be continued.
Like this it was ok though.

Nathan Chasse - 2018-04-18

SAME I was like "how could you do this to meeeeee how can I sleep now"

Raahil Shammin - 2018-09-30

And here we see a wild Meme-Matician, someone who likes memes and cares about math. They are endangered species, as many are overtaken by one interest, but some still remain.

Mr. Henri - 2020-01-13

I stopped to look for the next video

Daniel Astillero - 2018-03-24

"Being good at math is not about being right the first time. It's about having the resilience to carefully look back and understand the mistake (and) understand how to fix them." ~3B1B

Flerovium - 2018-03-26

Yes but it's very worth mentioning it in the context of math in my opinion.

Julius Fučík - 2018-09-09

This is true for how to live LIFE even.

Colette Johnson - 2019-02-15

@Julius Fučík Studying Math and Physics has the added benefit of keeping most people open-minded and relatively humble--you can't fake a right answer, and no one is always right!

Forfunckle Studios - 2019-08-12

Yeah that's exactly what he said.

David Amato Mantegari - 2019-08-12

@Paskal S. This applies to pretty much any serious fields.

EmissaryOfSmeagol - 2018-03-25

11:24 plot-twist of the century
13:36 plot-twist of the millennia

Flynn Curtis - 2019-03-26

Millennia is plural. Unless you're talking about a collection of millennia, you should be using millennium.

NoriMori - 2019-07-04


Serenity Departed - 2019-08-10

I'm sorry sir, but I have seen a better math plot twist. Unfortunately it's in russian https://www.youtube.com/watch?v=DkFam4ELD38

Marshall Kimber - 2019-10-21

EmissaryOfSmeagol 25:01 plot twist of the eon

SCP 055 - 2019-12-31


Harkler - 2018-03-25


3B1B: “And this time, I promise, there are no bugs.”

ant chooses this exact moment to defiantly walk across my phone’s screen

Abdega - 2018-03-24

Talking about colors and Stoke’s theorem was a perfect setup for Green’s Theorem

Sharon Murphy - 2018-04-25

Man, I need to go back and review
vector calculus

Tiburca Rodrigues - 2018-09-16

@Zimty In Chapter 7 of Arnold's "Mathematical Methods Of Classical Mechanics" there's a nice description too. It recquires a minimum background. In the last paragraph of "Ordinary Differential Equations" (the Roger Cook translation) there's a nice explanations of winding numbers and the degree of a mapping in the context of Ode's

Raahil Shammin - 2018-09-30

Is it weird that I had to actually say,”Ba-Dum-Tss” out loud in order to realize that it was the drum noise?

James Bra - 2019-05-06

actually green's theorem is used to derive stoke's theorem. also greens theorem can be used in some pretty intense isoperimetric problem

BlueGiant69202 - 2019-08-08

@ZerGreenOne Why not just use the Fundamental Theorem of Geometric Calculus?

Abdega - 2018-03-24

TFW colorblind…

Vincent T - 2018-04-05

Abdega I

Glitch 3 - 2018-05-25

I am a mild deutan and the colors looked fine to me. :P

Rami - 2018-08-08

Abdega if you have an android phone, find out how to unloxk the developer settings, then you'll be able to use the color blind settings on your phone

If it helps at all I have no idea, but its worth a shot

I Don't Have A Name - 2019-10-19

@Damien Busatto-Gaston I really didn't want to write r/IAmVerySmart.

שחר א. - 2020-02-12

Objects in Motion which is incredibly bad for the winding numbers used in the second half of the video

Ethanol 314 - 2018-03-24

The real reason this is worth 20 minutes is because 3 Blue 1 Brown uploaded it.

The Zip Creator - 2018-04-15


The-Thong Nguyen - 2018-04-29

Internet Explorer!

General Relativity Friedmann Equation - 2019-06-19


Evi1M4chine - 2019-10-19

That is literally … circular reasoning! :D

hawran hrafnain - 2019-11-27

And what about the imaginary reason? ;

3Blue1Brown - 2018-03-24

Even though we described how winding numbers can be used to solve 2d equations at a high level, it's worth pointing out that there are a few details missing for if you wanted to actually implement this. For example, in order to determine how often to sample points, you'd want to have some bounds on the rate at which the direction of the output changes. We will perhaps discuss this more in a follow-on video.

Also, to answer what a few people have asked in the comments, it's certainly the case that you can have a zero that this algorithm doesn't find. For example, combining regions whose boundaries have winding numbers +1 and -1 will give a region whose boundary has winding number 0, and hence which wouldn't be searched. For that matter, this is very much true in the 1d case as well; you could have a region with many zeros where the function is positive at both ends. You thought you could get an algorithm guaranteed to find all zeros of any continuous function? We're only human!

WeinSim - 2019-01-12

Hey is there some Website or online interaction program where you can type in functions and this function is run?

TroubleClef _ - 2019-02-08

Why is teacher pi yellow?

Matouš Pikous - 2019-02-24

Seems to me that the algorithm doesn’t work. What if there is a white spot in the loop and when you reach out from the center of the plane, you are getting to zero. It seems to me that this scenario is plausible, if not, why?

Anon54387 - 2019-09-06

21:10 So what's the point if the gray areas where zeros aren't found may or may not have a solution?

Charles Kogan - 2019-11-25

What about a 3d function

Shreyansh Darshan - 2018-03-24

To be continued... now!
gets goosebumps

Thiago Machado - 2018-03-24

also when the theory was wrong/incomplete.. it's like.. math on-the-fly XP

Tobyrobot - 2018-03-24

Shreyansh Darshan
I almost had a heart attack when I thought it was a cliff hanger

Antonio Lewis - 2018-03-24

I was about to ragequit math at that cliffhanger

DynestiGTI - 2018-03-24

Stop playing with our emotions 3blue1brown!

big pp - 2018-03-24


Dan Dart - 2018-04-16

"To be continued"
Me: Whoa whoa whoa whoa whoa
Me: Phew

ROOOMBA - 2018-03-24

I actually cried out in pain when it said "to be continued"

MyLittleWalrus - 2018-03-24

When the "To be continued" came up at 13:32, I was sad that the video was over, but satisfied with what I had seen so far. Cheeky surprise!

Amazing video as always, Grant.

Dragon Curve Enthusiast - 2018-03-25

... and Sridhar :-)

Orion D. Hunter - 2019-03-17

"To be continued... NOW!"

Tomas Roque - 2018-03-24

I really love how you are able to explain such concepts so clearly that even a highschooler who speaks English as a second language like me can understand them. Is anyone else here in the same position as me?

Elliot Jenner - 2018-03-24

This is interesting. I do hope you do more "Essence Of" series, though. I often send students to those when they have issues with calc or linear algebra. It would be great to have more to use.

Pi Man - 2018-03-24

In his Patreon page interview introducing the two new team members, Sridhar and Ben, Grant throws in right at the very end that Sridhar will be picking up the baton on the Essence of Probability series, and Ben will tackle Essence of Set Theory (I think). Grant himself will do an entire follow-on Essence of Linear Algebra series to add to/enhance what topics covered in Essence of L.A. Part I. With now three people creating content, the flow should increase commensurately.

Robin Schmidt - 2018-03-25

this sounds great! i also would love to see an essence of multivariable calculus. should lend itself well to visualization. ...and complex calculus....and tensors! keep the good work going! <3 this is really the best math channel, i 've ever seen!

Georges Miranda Spyrides - 2018-03-25

Actually, Grant does multivariable calculus by Khan academy! https://www.youtube.com/watch?v=TrcCbdWwCBc&list=PLSQl0a2vh4HC5feHa6Rc5c0wbRTx56nF7

Joel Dowdell - 2018-03-24

Watching the end bit and seeing the top of the "infinite list of all topics," my reaction was, "MAKE VIDEOS ON EVERY ONE OF THESE!"

NatNiks - 2019-08-24

And the Gödel will start to complain that .... Never mind.

Amir Saboury - 2018-03-25

i just became a patreon of yours.
your videos just tickle every cell in my brain and i love it.
keep up the good work :)

Nathan Phipps O'Neill - 2018-03-24

That "to be continued" had me checking the length of the video on the edge of my seat. Never wanted to hear more about math in my life. Inspiring stuff and fabulous animations. Thank you

Anirudh Giri - 2018-03-24

My heart broke when I saw the "To be continued..", but then I was filled with immense joy when the ..now popped up. Grant, you sneaky troll!

Daniel Shapiro - 2018-03-24

23:56 He uses TAU in his programming! I really hope Matt Parker watches this! haha! Great video as always

Sheshan Patel - 2018-03-24

When are you uploading probability series?

Samarth Sai - 2018-03-24

That would help so much for understanding quantum mechanics.

romanemul1 - 2018-03-24

you have to be patreon to see them.

Samarth Sai - 2018-03-24

So are they available for patreons or does he accept requests only of patreons.

Pi Man - 2018-03-24

Patreons have had sneak previews of (I think) the four he's made for that series. In his Patreon video introducing the new team members, he mentioned at the end that Sridhar will now assume command of the rest of the Essence of Probability series. Progress should now pick up with three guys contributing content. This and the previous video have been their respective "debuts"

Sheshan Patel - 2018-03-25

Ahmed Usman yes I am.

GAMERS FOREVER - 2018-04-14

I had one question: If you have a rod of length "l". And you broke it into 3 parts. Then find the probability that they will make a triangle.
If possible plzz make a video on it I want to see how you explain it. I love your channel.😍😍😍😍😍🔉

17lvlham - 2019-10-13

Very interesting question.
As I see, core of the solution is to find number of possible points placement, which forms parts with sizes capable to form triangle (sum of two ≥ third), and divide it by number of all points in the original rod.
I'm not even nearly a mathematician (just EE), but I have very strong desire to try.
Bet it will easy for discrete case, then I'll try to convert it into continuous one.

Shivraj - 2018-03-24

I was like noo thats not right, it isn't always like that
few minutes later:
top 10 anime plot twists
edit : why don't you put an ad after the 'clicky stuff'

Anant Mishra - 2018-03-24

17:49 Gave me Goosebumps!

Shreyas Sarangi - 2018-03-26

When the music stopped after the initial algorithm failed: 3b1b.exe has stopped

Anurag Juyal - 2018-03-28

13:35 epic 😂😂

Hasiumcreeper - 2018-03-25

13:33 I wish more shows did this.

Lukas Pfeifenberger - 2018-04-03

As someone who fell in love with math a long time ago, I have to say that this is one of the most beautiful and enjoyable videos I have ever seen!

Antonio Lewis - 2018-03-24

The cliffhanger in the middle was going to make me verbally attack you. Good thing you didn't hype up the stakes and leave us hanging..

columbus8myhw - 2018-03-25

Why is 3B1B consistently high art

chase tuttle - 2018-03-24

"To be continued", awwww. "Now", oh yay!

rodrigo - 2018-03-24

This 24 minutes video felt like it was 5 minutes long or so

plovet - 2018-03-30

Agree. I did a double-take when the video was over. Time flies by fast when you are having fun, or watching an excellent math video! :)

Niko L - 2018-03-29

"to be continued..." WHAT?!
"... now" oh...
I both love and hate you

José Alvim - 2018-03-24

"To be continued... Now"

Parallel Apple - 2018-03-24

"Being wrong is a regular part of doing math"
Sneaking growth mindset into a video? Nice to see all that discussion on Ben, Ben and Blue has paid off.

Sophia Campana - 2018-03-25

I love these videos! You impress me every time.

Connor Armstrong - 2018-03-25

This is the best channel on YouTube. So we'll made and thoughtfully executed. You and your team inspire curiosity, thank you!

F & Yu - 2018-03-24

13:32 My heart litterally stoped for a few seconds...

Joseph Pentony - 2020-02-22

F & Yu stopped*

Akshat Kumar Agarwal - 2018-03-24

I really hope you get paid enough for the amazing work that you're doing. Thank you for this. You put in a lot of effort and hardwork and it is visible in your videos. I wish you loads of success in life.

Aniket Sharma - 2018-03-25

Loved the video! You are my favourite mathematics teacher ever. You bring a new perception of mathematics.

Mohammed Sharukh - 2018-03-24

13:38 you got me lol

Piffsnow - 2018-03-24

That... that was beautiful ! Thank you. Thank you for the being that clear in your explanations, for always choosing so interesting content, for the neat animation well appreciated, for the colors in this video and for the kind message at the end.
I'm a maths teacher and your channel blew me away quite a few times ! And I love that. :)
So thank you again !

Yunus KALKAN - 2018-03-25

13:33 To be continued...
-Ohhh maaaan!
Realizing there is 10 minutes more and the word "now." comes up = Absolute happiness.

Gurkirat Singh - 2018-03-27

You know, all of this reminds me of cauchy's residue theorem. The zeroes are like poles and computing the winding number is the same as performing a contour integration.
If the contour integral is an integer multiple of 2πi, then the area contains a pole (or multiple).

Though the similarity is really nice.
Can anyone please elaborate on how it is related?

Joseph Parkes - 2018-11-19

I love these videos, and I am terrific at math, but the vids make me realize how much of a NOOB I actually am in comparison to other, much more brilliant people.

Now, I wonder if the integral has to do with the sine of 2π. But, as a disclaimer, I know NOTHING about what you are doing with math.

MegaTuroc - 2018-03-24

You scared me when you said 'To be continued' I thought that was the video and got really scared

Adam Xue - 2018-03-24

I was scared when I saw that To be continued XD
Immediately checked on video timer, and was like, PHEW
Nice troll XD

Abhi Shah - 2018-03-24

Honestly, I would definitely watch a movie made by you..
And I'm sure everyone else will too..
Soo, can we get a movie? :3

Pi Man - 2018-03-24

He DID recently speak to a group at Pixar. (Jus' sayin' . . . )

Red Salmon - 2018-03-24

Thanks for reminding me how beautiful mathematics is

ProbablyNothing - 2018-03-25

There is so much beautiful topology and complex analysis in this video. I love it. I'm a grad student in mathematical/theoretical physics and I really appreciate what you're doing to educate people.

josh mcgee - 2018-03-25

hah the part when the music cut out was hilarious