But what is the Riemann zeta function? Visualizing analytic continuation

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  • Опубликовано: 10 ноя 2024

Комментарии • 3,9 тыс.

  • @jmcsquared18
    @jmcsquared18 8 лет назад +11901

    As a mathematician, I was always deeply aggravated when other mathematicians on RUclips or elsewhere said thing such as, "the sum of all the integers converging to negative one over twelve" or "an infinite sum of ones is negative one-half" and these are just true things they found out, and are simply mind-blowing and counterintuitive. Yet their methods of telling it to the audience were clearly way out of context. Mathematics should not be mystical, such that it becomes ungraspable; it should be explained and appreciated for its rigor and intuitive creativity combined. Thank you for this content, I appreciate it because it is detailed, but not overly rigorous for the sake of rigor alone, so that it becomes inspirational. That is what education in science and math should be: inspiring others to want to learn, rather than shoving the information in their skulls. Education in science and mathematics is something this country lacks in, and content creators like you can help change that.

    • @3blue1brown
      @3blue1brown  8 лет назад +1791

      Wow, thanks so much. I definitely agree with the statement "Mathematics should not be mystical". It seems commonplace in outreach to use surprising facts to capture an audience that might not usually care about math, and insofar as this bring in more people who wouldn't otherwise be looking, that might be a net positive. But I do worry that tossing out only mysteries without arguments might have an overall negative effect on the public perception of mathematics.

    • @jacobkantor3886
      @jacobkantor3886 8 лет назад +153

      +3Blue1Brown Idk, whenever I finish one of your videos (or learning any super interesting new math topic) It feels totally mystical and unreal, even though I just saw or worked out the specific details on how and why whatever is happening is happening. It's kind of a "holy shit, this makes so much sense it can't be real".

    • @jmcsquared18
      @jmcsquared18 8 лет назад +152

      Jacob Kantor I will admit something: mathematics, indeed all of the good sciences, have a lesson in humility at every stop. Sometimes, when I learn of some crazy fact, like that there are as many rational numbers as there are integers, which on the surface can't be true, but is proven without a doubt mathematically, I have to understand the context in which the statements are said. It is oftentimes impossible to understand mathematics or physics out of context, and trying to might make it appear nonsensical.
      For instance, in the context of set theory, we say two sets have the same size if and only if you can find a way to pair each element of one set with exactly one of another. It isn't the same strictly as counting the elements, but it's the more general definition that we need because we deal with infinite sets; you can't even properly say that two infinite sets have the same "number" of elements because the number of elements each one has isn't even a finite number!
      You then you realize that, with infinities, weird stuff can happen like one set having the same number of elements as another one that it is properly contained in. So context is everything in mathematics, because we often deal with abstractions that we intuitively attempt to relate to, but oftentimes are nothing like what we are used to.

    • @MasterHigure
      @MasterHigure 8 лет назад +114

      I think he said it quite well in the "Who cares about topology" video, that children and laypeople might be told how to make a Möbius strip, and perhaps to cut it down the middle, but then they stop there without giving it any context. Why the Möbius strip is relevant to anyone, apart from its slightly funny one-sidedness is not usually discussed in these settings.
      Also, I'm very happy that this video is a bit critical towards the 1+2+3+4+... = -1/12 thing. I mean, the evidence as shown in this video clearly point towards a connection, but many other sources claim it's a direct equality. It is, in my opinion, the single bad thing Numberphile have ever done, for instance. As a person frequenting the math stackexchange, we regularly have people who come in there after stumbling across that one video and wonder what is going on, and whether they have misunderstood convergence completely.

    • @RoboBoddicker
      @RoboBoddicker 8 лет назад +48

      Speaking as a non-math person, 3Blue1Brown's description here was beautiful and elegant and made sense. But it's also 20 minutes long, and that doesn't even include an explanation of complex numbers or the complex plane. Plus, just creating the visualizations he uses requires a complete understanding of the material. So I think it's a bit unfair to call out Numberphile for being superficial - it's just a different format. Hell, I probably never would have even considered watching this video had I not learned the basic ideas from Numberphile.

  • @sorenlily2280
    @sorenlily2280 8 лет назад +6189

    "This video is long enough as it is"
    Dude you could make a video that was 2 hours long and I would watch every second.

    • @arongil
      @arongil 8 лет назад +72

      So true. :)

    • @zactron1997
      @zactron1997 8 лет назад +90

      Stuart Smith I'm waiting for a 2 hours video so badly, popcorn and notebook in hand, I await

    • @asielsmith6007
      @asielsmith6007 8 лет назад +10

      same, but he does a lot anyway

    • @Vaaaaadim
      @Vaaaaadim 8 лет назад +27

      I'd watch it as well. But I think that most people wouldn't and also "youtube's algorithm" might not favor it.

    • @JayLikesLasers
      @JayLikesLasers 8 лет назад +26

      Yeah, I'd be happy if he makes a feature-length about some journey into a mathematical concept.

  • @austinburrington6434
    @austinburrington6434 3 года назад +1766

    This guy should get a million dollars for making math intuitive and incredibly interesting.

  • @刘浩然-t4o
    @刘浩然-t4o Год назад +185

    I was a second year student in junior high school when I first watched this video. I was really intrigued by it, and started to dream of being a mathematician when I grow up. And now I am a sophomore in Peking University, majoring in math.
    I was reading Stein’s complex analysis just now, and the sixth chapter is about Zeta function and its analytic continuation. It suddenly reminded me of your video.
    Thank you for leading me to the amazing world of mathematics. 😊

    • @Albert-qp9ss
      @Albert-qp9ss 16 дней назад

      Woah, that's awesome! I'm currently a sophomore at your neighboring university haha, and was revisiting some of Grant's videos to better understand and appreciate some concepts. It's awesome how his videos span cultures and countries, yet we all are remarkably close to each other :)

  • @paulshi2821
    @paulshi2821 2 года назад +564

    I'm mind blown by the fact that this is FREE to watch for EVERYONE. Grant is truly making the world a better place.

    • @dann5480
      @dann5480 Год назад +3

      Pay him then!

    • @rivas97
      @rivas97 Год назад +1

      @@dann5480
      It's free!

    • @ИмяФамилия-е7р6и
      @ИмяФамилия-е7р6и Год назад +1

      What does "free" mean? Or does your mother pay for Internet access?

    • @eoneom
      @eoneom Год назад +4

      ​@@ИмяФамилия-е7р6и no, your mom

    • @quelqun-fw9lp
      @quelqun-fw9lp 8 месяцев назад

      ​@@ИмяФамилия-е7р6и he is using star li k free wifi i suppose but for me mobile data still very zxpensive

  • @MeanGreene87
    @MeanGreene87 2 года назад +54

    I’m a welder and have never used any of this. But it’s some of the most interesting and informative videos I’ve ever seen. I hope to one day understand all of this and hopefully be rich from it lol.

  • @hamdiabdelaziz7605
    @hamdiabdelaziz7605 5 лет назад +2475

    Imagine if Riemann can see this beautiful explanation and animation of his function.

    • @andy-kg5fb
      @andy-kg5fb 3 года назад +103

      @Kadir Garip wait what if all the mathematicians over the years have created an organisation of sorts in heaven and solved the millennium prize problems as well as other difficult problems.

    • @IsomerSoma
      @IsomerSoma 3 года назад +223

      He probably saw this and more in his head.

    • @ArnavBarbaad
      @ArnavBarbaad 3 года назад +159

      I'm sure Riemann's imagination showed him far, far more than whatever Grant can possibly animate

    • @marcelduartedasilvaxavier3749
      @marcelduartedasilvaxavier3749 3 года назад +14

      @@ArnavBarbaad Perfect!

    • @eclipse1353
      @eclipse1353 3 года назад +6

      Who knows... maybe, we all get to live multiple lifes...

  • @toddtrimble2555
    @toddtrimble2555 6 лет назад +409

    I'm a mathematician, and I must say that I'm really impressed by the quality of these videos. The visuals are just gorgeous, and the explanations are very sensibly done (and also nicely paced). You don't get bogged down in technical details, but there's lots of good intuition being developed and the mathematics itself is very meaty. Keep up the excellent work!

    • @dann5480
      @dann5480 Год назад

      No you're not.

    • @nuruzzamankhan1610
      @nuruzzamankhan1610 5 месяцев назад

      ​@@dann5480Yeah, I don't think any real mathematician would go around screaming 'Im a mathematician, I like this video, now gimme attention"

  • @MiguelMartinez299792458
    @MiguelMartinez299792458 2 года назад +25

    I teach complex analysis to physics students. I stumbled upon this channel following a recommendation from one of the students. I honestly found this a very impressive video on a beautiful subject. Truly excellent job! I wish I could make things half as visual on the blackboard...

  • @laradimello5791
    @laradimello5791 5 лет назад +230

    My Analysis professor recommended this video for a better understanding of Riemann's function and oh boy I'm pleased, now I truly understand! I'm so happy to live in a time where universities and youtube can complete one another.

    • @58585050
      @58585050 3 года назад +2

      Could you solve it?

    • @pedrodahmer8951
      @pedrodahmer8951 2 года назад

      @@58585050 tenho certeza que sim, americano!

  • @MrDaanjanssen
    @MrDaanjanssen 8 лет назад +86

    This is easily becoming one of my favorite channels on RUclips, thank you so much for all the videos

  • @thats_so_laven
    @thats_so_laven 2 года назад +6

    Man I just love you so much Grant
    Like, the level of effort and detail and dedication each video of yours has and the patience and obvious passion you have for teaching your viewers about the subject simply continues to floor me. I am in constant amazement of you.

  • @Piffsnow
    @Piffsnow 8 лет назад +72

    I'm a maths teacher and I always learn so much from you videos ! And... oh god it's beautiful !
    It's a feast each time.
    Thank you. :)

  • @jeongminpark3828
    @jeongminpark3828 5 лет назад +1295

    Please do the video about
    relation between zeta function and prime pattern

    • @lamepickuplines
      @lamepickuplines 5 лет назад +60

      Jeongmin Park I agree I wanna see one on
      1/zeta(s) = Π_{p prime} (1-p^-s)
      when Re(s) > 1
      I’m a grad student getting my masters in math preparing for a PhD in analytic number theory I’m currently in complex analysis right now

    • @lonestarr1490
      @lonestarr1490 5 лет назад +36

      @@lamepickuplines Then you should know about this already :D

    • @lamepickuplines
      @lamepickuplines 5 лет назад +12

      Lone Starr lol I don’t know much analytic number theory.. I still haven’t had much experience in relating complex analysis and number theory...

    • @hvoyaaudio
      @hvoyaaudio 5 лет назад +17

      there's a great book about that (and more) called Prime Obsession

    • @shambosaha9727
      @shambosaha9727 4 года назад +4

      @hvoya audio By John Derbyshire?

  • @gillfortytwo
    @gillfortytwo Год назад +4

    It's a great feeling coming back every 6 months or so to these vids and grasping these concepts a lil bit more, and feeling more grateful for how your visuals

  • @SSJProgramming
    @SSJProgramming 8 лет назад +25

    This is single-handedly the best video I've ever seen on the Riemann zeta function, as well as how analytic continuation is used here. I've truly gained a new appreciation for this function and its hidden beauty. Its connection with primes, as well as the implications for divergent sums is very deep. I look forward to more videos, cheers :)

  • @johnchessant3012
    @johnchessant3012 7 лет назад +1744

    8:58 - "focus on one of the marked points"
    Me: *focuses on (1 + 0i)*
    *facepalms*

  • @travorliu1192
    @travorliu1192 4 года назад +110

    The first time I watched this was in April of 2019, when I was learning the single-variable calculus with and could not wait to dive into the world of complex analysis. When I found this video again in 2020, I have learned all the relevant knowledge and analytically continued zeta function using a contour integral, eventually understanding the knowledge behind this amazing video. Thank you for creating such inspiring & fantastic video!

    • @Ganerrr
      @Ganerrr 2 года назад +9

      its 2022 you've proved it now right

    • @harry_page
      @harry_page Год назад +7

      @@Ganerrr Now it's 2023 and he's single-handedly solved all of the Millenium problems

    • @NamanNahata-zx1xz
      @NamanNahata-zx1xz Месяц назад

      ​@@harry_page He's him

  • @neurophilosophers994
    @neurophilosophers994 5 лет назад +15

    This is the best channel I’ve ever watched. Feynman always said if you can’t explain something simply enough you don’t truly understand it. The fact that you can explain these so well tells me you definitely understand it and I’m happy that i can to some extent.

    • @pokeman123451
      @pokeman123451 4 года назад

      "What I cannot build, I do not understand" and "Know how to solve every problem that has been solved" - Feynman's blackboard at the time of his death, 1988 (I think)

  • @mikesu8475
    @mikesu8475 5 лет назад +106

    You make me fall in love with mathematics, if one day I become a mathematician, you’re my mentor and inspiration! Respect from China.

    • @paulcherry5539
      @paulcherry5539 3 года назад +3

      go for it mike mathematics is the key to the universe after all

    • @klaus9356
      @klaus9356 3 года назад

      its been 2 years, did you do it?

    • @mikesu8475
      @mikesu8475 3 года назад +10

      @@klaus9356 I did a math minor along with my physics degree ☺️

    • @klaus9356
      @klaus9356 3 года назад

      @@mikesu8475 thats great. hopefully I will start my math career in university in just 2 years

    • @nonbiological
      @nonbiological 3 года назад

      @@mikesu8475 how old are you now!?

  • @xniyana9956
    @xniyana9956 2 года назад +3

    I've watched several videos on this topic from a lot of great math channels but this is the first one where I actually understood what the Reimann zeta function is and how it relates to the Reimann hypothesis. This is a great channel for simple folk like me that need pictures to understand stuff ☺

  • @jmiquelmb
    @jmiquelmb 8 лет назад +126

    This video is so good, that I've spent more than one hour online learning about the complex plane for first time

  • @samvargas2868
    @samvargas2868 5 лет назад +2269

    Me knowing Calculus:
    3Blue1Brown: "If you know calculus, you know we can take the derivative at any of these [complex] inputs"
    Me nervous: oh yeah, sure! Of course!

    • @donlansdonlans3363
      @donlansdonlans3363 5 лет назад +25

      Same

    • @TheAbele992
      @TheAbele992 5 лет назад +40

      Guess you don't know calculus then.

    • @EebstertheGreat
      @EebstertheGreat 5 лет назад +179

      The limit of a complex-valued function is defined in the same way as the limit of a real-valued function except that the real absolute values are replaced by complex absolute values (complex modulus). If you are familiar with the ε,δ-definition of a limit, then basically we are replacing the intervals on the line with disks on the complex plane. And if you can take the limit of a function, you can apply the exact same definition of a derivative for functions on the complex numbers that you can for functions of the real numbers. However, it turns out this limit only exists, and thus the complex derivative only exists, if the function satisfies a particular pair of conditions called the Cauchy-Riemann equations. A geometric interpretation of these equations is that the angle of lines intersecting anywhere but the origin is preserved under the transformation defined by the function.
      So it is not so straightforward. The upshot is that if the complex derivative exists, then it's continuous, and in fact derivatives of all orders exist. More than that, the function is analytic (at every point, there is a Taylor series that converges on some neighborhood of the point and is equal to the value of the function there), which is the relevant property in this video. These facts are not trivial at all, and they are central theorems in complex analysis, along with the identity theorem mentioned in the video (that analytic continuations are unique).

    • @samvargas2868
      @samvargas2868 5 лет назад +33

      @@EebstertheGreat thank you so much. This is hard for me to understand, but your explanation does help clairfy!

    • @ejb7969
      @ejb7969 4 года назад +10

      @@EebstertheGreat I hope that the limit as n approaches infinity of the sequence of my levels of understanding this as I re-read it for the nth time approaches actually understanding this.

  • @geoffstrickler
    @geoffstrickler 3 года назад +2

    This is the most easily understandable explanation of numerous things, including analytic continuation, complex analysis, and the continued zeta function that I’ve ever seen. And I say that as someone 50+ yrs old who has had a tremendous facility with math since I was at least 4 and generally taught myself most mathematics before I learned it in school, including many aspects of trig and calculus. And by “taught myself”, I actually mean figured it out on my own, not that I picked it up by studying a book. Books largely just connected my self-learning with accepted terminology, but did offer expanded views and results of what I figured out on my own. It wasn’t until calculus that I started encountering a significant amount about math that I hadn’t figured out on my own.

  • @johnskeff9617
    @johnskeff9617 8 лет назад +129

    It's if you can PROVE or DISPROVE the hypothesis. You get the money for both and both results would be equally ground breaking.

    • @duckymomo7935
      @duckymomo7935 8 лет назад +1

      so we know the hypothesis but we don't have a formal proof for it?

    • @phucminhnguyenle250
      @phucminhnguyenle250 8 лет назад +37

      Well, except if you can prove it then you can really prove hundreds of theorems. If you disprove it, then not only many theorems are disprove but also many important ones remain unproved. So both are ground breaking but the latter is kind of ugly.

    • @mikejones-vd3fg
      @mikejones-vd3fg 7 лет назад +1

      what is the problem?

    • @AllHailZeppelin
      @AllHailZeppelin 7 лет назад

      Right, but it probably IS true...

    • @jaredronning3020
      @jaredronning3020 7 лет назад +15

      You'd get the money if you disprove it. Disproving it doesn't necessarily involve finding a counter example (and I doubt that's how it would happen). For example, the existence of transcendental numbers was proved before any specific numbers were proved to be transcendental (over Q).

  • @pieshower
    @pieshower 5 лет назад +47

    You sir, FEED my craving for math in such a easy way to understand, especially the visuals. Every video I watch I leave astonished and amazed. I love math so much.

  • @marleigh5606
    @marleigh5606 3 года назад +677

    This is so neat but it’s 4:30 am I can’t help

    • @trakyaci
      @trakyaci 3 года назад +31

      I laughed at this comment for 4 minutes. I know that's not a big number, but consider that the usual response to something funny online is just a louder breathing out.

    • @maddieH24
      @maddieH24 3 года назад +7

      Its 1am on Christmas morning rn youre right this is so neat

    • @nsambataufeeq1748
      @nsambataufeeq1748 3 года назад +5

      It's 4:27am here

    • @tysparks598
      @tysparks598 3 года назад +3

      @@nsambataufeeq1748 6:26 am here

    • @pas-giaw6055
      @pas-giaw6055 3 года назад +1

      Cheese cat

  • @alexh3601
    @alexh3601 8 лет назад +122

    a 22 minute 3blue1brown video??? Yes please.

  • @kjpmi
    @kjpmi 8 лет назад +68

    I've seen your channel before but this is the first video I've watched in depth. I kick myself every day for not studying harder when I was younger. Math still scares me but I'm trying to change that.
    When I watch this video I feel this emotion that I don't have words to describe. You have an amazing gift for clarity. I just wish I had the words to describe what I feel and this comment doesn't do it justice. All I can say is thank you!

  • @mikeg3660
    @mikeg3660 3 года назад +3

    You are a gifted teacher to show the beauty and of math and how thinking outside the box (analytic continuation) opens us up to seeing around the bend by using our imagination.

  • @DLF-dv7ij
    @DLF-dv7ij 5 лет назад +2049

    All I understood was why who ever solves this gets a million dollars

    • @philosophicalinquirer312
      @philosophicalinquirer312 5 лет назад +546

      All I understood is I wont be getting a million dollars.

    • @D_D-_
      @D_D-_ 5 лет назад +285

      All I understood was that this will be the hardest way to get a million dollars

    • @wisdom6458
      @wisdom6458 5 лет назад +72

      @@philosophicalinquirer312 With this attitude you certainly won't.

    • @7vividfunster53
      @7vividfunster53 5 лет назад +19

      *You guys are getting Understood!!*

    • @chrisreven7271
      @chrisreven7271 4 года назад +13

      Because a group with a million dollars said "I'll give a million dollars to whomever solves this"

  • @cleramisheep
    @cleramisheep 5 лет назад +8

    As an electrical engineering graduate I find your contents very easy to understand, it would take me years to understand these kind of materials from reading books, thank you!!

  • @cmggun1709
    @cmggun1709 3 года назад +16

    Dear Grant Sanderson, Thank you for making these videos...for helping us see beauty of maths in its true nature. They are truly brilliant. I would like to request a video on Gamma and Beta functions and what it would mean to visualize them. I'm a beginner to this subject and as of now they seem like just some equations and theorems to me.
    Thank you.

  • @UnPuntoCircular
    @UnPuntoCircular 8 лет назад +380

    Love it! Thanks for the hard work put on these videos. This is how math should be first approached.

    • @lafyguy
      @lafyguy 6 лет назад

      UnPuntoCircular x. T

    • @meta04
      @meta04 6 лет назад +2

      3blue1brown swearing at 10:04? Unheard of.
      Also, being in calculus, I tend to be able to explain a lot of mathematical stuff to people in an understandable way, but I had no idea what "analytic continuation" is until I watched this video. Kudos 3b1b!

    • @2neutrino
      @2neutrino 6 лет назад +1

      "Damn" is hardly a swear word

    • @rmisegal
      @rmisegal 6 лет назад

      UnPuntoCircular ענפיץמנךמך
      יליכעעחעילל

  • @geraldmerkowitz4360
    @geraldmerkowitz4360 8 лет назад +12

    This is so good looking ! Your animations give a new perspective to math and reveal its beauty. That's awesome !

  • @erililil
    @erililil 3 года назад +12

    This has so many layers of abstractness. I love it.

  • @tommasosvalduz5226
    @tommasosvalduz5226 5 лет назад +6

    I really like your explanations. Their simplicity allows those who want to study maths to glimpse the beauty of the more complex concepts, like in this case, or to learn to see simpler ones from other perspectives. This is not just motivating, but actually inspiring. Thank you!

  • @JedsAnimations
    @JedsAnimations 8 лет назад +14

    I love it when I can learn things that aren't being taught at school! Thank you for making this!

  • @stephanc7192
    @stephanc7192 3 года назад +1

    This is a beautiful and such elagant demonstration and explanation!
    It feels like sometimes "they" want to pull wool over our eyes and leabe out thing like complex numbers in the zeta function and analytic continuation.
    Thank you so much for showing it so clearly!
    Kind regards
    Stephan

  • @eunhyoukshin7777
    @eunhyoukshin7777 6 лет назад +303

    10:06 I need a T-shirt of that pi-creature saying 'damn!'

  • @techdeth
    @techdeth 7 лет назад +543

    Thanks VSauce for introducing me to you, your content gets over my head relatively quickly, but I'm so fascinated. Thanks man

  • @sangeethas.1103
    @sangeethas.1103 3 года назад +2

    Wow, your videos are truely remarkable and outstanding, I am a Physicist with innate love for maths and its hidden truths, and seeing your videos give me more and more interest in finding the patterns and hidden meaning inside the numbers, one of the most fascinating world of complex analysis is it rounds around the unification presented by euler identity, which unifies power series, derivative and rotation mapped on to polar coordinates, this unification is so beautiful and one of the best ways to see numbers, is by unifying the concepts hidden in them.Sir I wish to see more of videos on complex analysis and euler equation with the above said unifications.

  • @iau
    @iau 7 лет назад +241

    This teaches me that 1+2+3+...=-1/12 is not really what it looks like at first glance.
    However, I would not have seen this video, nor understood any of this if I hadn't been click baited with 1+2+3+...=-1/12 by other videos.
    Funny how this works.

    • @99bits46
      @99bits46 5 лет назад +5

      it can be proved alternately with binomial theorem

    • @heyman4466
      @heyman4466 5 лет назад

      @@99bits46 Explain

    • @99bits46
      @99bits46 5 лет назад +8

      @@heyman4466 sorry i meant binomial formula. See ramanujan's original proof for this sum

    • @aashishpathak3628
      @aashishpathak3628 5 лет назад +1

      It is possible , shown by Ramanuzam sir.

    • @oneoveronethirtyseven9161
      @oneoveronethirtyseven9161 5 лет назад +11

      @@99bits46 Ramanujan's way still doesn't treat it as a sum in the traditional sense though. Though I do find it fascinating that Ramanujan summation agrees with the analytic continuation of the zeta function.

  • @marcusa2006x
    @marcusa2006x 7 лет назад +10

    Please do a series on complex analysis!! Your videos are the best!

  • @kuretaxyz
    @kuretaxyz 2 года назад +2

    I have watched all of your videos and I sometimes come back and re-watch one of them at random, just like listening to a good song I liked.

  • @LKalavera
    @LKalavera 7 лет назад +88

    PLEASE make the prime number part of this. Thanks a lot.

    • @3blue1brown
      @3blue1brown  7 лет назад +71

      It's on the list, but I'm struggling with how to do it in a way that's not too formula-heavy. It raises many interesting questions for what exactly to cover. Trust me, when I think it will make a quality video, I'll make it.

    • @LKalavera
      @LKalavera 7 лет назад +6

      I trust you! Can't wait :)

    • @stevearmstrong7023
      @stevearmstrong7023 5 лет назад

      Agrre please provide

    • @nahidhkurdi6740
      @nahidhkurdi6740 5 лет назад +2

      Much more difficult to animate, I think.

    • @Hi-6969
      @Hi-6969 3 года назад

      @@3blue1brown Still on your list, right?

  • @Yashpandey467
    @Yashpandey467 8 лет назад +332

    understanding it is way more beautiful and mind blowing than that million dollar prize!!

    • @3blue1brown
      @3blue1brown  8 лет назад +146

      Man, y'all are the best, motivated by actually learning what's going on more than "hook" of talking about the Millenium prize problems.

    • @ashrayaindrakanti984
      @ashrayaindrakanti984 8 лет назад +5

      I'm doing a project on complex functions. Like in linear algebra, this is filling the visual understanding to my arithmetic knowledge, which I think was your intention. What I wouldn't give for a 5 minute conversation

    • @tonynixon9715
      @tonynixon9715 7 лет назад +4

      3Blue1Brown I would love to know if you can do essence of complex analysis series and an essence of abstract algebra series.
      I would also like you to upload a video on the pi²/6

    • @hexane360
      @hexane360 7 лет назад +10

      Yeah but you get both. . .

    • @arbitrage2141
      @arbitrage2141 7 лет назад

      Yash Pandey Isnt it though???

  • @theherk
    @theherk 3 месяца назад +1

    I still come back and watch this regularly. One of the best explainer videos ever made.

  • @AndroidT01187
    @AndroidT01187 8 лет назад +69

    This is one of the most amazing mathematics videos that I have seen, I'd have to say that this is my favorite video from you by far. I must ask you of your current progress on The Essence of Calculus series you are working on and if you have any possible idea when you will upload it (or if it's a surprise,) because I'm really looking forward to those videos since I am currently taking a high school calculus class.

    • @3blue1brown
      @3blue1brown  8 лет назад +42

      The plan is to publish the series by early April.

    • @yaoliu7034
      @yaoliu7034 8 лет назад

      that sounds very interesting. i wish i could join your team :)

    • @beefmomma
      @beefmomma 7 лет назад +1

      3Blue1Brown please do more millennium prize problems! this video was fantastic

  • @velvetdrgn
    @velvetdrgn 6 лет назад +4

    Just learned about derivatives in school this year, usually when watching one of your videos I have to do a bit of googling to really understand what you're saying, but it feels nice to actually recognise something from math in one of your vids

  • @3716-e9o
    @3716-e9o 4 года назад +2

    I hv watched many videos about Riemann hypothesis, this is the best to show how analytic continuation works.

  • @mechwarreir2
    @mechwarreir2 8 лет назад +147

    Here's a challenge, try doing a visualization video on a topic of Abstract Algebra or Algebraic Geometry, these fields are just so impenetrably abstract.

    • @hanniffydinn6019
      @hanniffydinn6019 8 лет назад +11

      mechwarreir2 actually no algebraic geometry is what numbers really are, so far easier to grasp than dumb down versions of it like complex, quaternions etc... It is ust we are taught incorrectly in the first place!
      Everything is easier and simplier with algebraic geometry. As reality is really multidimensional.

    • @asnierkishcowboy
      @asnierkishcowboy 8 лет назад

      Take motives for example. They enable you to decompose an object in to smaller motives, just like molecules are made out of atoms. And maybe a motive occurs in two different objects, then they DO in fact have something im common :)

    • @SalixAlba256
      @SalixAlba256 8 лет назад +5

      A video on Reimann-Roch would be quite something.

    • @jean-patrickpelletier4162
      @jean-patrickpelletier4162 8 лет назад +2

      Algebraic manifolds

    • @BareClause
      @BareClause 8 лет назад +3

      What about schemes?

  • @EtzEchad
    @EtzEchad 8 лет назад +975

    I have found an elegant proof of the Riemann Hypothesis. It is a little too long to write down here in a RUclips comment though.

    • @hexxedhd6120
      @hexxedhd6120 8 лет назад +223

      David Messer the Fermat memes

    • @ujjwalrana5899
      @ujjwalrana5899 8 лет назад +33

      David Messer then publish it

    • @patrickhodson8715
      @patrickhodson8715 8 лет назад +16

      Lawl

    • @hephsiba
      @hephsiba 7 лет назад +38

      Louis de Branges de Bourcia says he has a proof, and it might be that he has. Unfortunately, no one is in a position to check it because no one understands the technicalities - which took a lifetime to develop.

    • @David_Last_Name
      @David_Last_Name 7 лет назад +16

      +David Messer I believe you bro.

  • @derekokeeffe9919
    @derekokeeffe9919 2 месяца назад

    Your way of communicating complex ideas is just amazing

  • @vh73sy
    @vh73sy 5 лет назад +19

    incredible, breath-taking, so accurate, colorful and concise

  • @mitch0070
    @mitch0070 7 лет назад +38

    Best line in the video: "If that doesn't make you want to learn more about complex functions, you have no heart!" I love it!!

  • @Michiel_de_Jong
    @Michiel_de_Jong 3 года назад +47

    My conjecture:
    The chance that the Riemann hypothesis is true is bigger than the chance me proving it.

    • @blaz2892
      @blaz2892 3 месяца назад +3

      This is tautological by the laws of statistics. P(A|B) >= P(B). Your conjecture is now a theorem. :)

  • @Chaos------
    @Chaos------ 7 лет назад +12

    this makes my mind tingle in the most pleasant way imaginable.

    • @dankwarmouse6248
      @dankwarmouse6248 5 лет назад

      Spinoza described that sensation as "intellectual love" :)

  • @nujuat
    @nujuat 8 лет назад +16

    I think this is the first time I've seen an explanation of this that actually makes sense.

  • @vladherasymenko543
    @vladherasymenko543 3 года назад +5

    I studied analytic functions as a part of my degree program and, to be honest, it didn’t make much sense to me back then. But your animations are really intuitive. I bloody wish I had seen them, while I was taking this course 😕

  • @Fernito
    @Fernito 7 лет назад +150

    And here I am at 3:00 AM watching your videos, while I should be sleeping. But you know what? It's OK. You, sir, are a genius. You make advance math so easy to grasp. Today I have literally learned more than in the last, say, 2 years? (which I have, admittedly, kinda wasted from an intellectual point of view). Thanks for making these videos.

    • @computerscientist5953
      @computerscientist5953 6 лет назад +14

      spending time on self-development at 3 am is ok.
      spending time on useless shit (aka gaming, partying, drinking) at 3 am is NOT ok.
      So, you are fine!

    • @ramchandracheke
      @ramchandracheke 5 лет назад +1

      Same here at 4.30 Am

    • @computerscientist5953
      @computerscientist5953 4 года назад

      @Beyblade420 your body will say thank you in 20 years from now, trust me, I had a lot of friends who were party animals at your age

  • @bagusbrahmantya1009
    @bagusbrahmantya1009 3 года назад +4

    Thank you for doing great work in promoting advanced mathematics. The world needs more people like you, Sir.

  • @john_hunter_
    @john_hunter_ 2 года назад +2

    It was really easy to understand when you showed how the points of the grid map on to their new points.

  • @poisonpotato1
    @poisonpotato1 8 лет назад +5

    When I first found this channel I commented that he would make a great video to visualize this function. This was not disappointing at all

  • @rednax3788
    @rednax3788 8 лет назад +1295

    I just noticed that when ever you're doing a shot of one pi teaching the others, 3 of them are blue, and 1 is brown. 3 blue, 1 brown --> 3blue1brown

    • @rigille
      @rigille 8 лет назад +46

      Well noticed

    • @rednax3788
      @rednax3788 8 лет назад +130

      I feel really bad for only commenting this after watching a beautiful visualisation, and explination of the Riemann Zeta Function.

    • @Nachtgrabb
      @Nachtgrabb 8 лет назад +49

      it's the PI conspiracy! 3 integers 1+1+1 and 1 Rest 0,141...

    • @asterisqueetperil2149
      @asterisqueetperil2149 8 лет назад +5

      Well noticed, I was wondering why this name too.
      Now you noticed it, is it related to the movie "Pay it forward" ? :D

    • @davidwright8432
      @davidwright8432 8 лет назад +43

      I wondered if it meant that three of his grandparents had blue eyes, and one, brown!
      Or that three quarters (roughly) of the Earth is ocean covered (blue) and one quarter, brown (land). And I'm working on relating it to the Illuminati ... :)

  • @userprobablynotfound
    @userprobablynotfound 2 года назад +1

    I remember desperately trying to understand the Riemann Zeta function as I took Calc III in summer school. I was doing school online so most of my learning consisted of me trying to learn through RUclips videos and running every possible question and equation through Wolfram Alpha so I could try and understand the concept if I worked backward from the solution. None of it helped me to really get it. Your video did in 20 minutes what months of study and pulling my hair out couldn't... But what really gets me is I took that class in the summer semester of 2016. I should have just waited until the next spring 😑

  • @radicalmathematics9728
    @radicalmathematics9728 6 лет назад +718

    How did you learn to make your graphics? Your style of presentation is very nice.

    • @vladislav_artyukhov
      @vladislav_artyukhov 6 лет назад +50

      Secret of channel)

    • @cuber18fred
      @cuber18fred 5 лет назад +211

      He's a programmer. All of the visualization is a product of his programming skills.

    • @omerresnikoff3565
      @omerresnikoff3565 5 лет назад +132

      manim, the software's called manim and it's available in GitHub, although it's quite confusing to use (imo)

    • @rudolf-blue
      @rudolf-blue 5 лет назад +22

      It's on github, but the documentation isn't that great

    • @nivk9445
      @nivk9445 5 лет назад +12

      @@benharris3100 Better late than never ;)

  • @dAvrilthebear
    @dAvrilthebear 5 лет назад +44

    Thank you! Came for -1/12, stayed for the Rieman Zeta Function

  • @George14215
    @George14215 2 года назад

    Ok, this is the best explanation of the Reimann hypothesis that I've seen on RUclips. In particular the explanation of Analytic Continuation. Cheers!

  • @selimbaydar123
    @selimbaydar123 8 лет назад +4

    I literally love you for doing this one !

  • @kraamesh
    @kraamesh 5 лет назад +32

    Euler, Riemann, Gauss all may be the great mathematicians of all time. But I am pretty sure none can exceed your ability to teach the complex concepts in layman terms...In other words, even the great mathematicians cannot explain the concepts better than you already explained...you are a Magician, Musician, and an Artist

    • @ИмяФамилия-е7р6и
      @ИмяФамилия-е7р6и Год назад

      emmmm..
      Feynman?

    • @sagej0estar710
      @sagej0estar710 Год назад

      @@ИмяФамилия-е7р6иFeynman was a physicist

    • @sidneysilva7364
      @sidneysilva7364 Год назад

      Dear noble friends, professors, students, acquaintances of this simple channel, with my respect to everyone present here; what impact would it have on the Universe of Mathematics, by stating that some numbers cited are not prime? and the Twin Cousins do not exist?
      two; 19; 41; 59; 61; 79; 101; 139; 179; 181; 199; 239; 241; 281; 359; 401; 419; 421; 439; 461; 479; 499; 521; 541; 599; 601; 619; 641; 659; 661; 701; 719; 739; 761; 821; 839; 859; 881; 919; 941; 1019; 1021; 1039; 1061; 1181; 1201; 1259; 1279; 1301; 1319; 1321; 1361; 1381; 1399; 1439; 1459; 1481; 1499; 1559; 1579; 1601; 1619; 1621; 1699; 1721; 1741; 1759; 1801; 1861; 1879; 1901; 1979;
      However, the "Rielmann Hypothesis" completely loses its strength in the theories of past times, however this prize that the Clay Institute wants to pay, will not be able to pay for an unfounded theory, since these numbers are not prime, it can totally change the history of Mathematics, bringing Innovative Mathematics to the current era, my concept of what a prime number is, I sanctioned a Law that must always be respected; for every prime number, where it will be factored from the smallest to the largest, and from the largest to the smallest only with the prime numbers themselves, so it will be considered a prime number .... follows how my thesis will be:
      I will multiply only with prime numbers, respecting my law:
      3*5*7*11*13*17 = 255255
      255255 3
      85085 5
      17017 7
      2431 11
      221 13
      17 17
      1
      In this first example it was from smallest to largest;
      255255 17
      15015 13
      1155 11
      105 7
      15 5
      3 3
      1
      In this second example it was from the largest to the smallest, only this pattern can say that it is a prime number. Sir Sidney Silva.

  • @ViratKohli-jj3wj
    @ViratKohli-jj3wj 3 года назад +3

    Best video I have seen on Riemann Hypothesis ever

  • @Scy
    @Scy 8 лет назад +20

    I have watched dozens of videos about the zeta function, and this is the one that finally made me understand HOW it works. Even if you just explained WHAT it does. Especially the spiral angle bit just made everything fall into place.
    I still can't explain it to anyone, but I'm not a professor, so I don't need to!

  • @snozceno3879
    @snozceno3879 6 лет назад +36

    I came here because a dude said he has proof, I understand nothing but seeing someone can prove it feels awesome

    • @arcuesfanatic
      @arcuesfanatic 6 лет назад +23

      Unfortunately, that proof is being met with skepticism. This is because many of the same guy's recent proofs about other fields of mathematics have been shown to have many inaccuracies. On top of this, the presentation he did was very hand-wavy in how he described it. So they're looking at his written proof to see if it holds any water or not.

    •  5 лет назад +2

      arcuesfanatic any updates on this?

    • @erinstrickland3337
      @erinstrickland3337 5 лет назад +13

      @ It didn't hold up. It's still an open question

    •  5 лет назад

      Erin Strickland thanks

    • @thesmart4128
      @thesmart4128 5 лет назад +1

      It could've been quite the shock for the proof to actually be correct :)

  • @Annibals
    @Annibals 3 года назад +2

    You're a good teacher
    To those who have a gap in understanding in the matter you made them get a good idea of with first principles. I greatly admire anyone who is objective and gets to the matrix of anything.

  • @wwebadgerse
    @wwebadgerse 4 года назад +129

    Can we get a full 3 hour version of you just saying " 1 over 1 to the s, plus 1 over two to the s..." For no reason whatsoever?

  • @LanarFalcon1
    @LanarFalcon1 5 лет назад +129

    17:40 the longest running joke in mathematics

    • @vidyas.4531
      @vidyas.4531 4 года назад

      ?

    • @vernie7882
      @vernie7882 4 года назад +2

      Where was the joke?

    • @t_h_e_o4303
      @t_h_e_o4303 4 года назад +68

      Proof is trivial and left as an excercise to the reader

    • @epicswirl
      @epicswirl 4 года назад +10

      T_h_e_o exactly if you take a math course the teacher will always just say this is trivial let’s move on 😂

    • @ujjalmajumdar618
      @ujjalmajumdar618 4 года назад +13

      It is often said that everything a mathematician can prove is trivial. Because everything they prove becomes trivial.

  • @j.503
    @j.503 4 года назад +2

    You make difficult Mathematics topics interesting and accessible to a non-mathematician. That shouldn't be possible. Thanks.

  • @whentheanimals
    @whentheanimals 4 года назад +6

    17:44 the best definition of the "trivial solution" I've ever heard

  • @josephcoon5809
    @josephcoon5809 3 года назад +3

    5:40 You’ve basically described the basics behind string theory in which you wrap the “imaginary dimension on itself and have it exist separately from our conventional spatial dimensions. The micro spatial dimensions don’t “exist” to our physical realms, but they help explain the forces which affect our material reality by existing just outside of those three physical dimensions in their own micro dimensions.

  • @reenabalasarangi9331
    @reenabalasarangi9331 22 дня назад +1

    10:04 The most expressive this guy has ever been

  • @Viewer2812
    @Viewer2812 4 года назад +18

    "After the transformation, the lined make such lovely arcs before they abruptly stop. Don't you just want to... Continue those arcs?"

  • @pepegasadge2977
    @pepegasadge2977 8 лет назад +4

    Hey, this shit really deserves more views. Amazing visualizations. Amazing video. Amazing explanation.

  • @luvyoomgi
    @luvyoomgi Год назад

    Came here from a comment under PeakMath's Riemann Hypothesis Saga, and I love the way your illustrations and explanations compliment each other!
    Would be so cool to see a collaboration between the two channels on this topic!!!

  • @az8560
    @az8560 10 месяцев назад +11

    There's a secret part of this video. You just need to analytically continue it to the negative timestamps...

  • @livintolearn7053
    @livintolearn7053 6 лет назад +9

    Thanks a lot for all of the videos you've made!! They are all exceptional!Once again, THANK YOU SOOOOOOO MUCH!!!

  • @niiidar
    @niiidar Год назад

    I recently read the book "Prime Numbers and the Riemann Hypothesis" and had to tap out when they got to the last chapter and started heavily leveraging the riemann zeta function, of which I had no clue what was. This video beautifully complemented that book, and made it understandable. Thank you!

  • @zairaner1489
    @zairaner1489 8 лет назад +12

    Beautifull video, the only thing I missed was as comment on the fact that you cannot extend it to the whole complex plane because you need to take 1 out

    • @3blue1brown
      @3blue1brown  8 лет назад +19

      Yes, very good point. I never really like to think of poles as "taken out", since it's so nice to think of them as going to a particular point at infinity (i.e. Riemann sphere). But your point stands, it could've used a word or two.

    • @zairaner1489
      @zairaner1489 8 лет назад

      I actually considered to mention the point at infinity in my comment as well, which is new knowledge to me considering we just did that just this week in my lecture. But yeah perfect video-again!

    • @TOCZEKX
      @TOCZEKX 8 лет назад

      Awesome video. Wchich computer programme can visualise complex graph?

  • @HDQuote
    @HDQuote 8 лет назад +212

    this is almost psychadelic

    • @Stickyxgo
      @Stickyxgo 8 лет назад +29

      What is a psychedelic experience if not reality.

    • @Sivolc11
      @Sivolc11 8 лет назад

      uhhh... psychedelic delusions/human cognition in an error state, obviously?

    • @xHubert0
      @xHubert0 8 лет назад +12

      +pxxner This is the most beautiful thing I have read today, thanks mate

    • @marksmod
      @marksmod 8 лет назад +1

      dafuq?

    • @marksmod
      @marksmod 8 лет назад +5

      Your brain is a function, and LSD fucks shit up such that you start seeing some of the underlying Functions in action.

  • @kemsekov6331
    @kemsekov6331 2 года назад +1

    On the third try of watching this video with several months spans I finally understood everything.
    This was awesome

    • @chrissdehaan
      @chrissdehaan 2 года назад

      So you're about to be $1,000,000 ricer too~

  • @JMEssex
    @JMEssex 8 лет назад +14

    How come nobody has attempted to explain the Riemann Zeta function in terms of the Quaternions?
    William Rowan Hamilton in 1843 describes quaternions as a number system that extends the complex numbers, and a quaternion as the quotient of two directed lines in a three-dimensional space or equivalently as the quotient of two vectors.
    a + bi + cj + dk.

    • @nathanisbored
      @nathanisbored 8 лет назад +7

      my understanding is that with quaterions, you lose multiplicative commutivity, so their use is limited. it's good for modeling 3D vectors and things, but for analyzing the domains of functions, the sweet spot is with complex numbers.

    • @Chaos------
      @Chaos------ 6 лет назад

      He did a video on quaternions recently I got the same impression. The animation's he use's are exact replications of the ones he used to explain quaternions, so there has to be a deeper connection between them.

  • @ashwinkumar7182
    @ashwinkumar7182 7 лет назад +6

    My eyes became moist after watching this.
    This is so beautiful!

  • @flightlesswizard
    @flightlesswizard Год назад +2

    "Pretty much any function with a name is analytic." Absolute value would like to speak with you.

  • @SpiffyCheese2
    @SpiffyCheese2 7 лет назад +296

    20:30 Markus Persson is a 3Blue1Brown Patreon support. Now we know that Notch(the creator of Minecraft) likes math!

    • @moshecarmeli9364
      @moshecarmeli9364 6 лет назад +6

      ThatMathNerd עעעעעעעעעעעעעעעעעע עיעעעעעעי יייייייייייייייייייייייייייייייעעיייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייי

    • @sebastianjakov4895
      @sebastianjakov4895 6 лет назад +3

      Wooooooot??!! Mind blows up

    • @rshua
      @rshua 5 лет назад +3

      Hunh interesting

    • @jakistam1000
      @jakistam1000 5 лет назад +79

      So you're saying someone who's a pretty good coder likes math? How suprising!

    • @crackedemerald4930
      @crackedemerald4930 5 лет назад +19

      @@jakistam1000 considering the solidity of Minecraft's coding, it kinda doesn't surprise me

  • @nicogerst6007
    @nicogerst6007 2 года назад +234

    If every teacher/professor could explain subjects this well, we'd already be harvesting energy from black holes and teleporting to mars. 110% respect!

    • @jwcrawley
      @jwcrawley 2 года назад

      And at its converse, bad teachers do more harm to learning and understanding than most other things.

    • @rathorefamily
      @rathorefamily Год назад +12

      It doesn’t make much sense to agree with someone more than 100%, but maybe we could analytically extend the definition of agreement 😅

    • @octs609
      @octs609 10 месяцев назад

      analytically continue percentages baby@@rathorefamily

  • @pullrequest1296
    @pullrequest1296 3 года назад +2

    12:20 can't help crying. How beautiful the math is!

  • @basuam
    @basuam 7 лет назад +16

    Thank you very much for this beautiful explanation. I found this extremely intuitive because you have created really beautiful visualizations. Thank you very much for this HARD WORK. You have made Mathematics really sexy

  • @josephcoon5809
    @josephcoon5809 3 года назад +6

    15:30 Instead of “extending” the grid lines to the right of x=1, consider why that boundary exists. It is the consequence of the multiplicative identity being 1. With that in mind, consider 1 as being a boundary that can be represented in the same way that i is a boundary in a complex plane resulting in limiting the result of a complex power to a unit circle.
    Treat x=1 like i, and wrap it around the origin. This will allow you to see how the “left” and “right” of the graph are related by translating “left” and “right” to “inner” and “outer.” Once you have done that, you can “flip” it “down” and add a third dimension that represents the amount of “space” “inside” the barrier of 1 as a curved space outside the other two dimensions.
    From a physics standpoint, you’ve basically described the transition from flat space to curved space as described by Einstein’s general theory of relativity. You can also view it as the transition from from real space to singularity space inside the event horizon of a black hole, where 1 represents the event horizon.
    From this perspective, the Riemann challenge becomes an exercise in describing the singularity (represented by 0 in this challenge) which is the intersection between General Relativity and Quantum Theory: The Grand Unified Theory.

    • @exurb8a502
      @exurb8a502 3 года назад +1

      🤣🤣🤣🤣🧐🧐🧐🧐🧐🧐

    • @josephcoon5809
      @josephcoon5809 3 года назад

      @@exurb8a502 🤣😅🤣🤣😂😂🤣🤣🤣🤣

  • @amandaroberts6535
    @amandaroberts6535 Год назад +1

    The angle-preservation of analytic continuance reminds me so much of the conformal models of non-Euclidean geometry. Like how angles in a Poincaré disk match what it looks like.

  • @Fanny10000
    @Fanny10000 3 года назад +3

    Animations are absolutly great ! I feel I can "see" analytic functions as I never have!