But what is the Riemann zeta function? Visualizing analytic continuation

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Unraveling the enigmatic function behind the Riemann hypothesis
Help fund future projects: / 3blue1brown
An equally valuable form of support is to simply share some of the videos.
Special thanks to these supporters: 3b1b.co/zeta-thanks
Home page: www.3blue1brown.com/
Posters/shirts for this visualization at 3b1b.co/store
Music by Vince Rubinetti: / riemann-zeta-function
Check out some of Vince's other work here: www.vincentrubinetti.com/
For those who want to learn more about complex exponentiation, here are a few resources:
- My video on the topic: • Euler's formula with i...
- Mathologer's: • e to the pi i for dummies
- Better Explained: goo.gl/z28x2R
For those who want to learn more about the relationship between 1+2+3+4+... and -1/12, I'm quite fond of this blog post by Terry Tao: goo.gl/XRzyTJ
Also, in a different video "What does it feel like to invent math", I give a completely different example of how adding up growing positive numbers can meaningfully give a negative number, so long as you loosen your understanding of what distance should mean for numbers: • What does it feel like...
Interestingly, that vertical line where the convergent portion of the function appears to abruptly stop corresponds to numbers whose real part is Euler's constant, ~0.577. For those who know what this is, it's kind of fun to puzzle about why this is the case.
Thanks to these viewers for their contributions to translations
Hebrew: Omer Tuchfeld
------------------
3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with RU-vid, if you want to stay posted about 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: goo.gl/WmnCQZ
Various social media stuffs:
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Twitter: / 3blue1brown
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Опубликовано:

5 май 2024

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Комментарии : 3,9 тыс.

@jmcsquared18 7 лет назад

As a mathematician, I was always deeply aggravated when other mathematicians on RU-vid 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 7 лет назад

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 7 лет назад

+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 7 лет назад

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 7 лет назад

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 7 лет назад

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 7 лет назад

"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 7 лет назад

So true. :)

@zactron1997 7 лет назад

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

@asielsmith6007 7 лет назад

same, but he does a lot anyway

@Vaaaaadim 7 лет назад

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

@JaySmith91 7 лет назад

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

@austinburrington6434 3 года назад

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

@OrthodoxPerspective 2 года назад

he does

@paulmckinney5968 2 года назад

He makes it easy for me to understand I love it.

@amritaanshnarain7524 Год назад

@vadimZ1000 9 месяцев назад

Every year

@paulshi2821 2 года назад

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 10 месяцев назад

Pay him then!

@rivas97 9 месяцев назад

@@dann5480 It's free!

@user-iw1qn3mt7e 9 месяцев назад

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

@eoneom 9 месяцев назад

@@user-iw1qn3mt7e no, your mom

@I_killed_that_beard_guy 8 месяцев назад

You are paying with ads or RU-vid premium

@hamdiabdelaziz7605 4 года назад

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

@andy-kg5fb 3 года назад

@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 3 года назад

He probably saw this and more in his head.

@ArnavBarbaad 2 года назад

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

@marcelduartedasilvaxavier3749 2 года назад

@@ArnavBarbaad Perfect!

@eclipse1353 2 года назад

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

@johnchessant3012 6 лет назад

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

@marcopicerni1304 4 года назад

I'm dead ahahahhahahahahahahahah

@mashfiqrizvee2537 4 года назад

i laughed so hard lol

@blackcat5771 4 года назад

LOL

@blackcat5771 4 года назад

@@espositogregory chill

@mashfiqrizvee2537 4 года назад

@@espositogregory chill

@user-cr5tq1gf2g Год назад

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. 😊

@marleigh5606 3 года назад

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

@trakyaci 3 года назад

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 3 года назад

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

@nsambataufeeq1748 3 года назад

It's 4:27am here

@tysparks598 3 года назад

@@nsambataufeeq1748 6:26 am here

@pas-giaw6055 3 года назад

Cheese cat

@samvargas2868 4 года назад

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 4 года назад

Same

@TheAbele992 4 года назад

Guess you don't know calculus then.

@EebstertheGreat 4 года назад

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 4 года назад

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

@ejb7969 4 года назад

@@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.

@laradimello5791 4 года назад

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 3 года назад

Could you solve it?

@pedrodahmer8951 Год назад

@@58585050 tenho certeza que sim, americano!

@thats_so_laven 2 года назад

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.

@MeanGreene87 2 года назад

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.

@toddtrimble2555 5 лет назад

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 10 месяцев назад

No you're not.

@jeongminpark3828 4 года назад

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

@greatstuff5 4 года назад

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 4 года назад

@@greatstuff5 Then you should know about this already :D

@greatstuff5 4 года назад

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 4 года назад

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

@shambosaha9727 4 года назад

@hvoya audio By John Derbyshire?

@MiguelMartinez299792458 2 года назад

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...

@gillfortytwo Год назад

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

@DLF-dv7ij 4 года назад

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

@philosophicalinquirer312 4 года назад

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

@D_D-_ 4 года назад

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

@hassanakhtar7874 4 года назад

And their names never forgotten

@wisdom6458 4 года назад

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

@7vividfunster53 4 года назад

*You guys are getting Understood!!*

@eunhyoukshin7777 5 лет назад

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

@Mono_Autophobic 3 года назад

Yeah👍

@geoffstrickler 2 года назад

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.

@xniyana9956 2 года назад

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 ☺

@EtzEchad 7 лет назад

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

@hexxedhd6120 7 лет назад

David Messer the Fermat memes

@ujjwalrana5899 7 лет назад

David Messer then publish it

@patrickhodson8715 7 лет назад

Lawl

@hephsiba 7 лет назад

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.

@BiophysicalChemist 7 лет назад

David Messer Andrew Wiles is not amused.

@Piffsnow 7 лет назад

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. :)

@user-iw1qn3mt7e 9 месяцев назад

elementary math teacher, I hope? in other words, an arithmetic teacher?

@cmggun1709 2 года назад

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.

@travorliu1192 4 года назад

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!

@GanerRL 2 года назад

its 2022 you've proved it now right

@harry_page Год назад

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

@SSJProgramming 7 лет назад

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 :)

@MrDaanjanssen 7 лет назад

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

@stephanc7192 3 года назад

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

@mikeg3660 3 года назад

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.

@pieshower 4 года назад

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.

@kjpmi 7 лет назад

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!

@3blue1brown 7 лет назад

+kjpmi Wow, thanks for sharing. This was very motivating to read.

@fossilfighters101 7 лет назад

@Michiel_de_Jong 2 года назад

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

@bagusbrahmantya1009 2 года назад

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

@mikesu8475 5 лет назад

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 2 года назад

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

@klaus9356 2 года назад

its been 2 years, did you do it?

@mikesu8475 2 года назад

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

@klaus9356 2 года назад

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

@nonbiological 2 года назад

@@mikesu8475 how old are you now!?

@Yashpandey467 7 лет назад

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

@3blue1brown 7 лет назад

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 7 лет назад

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 7 лет назад

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 7 лет назад

Yeah but you get both. . .

@arbitrage2141 7 лет назад

Yash Pandey Isnt it though???

@STohme 3 года назад

Very nice talk, your presentation is almost accessible to the general public. You explain very well that the sum of Ramanujan series is not strictly speaking equal to -1/12 but it takes its sens as an analytic continuation of an holomorphic function beyond its convergence domain on the complex plane. Excellent video. Many thanks.

@erikliljeberg3291 3 года назад

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

@johnskeff9617 7 лет назад

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 7 лет назад

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

@phucminhnguyenle250 7 лет назад

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.

@BiophysicalChemist 7 лет назад

John Skeff Nope, you only get the money if you prove it. A disprove would be finding a counter example, but anyone who has intuition for complex analysis would tell you they have no doubt that it is true. The solution is most likely pretty simple too, it's just that no one has found the right way of thinking about the problem that makes that solution clear.

@mikejones-vd3fg 7 лет назад

what is the problem?

@AllHailZeppelin 6 лет назад

Right, but it probably IS true...

@alexh3601 7 лет назад

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

@nicogerst6007 Год назад

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 Год назад

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

@rathorefamily 10 месяцев назад

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

@octs609 3 месяца назад

analytically continue percentages baby@@rathorefamily

@niiidar 10 месяцев 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!

@jmiquelmb 7 лет назад

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

@tommasosvalduz5226 5 лет назад

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!

@veggiet2009 4 года назад

I would just like to say that I love this channel, I love math, and visual representation. But also I can go back to videos that I've already listened to, and just put them on in the background and just listen, and it's great to work to

@reggyreptinall9598 2 года назад

Great channel thanks for your work 🙌🙌 the episode about Euler's equation was truly remarkable.

@UnPuntoCircular 7 лет назад

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

@lafyguy 6 лет назад

UnPuntoCircular x. T

@meta04 6 лет назад

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 6 лет назад

"Damn" is hardly a swear word

@rmisegal 5 лет назад

UnPuntoCircular ענפיץמנךמך יליכעעחעילל

@cleramisheep 5 лет назад

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!!

@sohambhattacharjee951 3 года назад

Thank you for making this so sensible and relatively easier to understand than it actually is.

@sangeethas.1103 2 года назад

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.

@neurophilosophers994 5 лет назад

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 3 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)

@AsiaCrasie 5 лет назад

This is the coolest, most informative, easy to understand video I have watched in three years. Bless your soul.

@matthewma3716 3 года назад

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

@George14215 2 года назад

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

@mechwarreir2 7 лет назад

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 7 лет назад

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 7 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 7 лет назад

A video on Reimann-Roch would be quite something.

@jean-patrickpelletier4162 7 лет назад

Algebraic manifolds

@BareClause 7 лет назад

What about schemes?

@velvetcake5425 5 лет назад

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

@vitorschroederdosanjos6539 4 года назад

Finly someone that explains it in a very visual and relatively uncomplicated way Thank you very much

@kuretaxyz 2 года назад

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.

@geraldmerkowitz4360 7 лет назад

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

@livintolearn7053 6 лет назад

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

@luvyoomgi 8 месяцев 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!!!

@Fanny10000 3 года назад

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

@mitch0070 7 лет назад

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!!

@JedsAnimations 7 лет назад

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

@vladherasymenko543 3 года назад

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 😕

@mattm7482 4 года назад

The main thing I appreciate apart from the math after watching the video is how much effort the producers put into the visualization. Thanks for making Math fun and pretty (literallly).

@rednax3788 7 лет назад

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 7 лет назад

Well noticed

@rednax3788 7 лет назад

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

@Nachtgrabb 7 лет назад

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

@asterisqueetperil2149 7 лет назад

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

@davidwright8432 7 лет назад

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 ... :)

@vh73sy 5 лет назад

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

@Annibals 2 года назад

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.

@flightlesswizard 9 месяцев назад

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

@marcusa2006x 6 лет назад

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

@techdeth 6 лет назад

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

@adarshsathian2862 6 лет назад

Noah Price This is not Vsauce

@tyronechickin1180 6 лет назад

Read his comment again Madara

@redhatt6747 6 лет назад

Noah price same here

@ziusudra970 6 лет назад

HEY VSAUCE 3BLUE1BROWN HERE

@MegaMoh 6 лет назад

No, this is patrick

@190marius 3 года назад

Such a beautiful video! It may inspire a lot of people to understand the problem, and who knows maybe it will inspire the person who is going to demonstrated or negate the Riemann hypothesis. Love from Rumania!

@userprobablynotfound 2 года назад

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 RU-vid 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 😑

@iau 7 лет назад

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 5 лет назад

it can be proved alternately with binomial theorem

@heyman4466 5 лет назад

@@99bits46 Explain

@99bits46 5 лет назад

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

@aashishpathak3628 5 лет назад

It is possible , shown by Ramanuzam sir.

@oneoveronethirtyseven9161 5 лет назад

@@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.

@selimbaydar123 7 лет назад

I literally love you for doing this one !

@josephcoon5809 3 года назад

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.

@greglambert6199 2 года назад

Assuming the reimann hypothesis, this video was so well done. Thank you.

@AndroidT01187 7 лет назад

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 7 лет назад

The plan is to publish the series by early April.

@yaoliu7034 7 лет назад

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

@beefmomma 7 лет назад

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

@Chaos------ 6 лет назад

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

@dankwarmouse6248 4 года назад

Spinoza described that sensation as "intellectual love" :)

@adammehdi4934 2 года назад

A great companion to Roger Penrose's Road to Reality chapters 7 & 8. I came for a visual intuition of analytic continuation, and left so very satisfied.

@rbr1170 2 года назад

This is amazing beautiful!!! I watched it a few times before withour prior knowledge of calculus but know with more idea of its foundations, just wow!

@poisonpotato1 7 лет назад

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

@LKalavera 7 лет назад

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

@3blue1brown 7 лет назад

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 7 лет назад

I trust you! Can't wait :)

@stevearmstrong7023 5 лет назад

Agrre please provide

@nahidhkurdi6740 5 лет назад

Much more difficult to animate, I think.

@Hi-6969 3 года назад

@@3blue1brown Still on your list, right?

@EWischan Год назад

I want to give this an additional thumbs up every time I watch it. It's so good.

@demerion 2 года назад

I think this is the second time I've watched this video (RU-vid says I didn't watch it before, but that's unreliable). The first time was a few years ago when I randomly stumbled upon the topic. I didn't really understand what was happening but found it interesting. Today I watched it again and somehow understood the underlying concepts and I now have a good grasp of what the Riemann hypothesis is. I also understand what complex continuation does, though I don't understand how. I'm sure that I will come back to this and understand even more in the future.

@archivaltransfer 5 лет назад

love it, reminds me of how Smith derived the chart for capacitive reactance and induction

@jacksonstarky8288 4 года назад

This was the video that pulled me into your channel... because Riemann. I haven't taken a formal mathematics course since 1992, but I've always loved mathematics recreationally (thanks in large part to Martin Gardner). I'm still pondering the role of gamma (the Euler-Mascheroni constant) in the cutoff of the non-continued zeta function; I know it's connected through the gamma function that I still can't wrap my brain around (probably because you haven't done a video on the gamma function or L-functions yet). Trying to be patient... but your videos have spoiled me, and Wikipedia is no longer an adequate math teacher.

@j.503 3 года назад

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

@michaelmahoney5677 4 года назад

My goodness, a video that actually explains this properly. Thank you!

@nujuat 7 лет назад

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

@mrping2603 7 лет назад

This channel definitely needs more subscribers

@emilyalice948 3 года назад

I learn maths as a hobby and your videos inspire me to start along new paths but its not until studying the material in full that I can truly appreciate the additional dimension of richness and lucidity your visual intuition brings, forever grateful thank you 🙏

@diogomendonca5821 3 года назад

I love how I always feel geeking out in these videos.

@dAvrilthebear 4 года назад

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

@Fernito 6 лет назад

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 5 лет назад

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 5 лет назад

Same here at 4.30 Am

@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

@davidwagner6116 2 года назад

Yay for complex analysis! Thanks 3blue1brown!

@Viewer2812 3 года назад

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

@Lesideesfroides 6 лет назад

Amazing as always

@basuam 6 лет назад

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

@amandaroberts6535 9 месяцев назад

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.