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.
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.
+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".
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.
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.
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.
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. 😊
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 :)
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.
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.
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!
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...
@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.
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
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!
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
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!
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).
@@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.
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.
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 :)
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 ☺
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.
"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)
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!
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.
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.
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.
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!
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.
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!!
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.
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.
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!
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
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
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
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!
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 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.
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.
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.
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 :)
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.
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).
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
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.
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.
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.
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.
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 ... :)
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 😕
I love how obviously passionate about math the narrator is in all of these videos, but I gotta be honest i only watch this for the animations (and the soothing voice). I can’t even begin to understand how one animates this stuff.
I would like to say that this video got me into math as a whole not just as a tool. I already enjoyed math but this has made me study way more math than i ever thought I would. thank you.
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.
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
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.
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 😑
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.
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!
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!!!
Am not a mathematician and left it way back in engineering and never appreciated calculus beyond having to solve for it on exams. This visualization of Math and Zeta function was amazing and very addictive. Now am on a total binge watching of math videos at the cost of my work :) Seriously, if Math was taught this way, probably would have chosen a different line to work on! Great work breaking this down to normal folks.
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 🙏
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
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.
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!
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
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.
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.
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.
+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".
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.
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.
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.
"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.
So true. :)
Stuart Smith I'm waiting for a 2 hours video so badly, popcorn and notebook in hand, I await
same, but he does a lot anyway
I'd watch it as well. But I think that most people wouldn't and also "youtube's algorithm" might not favor it.
Yeah, I'd be happy if he makes a feature-length about some journey into a mathematical concept.
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. 😊
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 :)
This guy should get a million dollars for making math intuitive and incredibly interesting.
he does
He makes it easy for me to understand I love it.
p
Every year
I'm mind blown by the fact that this is FREE to watch for EVERYONE. Grant is truly making the world a better place.
Pay him then!
@@dann5480
It's free!
What does "free" mean? Or does your mother pay for Internet access?
@@ИмяФамилия-е7р6и no, your mom
@@ИмяФамилия-е7р6и he is using star li k free wifi i suppose but for me mobile data still very zxpensive
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.
I understand it but I'm still quite broke lol.
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.
Could you solve it?
@@58585050 tenho certeza que sim, americano!
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!
No you're not.
@@dann5480Yeah, I don't think any real mathematician would go around screaming 'Im a mathematician, I like this video, now gimme attention"
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...
Imagine if Riemann can see this beautiful explanation and animation of his function.
@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.
He probably saw this and more in his head.
I'm sure Riemann's imagination showed him far, far more than whatever Grant can possibly animate
@@ArnavBarbaad Perfect!
Who knows... maybe, we all get to live multiple lifes...
Please do the video about
relation between zeta function and prime pattern
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
@@suayhossien Then you should know about this already :D
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...
there's a great book about that (and more) called Prime Obsession
@hvoya audio By John Derbyshire?
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!
its 2022 you've proved it now right
@@Ganerrr Now it's 2023 and he's single-handedly solved all of the Millenium problems
@@harry_page He's him
8:58 - "focus on one of the marked points"
Me: *focuses on (1 + 0i)*
*facepalms*
I'm dead ahahahhahahahahahahahah
i laughed so hard lol
LOL
@@espositogregory chill
@@espositogregory chill
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. :)
elementary math teacher, I hope?
in other words, an arithmetic teacher?
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
This is easily becoming one of my favorite channels on RUclips, thank you so much for all the videos
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!
Same
Guess you don't know calculus then.
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).
@@EebstertheGreat thank you so much. This is hard for me to understand, but your explanation does help clairfy!
@@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.
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.
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 :)
This video is so good, that I've spent more than one hour online learning about the complex plane for first time
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 ☺
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.
"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)
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!
+kjpmi Wow, thanks for sharing. This was very motivating to read.
+
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.
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.
You make me fall in love with mathematics, if one day I become a mathematician, you’re my mentor and inspiration! Respect from China.
go for it mike mathematics is the key to the universe after all
its been 2 years, did you do it?
@@klaus9356 I did a math minor along with my physics degree ☺️
@@mikesu8475 thats great. hopefully I will start my math career in university in just 2 years
@@mikesu8475 how old are you now!?
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.
Love it! Thanks for the hard work put on these videos. This is how math should be first approached.
UnPuntoCircular x. T
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!
"Damn" is hardly a swear word
UnPuntoCircular ענפיץמנךמך
יליכעעחעילל
a 22 minute 3blue1brown video??? Yes please.
This is so neat but it’s 4:30 am I can’t help
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.
Its 1am on Christmas morning rn youre right this is so neat
It's 4:27am here
@@nsambataufeeq1748 6:26 am here
Cheese cat
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!!
I love it when I can learn things that aren't being taught at school! Thank you for making this!
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.
10:06 I need a T-shirt of that pi-creature saying 'damn!'
Yeah👍
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.
David Messer the Fermat memes
David Messer then publish it
Lawl
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 Messer I believe you bro.
This has so many layers of abstractness. I love it.
This is so good looking ! Your animations give a new perspective to math and reveal its beauty. That's awesome !
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!
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
Thanks VSauce for introducing me to you, your content gets over my head relatively quickly, but I'm so fascinated. Thanks man
Noah Price This is not Vsauce
Read his comment again Madara
Noah price same here
HEY VSAUCE 3BLUE1BROWN HERE
No, this is patrick
All I understood was why who ever solves this gets a million dollars
All I understood is I wont be getting a million dollars.
All I understood was that this will be the hardest way to get a million dollars
@@philosophicalinquirer312 With this attitude you certainly won't.
*You guys are getting Understood!!*
Because a group with a million dollars said "I'll give a million dollars to whomever solves this"
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.
understanding it is way more beautiful and mind blowing than that million dollar prize!!
Man, y'all are the best, motivated by actually learning what's going on more than "hook" of talking about the Millenium prize problems.
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
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
Yeah but you get both. . .
Yash Pandey Isnt it though???
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!
I still come back and watch this regularly. One of the best explainer videos ever made.
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.
it can be proved alternately with binomial theorem
@@99bits46 Explain
@@heyman4466 sorry i meant binomial formula. See ramanujan's original proof for this sum
It is possible , shown by Ramanuzam sir.
@@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.
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.
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.
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 :)
A video on Reimann-Roch would be quite something.
Algebraic manifolds
What about schemes?
I hv watched many videos about Riemann hypothesis, this is the best to show how analytic continuation works.
It's if you can PROVE or DISPROVE the hypothesis. You get the money for both and both results would be equally ground breaking.
so we know the hypothesis but we don't have a formal proof for it?
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.
what is the problem?
Right, but it probably IS true...
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).
Please do a series on complex analysis!! Your videos are the best!
Thank you for doing great work in promoting advanced mathematics. The world needs more people like you, Sir.
incredible, breath-taking, so accurate, colorful and concise
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
If every teacher/professor could explain subjects this well, we'd already be harvesting energy from black holes and teleporting to mars. 110% respect!
And at its converse, bad teachers do more harm to learning and understanding than most other things.
It doesn’t make much sense to agree with someone more than 100%, but maybe we could analytically extend the definition of agreement 😅
analytically continue percentages baby@@rathorefamily
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!!
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.
The plan is to publish the series by early April.
that sounds very interesting. i wish i could join your team :)
3Blue1Brown please do more millennium prize problems! this video was fantastic
Your way of communicating complex ideas is just amazing
PLEASE make the prime number part of this. Thanks a lot.
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.
I trust you! Can't wait :)
Agrre please provide
Much more difficult to animate, I think.
@@3blue1brown Still on your list, right?
this makes my mind tingle in the most pleasant way imaginable.
Spinoza described that sensation as "intellectual love" :)
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.
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.
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!
Same here at 4.30 Am
@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
Thank you! Came for -1/12, stayed for the Rieman Zeta Function
You make difficult Mathematics topics interesting and accessible to a non-mathematician. That shouldn't be possible. Thanks.
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
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
Well noticed
I feel really bad for only commenting this after watching a beautiful visualisation, and explination of the Riemann Zeta Function.
it's the PI conspiracy! 3 integers 1+1+1 and 1 Rest 0,141...
Well noticed, I was wondering why this name too.
Now you noticed it, is it related to the movie "Pay it forward" ? :D
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 ... :)
Best video I have seen on Riemann Hypothesis ever
I think this is the first time I've seen an explanation of this that actually makes sense.
I literally love you for doing this one !
It was really easy to understand when you showed how the points of the grid map on to their new points.
20:30 Markus Persson is a 3Blue1Brown Patreon support. Now we know that Notch(the creator of Minecraft) likes math!
ThatMathNerd עעעעעעעעעעעעעעעעעע עיעעעעעעי יייייייייייייייייייייייייייייייעעיייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייייי
Wooooooot??!! Mind blows up
Hunh interesting
So you're saying someone who's a pretty good coder likes math? How suprising!
@@jakistam1000 considering the solidity of Minecraft's coding, it kinda doesn't surprise me
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 😕
I love how obviously passionate about math the narrator is in all of these videos, but I gotta be honest i only watch this for the animations (and the soothing voice). I can’t even begin to understand how one animates this stuff.
My eyes became moist after watching this.
This is so beautiful!
Ashwin Kumar right 😢
stfu
My conjecture:
The chance that the Riemann hypothesis is true is bigger than the chance me proving it.
This is tautological by the laws of statistics. P(A|B) >= P(B). Your conjecture is now a theorem. :)
On the third try of watching this video with several months spans I finally understood everything.
This was awesome
So you're about to be $1,000,000 ricer too~
There's a secret part of this video. You just need to analytically continue it to the negative timestamps...
I would like to say that this video got me into math as a whole not just as a tool. I already enjoyed math but this has made me study way more math than i ever thought I would. thank you.
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.
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
emmmm..
Feynman?
@@ИмяФамилия-е7р6иFeynman was a physicist
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.
"After the transformation, the lined make such lovely arcs before they abruptly stop. Don't you just want to... Continue those arcs?"
Ok, this is the best explanation of the Reimann hypothesis that I've seen on RUclips. In particular the explanation of Analytic Continuation. Cheers!
17:40 the longest running joke in mathematics
?
Where was the joke?
Proof is trivial and left as an excercise to the reader
T_h_e_o exactly if you take a math course the teacher will always just say this is trivial let’s move on 😂
It is often said that everything a mathematician can prove is trivial. Because everything they prove becomes trivial.
You make my brain hurt so good.
rasmus hansen 😂😂😂😂
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 😑
I came here because a dude said he has proof, I understand nothing but seeing someone can prove it feels awesome
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.
arcuesfanatic any updates on this?
@ It didn't hold up. It's still an open question
Erin Strickland thanks
It could've been quite the shock for the proof to actually be correct :)
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?
He is not vsauce bruh😂
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!
How did you learn to make your graphics? Your style of presentation is very nice.
Secret of channel)
He's a programmer. All of the visualization is a product of his programming skills.
manim, the software's called manim and it's available in GitHub, although it's quite confusing to use (imo)
It's on github, but the documentation isn't that great
@@benharris3100 Better late than never ;)
Hey, this shit really deserves more views. Amazing visualizations. Amazing video. Amazing explanation.
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!!!
Am not a mathematician and left it way back in engineering and never appreciated calculus beyond having to solve for it on exams. This visualization of Math and Zeta function was amazing and very addictive. Now am on a total binge watching of math videos at the cost of my work :) Seriously, if Math was taught this way, probably would have chosen a different line to work on! Great work breaking this down to normal folks.
17:44 the best definition of the "trivial solution" I've ever heard
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 🙏
Thanks a lot for all of the videos you've made!! They are all exceptional!Once again, THANK YOU SOOOOOOO MUCH!!!
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
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.
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!
Awesome video. Wchich computer programme can visualise complex graph?
"Pretty much any function with a name is analytic." Absolute value would like to speak with you.
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
Animations are absolutly great ! I feel I can "see" analytic functions as I never have!
I want to give this an additional thumbs up every time I watch it. It's so good.
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.