i am but a simple man i see video about Ramanujan I click

@Mathologer

10 ай бұрын

That's definitely the way to go.

@Hi-Phi

10 ай бұрын

Your user name reaffirms this.

@1stlullaby484

10 ай бұрын

My man you're old !! So ancient!

Ramanujan is my favorite. He saw things in a way nobody else ever has. Maybe someday we will have another like him. I love his work it seems so natural yet is so deep.

@cv507

7 ай бұрын

vvhöö säß he´$ -8önn-? ? his indiän mäFF titchäiR xD make jökes the löck släyce v v

@warguy6474

5 ай бұрын

idk about natural 😅

I am just amazed by how Ramanujan's mind worked.

@walterrutherford8321

5 ай бұрын

There are people in history (and a few undoubtedly alive today) whose minds work at a freakishly different level: Archimedes, Euler, Gauss, Ramanujan, Escher, Tesla, …

@PC_Simo

2 ай бұрын

@@walterrutherford8321 Indeed. Also, Stephen Hawking.

Wow just the first 2 minutes of introduction is just amazing, what a great mind Ramanujan had, and beautifully explained professor!

@thej3799

10 ай бұрын

True that I'm figuring stuff out now and it's just like whoa I mean I get to say that it's part of my generation it's just like whoa you understand this guy figured this stuff out just one day I mean I did the same thing but but he already did it so it's not quite the same because maybe part of the collective conscious help me do it but the collective conscious wasn't there yet and this guy just fucking I don't know just pulled it out of the fucking chaos and made it order for us which is amazing and transcended and awesome and beautiful and everything that I love about the human species that I hope I absolutely hope we figure out how to save we figure out how to get along on this beautiful Earth if you sweep away the grime and the dirt the beautiful Earth is under there the green below and the blue skies above it's all there and we can just sweep it away you understand ✨ The mathologer said it exactly right this man is one of the smartest people that ever lived you could put them up with sir Isaac Newton. Two the smartest people that have ever probably existed on this earth at least that we know of My mom was smarter than me and my sister was the math genius of the family. I wish she was still here because who knows what she could have done by now she would have been 39. No offense but she probably would have had a KZread channel she was one of the first people that showed me how smartphone worked and how you could have KZread on a mobile we were at a beach and we were watching afroman videos but like she had the G2 the one with the flip out keyboard she was so smart I miss her so much and I guarantee you if you think I've come up with anything my sister I walked in one time when she was doing homework as a senior in high school she was in calculus and she was doing it in her head figuring out the answers and then showing her work afterwards and I'm the only person in this world that knows about this and someday I will tell her kids how damn smart their mom was. Even if that is my life purpose I hope they seek me out. Brenden and Trenton find me someday and I'm going to tell you how awesome your mom was I promise you I'm the only person that knows your grandfather is going to tell you a bunch of bullshit I promise you my sister your mother was sublime and God damn she was smart smarter than I ever was hell she might have been able to hang with the mathologer without having to Google a whole bunch of stuff like I do. ✨

@sonarbangla8711

10 ай бұрын

Ramanujan had many identities with exponential functions, burying Euler's famous identity.

I had worked on this problem in my pre-university days. Though I eventually had to see the solution out (couldn’t solve it), it was my first deep exposure to Ramanujam’s mind and mathematical thinking. (The first time mathematical connect was obviously the introduction to limits concept). That was when I truly understood why he is called “the man who knew infinity”. No wonder, one of the greatest son of Indian soil 🙇

8:30 You can still use the exponential function in the following way: Let y = ce^f(x) y' = f'(x)ce^f(x) Therefore f'(x) = x and f(x) = ½x² + const So in the case of y' = xy we have y = ce^½x²

@simonmultiverse6349

10 ай бұрын

Those three chords played at the start sound EXACTLY like the first few seconds of Babooshka by Kate Bush.

@maxprofane

10 ай бұрын

​@@simonmultiverse6349Wth are you talking about? Well a few seconds later I've heard what you were talking about. Good catch!

@robaxhossain5653

8 ай бұрын

I also like to comment like you. Sometimes, he shows very immature style but truly genius. Some methods he shows is just copy paste. I never see his own style. Undoubtedly, Ramanujan developed his own method to solve series problem. That is why it was pretty difficult to capture the then mathematician.

Continued fractions are truly amazing and, for most people, mysterious mathematical objects. This is really a pity, because just as the natural base for logarithms is e rather than 10, and the natural measure of an angle is radians rather than degrees, the most natural representation of a real number, in a sense, is a continued fraction.

@toniokettner4821

10 ай бұрын

i don't think so. real numbers are not meant to be written down by their construction. they are incountably infinite, so humanity can only ever be able to write down countably many real numbers. but in application, humans don't care about the exact value. and the scientific notation does approximate numbers perfectly.

@lexyeevee

10 ай бұрын

if we're going that deep, arguably the natural measure of an angle might be its cosine

@RobinHillyard

10 ай бұрын

@@toniokettner4821 With all due respect, I do think that you're missing the point. phi (the golden ratio) for example can be written in continued fraction form with just one number: 1. Admittedly, that has another perfectly good precise representation.

@Sameer_S_Kulkarni

10 ай бұрын

@@RobinHillyard I think that's just one example you can provide for that side of the argument. Looking at the overall picture, the scientific decimal notation is more useful as well as aesthetic to look at, in most cases, compared to continued fraction representation. For example, multiplication using decimal notation is arguably simpler than using continued fraction representation...

@toniokettner4821

10 ай бұрын

@@RobinHillyard only very special numbers have a nice continued fracrion representation. mainly roots of integers

Am I the only one to notice the infinite sum on the last slide is wrong? It has the product of natural numbers in the denominator instead of just odd numbers. And it includes even powers of x as well.

@thej3799

10 ай бұрын

It depends on if you're counting spaces or integers

Absolutely amazing. I did not want the video to end!

@Mathologer

10 ай бұрын

Thank you very much :)

This is the most mind boggling video ever made on this equation. and definitely a masterclass. How Ramanujan thought this is simply impossible to fathom. It appears that when it comes to Maths first comes god and then comes Ramanujan.

@imtiazmohammad9548

10 ай бұрын

​@sarcastic_math343Neumann cannot even come close to Ramanujan

You make complicated-looking maths problems sound so easy! Please keep up the great work!

Best one Ramanujan's explanation I have never seen before . Awesome MATOHLOGER .

It would be nice to see a follow up filling in the details that were left on the table at the end of the video. A video on the connection between continued fractions, cutting sequences, and trajectories of billiard tables could also be a fun "spiritual successor". ;)

This is one of the clearest mathematical expositions I have ever heard.

Im a math failure. I'm here for your t-shirts 😅

@mrboombastic_69420

9 ай бұрын

No one is a math failure, only people who were not taught some small but critical math rules or ideas

@1stlullaby484

9 ай бұрын

​@@mrboombastic_69420😂bro you said 'only people...' i get u didn't mean that but 😂

@1stlullaby484

9 ай бұрын

If your definition of a math failure is failing in a math test then I'm too(failed in my mid term, 8th grade) , but I've graduated with math and now will be going for further studies because it's a really really good subject

@1stlullaby484

9 ай бұрын

On a different note, the person who commented above is at the least partially correct. Because it's always either we weren't taught the right way or our own fault for ignoring it or our studies. It's not a big deal, it's common we do sometimes neglect our studies unless you're a complete nerd. So, you were never a failure, it's just you not seeing the other way around (meaning you don't feel or think that you can turn it around)

@PC_Simo

7 ай бұрын

⁠@@1stlullaby484 What’s funny about ”only people…”? What am I missing, here? 🤔😅

Encore une vidéo prouvant le génie de Ramanujan. Démonstration hallucinante !!!! Bravo, comme d'habitude. Les vidéos mathématiques les plus géniales du net.

Another truly amazing video! Ramanujan was amazing, and so are you!

It has been nearly a decade since I sat in a maths classroom and it wasn't until I began to watch your videos that I realized how much I miss it.

Thank you for another amazing exposition! I had seen this identity a long time ago and always wondered how it could be derived. Totally agree that it is just a beautiful identity to look at! Ps. I noticed that the yellow integral identity holds for negative x as well. Fascinating that the error function has this sort of continued fraction expansion.

@Mathologer

10 ай бұрын

Glad you enjoyed this explanation :)

You know what is really magical? You have made your instruction and animation line up, to almost create a gamified experience. This was wonderfully engaging and explantory, well done! Mathematical Tetris :)

Your combination of complex theory, with a simple start, great graphics and robust thought was a true work of art here, very well done. I really appreciated the disclaimer section at the end as the needed warning of not being too glib.

Dear Professor, another shining explanation about a brilliant gem. Thanks a lot. Again: this is my favourite channel EVER!

Fantastic video! So incredibly clear how this all is derived, step by step, from some simpler warm up expressions. Great job!!!!

What an amazing identity. A really great video as well, breaking it all down in a very digestible way, thank you!

So many nice reminders of maths I haven't thought about in a while. Great pedagogical approach.

E ceva fenomenal , unii oameni se nasc geniali , a fost ceva deosebit acest film extraordinar -Multumesc mult Maestre -Romania!

what an incredible journey of revelations. I truly appreciate your work in showcase these amazing feats.

The animations of the equations are so perfect to illustrate things

ANOTHER MASTERPIECE LETS GO!

Thank you so much for your wonderful and inspiring videos!

21:40 Decompose both expressions as products of fractions by pairing each term in the numerator with the term below it in the numerator. On the right, you have that each fraction is > 1, so their product will always grow. On the left, however, they alternate between > 1 and < 1, so it's at least possible for it to converge.

@Mathologer

10 ай бұрын

That's it :)

@dylan7476

10 ай бұрын

If you group them in pairs though you get n*n/[(n-1)(n+1)] = n^2/(n^2-1), which is always > 1. Why does this not suggest the left fraction grows infinitely?

@Sameer_S_Kulkarni

10 ай бұрын

@@dylan7476 Exactly! I too had the same question. Can someone answer this?

@det1729

10 ай бұрын

@@Sameer_S_Kulkarni You can bound (2n)^2/((2n)^2 - 1) by 1+1/n^2 which can be further bounded by e^(1/n^2). This reduces the product into a sum, and since the sum of reciprocals of squares converges, you're happy :3 For the other product, you can write that as (1+1/1)(1+1/3)(1+1/5)... and now it's easy to see that 1/1+1/3+1/5... is a lower bound.

@shoam2103

10 ай бұрын

@@Sameer_S_Kulkarni there's always a mathologer video for it ;) But I'm not sure which particular one. Have to look it up..

On Ramanujan's channel this video is 20 seconds long and the explanation consists of him saying "I saw that this identity must be true"

For the challenge near the end related to the Wallis product, if you simply eliminate the 1 in the denominator then each factor is < 1 converging to 1, meaning the overall product is finite.

@gcewing

10 ай бұрын

It's finite, yes, but it's not what we want! The first factor is 2/3, which is already smaller than sqrt(pi/2), and it can only get smaller from there. In fact, my experiments suggest that it converges to 0.

@Ranoake

9 ай бұрын

You could do the same for the other product too, but it apparently does not converge anyway. Funny how adding 1s changes nothing even though it seems like it should. It all comes down to how to write the product, what is the general term? That determines if the 1s should stay or not, removing them is not allowed, it changes the terms of the product, and hence the overall value.

Is anybody else more excited about Mathologer videos than Hollywood videos. I can’t explain the excitement I feel when I get the notification.

Wow fascinated to see an incredible math art work by the god mathematician Ramanujan, explained by another great mathematician who simplifies all math to easy understanding.

Fantastic! Amazing. The last bit left unanswered questions, as intended. Everything else was clear.

Whoa, getting name-checked in a Mathologer video! Achievement unlocked!

@Mathologer

10 ай бұрын

Congratulations on coming up with that nice way of getting the fraction (and on unlocking another achievement level :)

Beautiful 😊 what a mind Ramanujan had

Wie immer super!

I'm just stoked that I immediately recognized the intro as Ramanujan. Not because I recognized the identity, or am familiar with the math, but simply because any time Mathologer breaks out the continued fractions, it's a one-way ticket to Ramanujan-town.

Wow, that knocked my socks off Professor Polster! Beautiful beyond belief! How can I get back to work now with this spinning in my head! What I need is some tea and some just sitting stupefied, savoring the aftertaste.

Your videos are greatly awaited and they are always worth waiting for. Your videos always generate love for Mathematics. I wish I had a teacher like you in the school days. Lots of love and respect to you. Always ❤

@wonderbars36

10 ай бұрын

For real. Great way to show so many topics in this one. The diff eq. in here was great and so much more accessible. Now I get what it can do a lot more clearly.

@Mathologer

10 ай бұрын

Glad you are enjoying these videos so much :)

trulyyyy magic, brilliant!!

Beautiful as always

I love how some of the transitions make the image of Ramanujan smile.

@Mathuyius

10 ай бұрын

that was so corny though

Absolutely loved the video.

Thank you for delivering so nice content, which helps more people experience the beauty of mathematics.

Love me some mathologer masterclasses... still waiting on that Kurosawa length Galois theory video :)

11:00 Also; √2 is also a component, in the left side of Ramanujan’s identity; it’s just in the denominator. The left side is, basically, just: (√π*√e)/√2. 🤔

Brilliant!! I am hooked on your math.

Thanks Mathologer.. what a mind Ramanujan had..

Wow 😮 Beautiful and very ingenious ❤

Ramanujan obtained a general expression using three variables that can generate an infinite number of such equations, like the ones sent in by some readers. Just define f(x) = x + n + a, giving f(x)2= ax + (n + a)2 + x f(x + n). Now you can carry out the same recursive trick we did above for any values of x, n and a. Our example above is obtained if we set a = 0, n = 1 and x = 2. Have fun generating your own infinite nested radicals!

@Sameer_S_Kulkarni

10 ай бұрын

What is f(x)2? Is it the derivative or the function squared?

@1AdityaSingh

10 ай бұрын

​@@Sameer_S_KulkarniThe notation f^2(x) can have different meanings depending on the context and the convention used by the author. Here are three possible interpretations: 1-) f^2(x) means (f(x))^2, i.e., the function is evaluated at x, squared. This is the most common interpretation of this notation. 2-) f^2(x) means f(f(x)), i.e., the function f is applied twice to x. This is less common, but it can be used in some contexts, especially in functional analysis and dynamical systems. 3-) f^2(x) means the second derivative of f(x), i.e., f''(x). This is a less likely interpretation, but it can be used in some contexts, especially in differential equations and calculus. To avoid ambiguity, it's always best to check the context and the conventions used by the author to determine the intended meaning of f^2(x).

@shoam2103

10 ай бұрын

@@1AdityaSingh But what did *you* mean in *this particular case*?

@shoam2103

10 ай бұрын

@@1AdityaSingh Or in other words, did you mean the derivative, squaring, repeated function application? Or even all of the above or a combination there of?

@Sameer_S_Kulkarni

10 ай бұрын

@@1AdityaSingh You are the author Aditya. You should know the context and the convention. I meant what did YOU mean in your comment. You suddenly started answering like chat GPT. That was hilarious 😂

Honestly it blows my mind when I see these mathematicians from India generated some of the most popular integrations, limits and whatnot with sheer simplicity yet mysteriously and seamlessly embedded them either in a poem or a mantra or a prose so not only the willing one is able to synthesize the literature written but able to practically implement the math encrypted in it. Even more interesting is that these mathematicians always dedicated their discoveries to God and let the discovery have an open access for all irrespective of their background without claiming to be the founder of the said discovery; precisely why I am convinced to believe why several of their discoveries that we today are studying/ using don't bear their names.

This video is a great journey 👍

One of the channels for which I will always give a thumbs up, even before watching 😅

Amazing video mathologer as usual. I really liked the tiny bits of sneak manipulations with calculus in the video.

@Mathologer

10 ай бұрын

Glad you liked it!

Ramanujan is my favorite. I simply cannot begin to comprehend how his mind worked.

Wow, I love this mix of algebra, calculus and clever manipulations ♥️ For me the most beautiful part of the derivation was 1/1/2/3/4/...=sqrt(pi/2) Thank you for your work 👌

I love your videos, thank you!!!

I've missed your videos for so long, good to see the king talk about another king of math..

@Mathologer

10 ай бұрын

Wow, thanks

Easily one of the top three math shirts of the channel right here.

Marvelous & lovely!

you are the best math youtuber keep going pls

You are awesome. Your explanations are always very good. 😄

Another great video! I always tell the class about your channel and Ramanujan

@Mathologer

10 ай бұрын

Thanks for sharing!!

@drgatsis

10 ай бұрын

@@Mathologer you're most welcome!

This is so astonishingly beautiful 😍

Nice video. Ramanujan is my favourite mathematician.

Just amazone, more on ramanujan pls

So brilliant~

Great video. The most amazing thing is that the root of pi divided by 2 can be represented as the sum of two numbers. An amazing result considering it is obtained from a normal distribution. Thank you, something to think about. Thanks again for the video :).

Please make more of such contents

Beautiful, Beautiful, Beatiful !!!!!

I think this is one of your best videos! It was quite a drama filled with lots of "aha!" moments, but also making sure to watch out for any sneaky moves (knowing that the Wallis Product is "conditionally convergent" primed me for the big reveal that things weren't quite what they seemed with taking the square root of it). I would say that the fantastic fractions segment was my favorite part. :)

@Mathologer

10 ай бұрын

Have not heard from you for a while :) Glad you enjoyed this video and thank you very much for your continuing help with answering questions. I also think this video worked out very well. By the looks of it, not a video that will be hugely popular. Still very much worth doing.

@MuffinsAPlenty

10 ай бұрын

@@Mathologer It's unfortunate that it's not super popular! And yes, this past academic year was quite busy. But hopefully this next year will be easier (the lie we all tell ourselves every year, right?)

Ramanujan was an amazing genius. Love from Sweden💛💙

@imtiazmohammad9548

10 ай бұрын

Yes, smarter than Euler

Long waited 😊

always great to have your mind blown.

It was Douglas R. Hofstadter's GEB that introduced me (and I guess many of us) to this fascinating man from India, while the intriguing man from Germany quasi introduced himself, through these nonpareil YT videos of his.

I ❤ MASTERCLASS😊

17:31 Slight mistake, the sum is supposed to be x/1 + x^3/1.3 + x^5/1.3.5 and so on.

@Mathologer

10 ай бұрын

The cut-and-paste autopilot demon strikes again :(

Another wonderful video.

Pretty cool how something this crazy is understandable with just a little bit of calculus and a couple of prior results.

Gorgeous🙏🏻

Mind blowing

Omg i just realized,,,uve been posting in the exact same style for 8 yrs👏👏

Great video. Although wasn't this fraction in particular discovered by Laplace and proved by Jacobi. Of course, Ramanujan os a genius to have rediscovered it all by himself, but I was really hoping we'd get more about similar fractions. Digging deeper I found a book by S Khrushchev which discusses a whole theory of continued fractions like these along with great and largely unknown work done by Euler in this field. I think it can be found online as a pdf.

@1stlullaby484

10 ай бұрын

Send me link please I would like to read as well

@leonhardeuler7647

10 ай бұрын

@@1stlullaby484 Sorry can't provide the link, but if you search orthogonal polynomials form Euler's point of view pdf, I think you'll find it online. If not, I'll try adding the link.

@Mathologer

10 ай бұрын

The book by Khushchev is great. Also have a look at my notes in the description of this video :)

Definitely another interesting video. Not only maths part was interesting, even the music at the end was very soothing too.

@Mathologer

10 ай бұрын

Glad you enjoyed it!

It's so beautiful.

absolutly genius

Amazing

Even without baby calculus, I have watched you enough, @Mathologer, to be able to keep up with how this works -- Danke sehr für das Video!

Cool!

Always good to see a new Mathologer video! Nice shoutout to my old friend John Baez.

@Mathologer

10 ай бұрын

I've been following his blog for years :)

After Euler he his the next Hero for math's

y'know, the warm up puzzle i actually once thought of in like 8th grade in geography class cause i was bored, but i had no knowledge of calculus, so lets just say it stumped me for a while (until i aproximated and than guessed e-1 cause you know, e is pretty famous), so seeing it as a warm up puzzle in this video made me feel a bit nostalgic, so thanks i guess

Neat!

Never imagined that there might be a relationship between calculus and continuous fractions🤯

You got me into continued fractions Mathologer. Now I have published work on folding continued fractions.

@Mathologer

10 ай бұрын

Folding continuous fractions. That sounds interesting :)

The animation during 'autopilot' mode is amazing! If I may: What software/automation stuff do you use?

After some bit long time ❤ waiting jusst for vedio