How did Ramanujan solve the STRAND puzzle?

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Күн бұрын

Today's video is about making sense of an infinite fraction that pops up in an anecdote about the mathematical genius Srinivasa Ramanujan.
00:00 Intro
04:31 Chapter 1: Getting a feel for the puzzle
08:27 Chapter 2: Algebra autopilot
12:37 Chapter 3: Infinite fraction
17:51 Chapter 4: Root 2
21:19 Chapter 5: Euclidean algorithm
30:15 Chapter 6: The best of the best: 17/12
36:34 Chapter 7: Outramanujing Ramanujan
This was supposed to be a short video but in the end turned out to be quite a tricky to sort out. Anyway, as it sometimes happens, I got carried away and now the video really covers a lot of ground : Pell equations, visualising continued fractions by dissecting rectangles into squares, the relationship between continued fractions and the Euclidean algorithm, the irrationality of root 2. Overall quite a few things that you won't find anywhere else :)
The way I tell the anecdote in this video is based on the following account by Ramanujan's friend Prasanta Mahalanobis: Current Science, Vol. 9 (3), pp. 74-75.
"On another occasion, I went to his room to have lunch with him. The First World War had started some time ago. I had in my hand a copy of the monthly Strand Magazine which at that time used to publish a number of puzzles to be solved by the readers. Ramanujan was stirring something in a pan over the fire for our lunch. I was sitting near the table, turning over the pages of the Strand Magazine. I got interested in a problem involving a relation between two numbers. I have forgotten the details but I remember the type of the problem. Two British officers had been billeted in Paris in two different houses in a long street; the two numbers of these houses were related in a special way; the problem was to find out the two numbers. It was not at all difficult; I got the solution in a few minutes by trial and error. In a joking way, I told Ramanujan, 'Now here is a problem for you'. He said, 'What problem, tell me', and went on stirring the pan. I read out the question from the Strand Magazine. He promptly answered 'Please take down the solution' and dictated a continued fraction. The first term was the solution which I had obtained. Each successive term represented successive solutions for the same type of relation between two numbers, as the number of houses in the street would increase indefinitely. I was amazed and I asked him how he got the solution in a flash. He said, 'Immediately I heard the problem it was clear that the solution should obviously be a continued fraction; I then thought, which continued fraction? And the answer came to my mind. It was just as simple as this.' "
There is a complete digital archive of The Strand magazine. You can find the page with the puzzle here: tinyurl.com/y2lnb8xf (page 790)
If you read the puzzle in the Strand you'll find that the problem is actually phrased somewhat differently to what Mahalanobis remembers and Mahalanobis also does not spell out the infinite fraction that Ramanujan came up with. And if you do the math(s) some of the other things he says also don't quite sound right. What I am presenting in this video is my best guess for what really happened.
In particular, the continued fraction that I am talking about in video is probably the most natural candidate for Ramanujan's infinite fraction, but others have argued that it could have been a different continued fraction (which I don't buy :) You can find these other infinite fractions here: 'Ramanujan's Continued Fraction for a Puzzle" by Poo-Sung Park tinyurl.com/yyfdscgr and here 'On Ramanujan, continued fractions and an interesting street number' by John Butcher tinyurl.com/yy6nv2yg
Solution to the red cross puzzle from Dudeney's book "Amusements in Mathematics" p. 168 :) imgur.com/a/bBuLOZN
Another interesting way to systematically search for solutions to the Strand puzzle is this: The equation we want to solve is 2 x^2=y^2+y. You can rewrite this as x^2 = y(y+1)/2. The formula on the right is just the formula for 1+2+3+...+y. So just keep adding 1+2+3+... and at every step check whether the number you get is a square ... :)
Other short formulas: 1) Expanding (1+√ 2)^n gives a number a+b√2. Then a/b is the nth partial fraction. 2) Play with powers of the matrix {{2, 1}, {1, 0}}
Some number Easter eggs are hiding on this slide • How did Ramanujan solv... :)
Link to the unlisted Marching Squares video: • Root 2 and the deadly ...
Here is a version of the t-shirt I am wearing: tinyurl.com/y5vgo7zb This one is about that other famous Ramanujan anecdote: tinyurl.com/y626c86x actually features prominently in another one of my videos.
The music in this video is by Chris Haugen, Fresh Fallen Snow (playing in the video) and Morning Mandolin (for the credits) and Nate Blaze 'Tis the season, all from the free KZfaq music library
Enjoy!
Burkard
14.9.2021: Thank you very much Michael Didenko for your Russian subtitles.

Пікірлер: 1 600

@hemaroy6439 3 жыл бұрын

Ramanujan was such a great mathematician that on 22nd Dec.,as his birthday is celebrated as national mathematics day in India.

@deborahkeesee7412 2 жыл бұрын

It's also near the average date of the Winter Solstice, which logically (to me) should fall on the *first* day of the year, so we should reset the calendar to make that happen. After all, if a Pope in 1582 can do that why not some actual scientists?? Imagine if the calendar consisted of 4 identical quarters of 30, 30 and 31 days each, adding up to 364 days every year plus one extra to get to 365 and another on Leap Years - how easy would that be! Weeks would start on Monday as most of the world already agrees, and the last one or two extras would be inserted between that last Sunday and the first Monday of the following year so that *every year* would look the same as well as being much more culturally neutral than now. I would think that the scientific and business worlds would love this kind of standardization and predictability even if doesn't appeal to traditionalists.

@mihailmilev9909 Жыл бұрын

@@deborahkeesee7412 huh u actually have a point there

@topilinkala1594 Жыл бұрын

@@deborahkeesee7412 French tried this type of the calendar after the revolution but it did not catch. Too much weight on church.

@tinfoilhomer909 Жыл бұрын

@@deborahkeesee7412 I don't understand why we can't have a 360 day year and just let the stars and seasons slide around.

@DendrocnideMoroides Жыл бұрын

@@tinfoilhomer909 then there is no point in having a year, why can't a day be equal to 20 hours?

@deepanshu_choudhary_ 3 жыл бұрын

Everyone: maths is boring :( Mathsloger : let me take care of it. ;) Btw your videos are very interesting and full of knowledge...... Love from india 🇮🇳❤❤

@phasm42 3 жыл бұрын

Truly the man who knew infinity.

@stoirtap12 2 жыл бұрын

Ramanujan solved the first Strand-type puzzle. Very impressive

@cooperjohn301 Жыл бұрын

was looking for this comment

@lorenzobianchi1896 3 жыл бұрын

Ramanujan is the classic kid that doesn't listen in class, forgets to take notes, does no homework but then FREAKIN' ACES the test because he found his own way of doing things... He will never cease to amaze me!

@user-cv1jb9xv2p 3 жыл бұрын

Sir Ramanujan was very polite and disciplined. He respected elders and the ethics of a place(school, office, neighbour....)

@lorenzobianchi1896 3 жыл бұрын

@@user-cv1jb9xv2p Of course, I meant it as a metaphor, didn't mean to disrespect him. Have a nice day!

@user-cv1jb9xv2p 3 жыл бұрын

I misinterpreted it. The times are wierd now. I staying much on social media, I think that's why it happened. Stay home, stay safe, eat healthy and do riddles.

@shoam2103 3 жыл бұрын

I think that's Einstein? Or not.. He just didn't ace the tests. Ramanujan was just exceptionally good at math, but bad at everything else. His teachers and community recognized it, and had great expectations.. He did *more* homework (his tutors gave him books and materials), took copious notes on his own, etc. So in a way, it's kinda the reverse of our modern day expectations of a brilliant mind.

@lorenzobianchi1896 3 жыл бұрын

@Robert Slackware tell me about it, story of my life!

@rohanshah6178 3 жыл бұрын

The beauty of mathematics lies in the way how seemingly unrelated threads interweave to create the fabric of utmost mathematical elegance. And Mathologer .......you do a great job untangling those threads and making us see and appreciate the beautiful connections lying underneath. Thank you so much.

@magicmulder 2 жыл бұрын

Yup. The most fascinating results are those that connect seemingly unconnected fields, like Taniyama-Shimura.

@adarsh5870 2 жыл бұрын

Only if Ramanujan lived longer we would have had mathematicians who would have had their PhDs with him and how much more he would have inspired the next generation. His intuition in mathematics is Insane its God-like.

@aniket385 Жыл бұрын

A large part of his earlier life was to personally rediscover the maths of 2000 years already done by previous generation due to his poor schooling till he arrived at present time .

@damianflett6360 3 жыл бұрын

>ramanujan answered instantly >takes 40 minute video to explain how This dude was insane

@itsbikidey 3 ай бұрын

Instantly means 5 - 10min

@the-boy-who-lived Ай бұрын

The question wasn't that hard though

@jonathangrey6354 3 жыл бұрын

Ramanujan was a freaking force. What a beast!

@katarinakraus120 3 жыл бұрын

Ramanujan was and is great.

@it6647 3 жыл бұрын

You forgot "alwæs will be"

@dougr.2398 3 жыл бұрын

Pratik Sonavane my! What antiquated spellynge!!!

@guitarguy4372 3 жыл бұрын

Well, not 'is'. Because he passed away already. RIP.

@aviralsood8141 3 жыл бұрын

@@SomeRandomGtaDude-zl3us A big part of Ramanujan's character was his independent and unique approach to mathematical thinking and proofs. There is no guarantee he would have been nurtured into a better mathematician if he had been made to memorise the tricks of the field like an average student. He might have lost his knack of finding clever and tricky insights out of thin air. Also it is a disservice to Gauss.

@sagarsonawane1698 3 жыл бұрын

@@SomeRandomGtaDude-zl3us he would never have shine today like how he is remember today. Education would have wasted his time and would have train him in particular direction. And not have found numerous ways of finding the answer

@joshkeegan3009 3 жыл бұрын

When you said he solved this instantly I couldn’t help but feel small

@idjles 3 жыл бұрын

Don’t feel bad at being small to a titan like Ramanujan.

@vincentconti3633 3 жыл бұрын

@@idjles it's true! We are not geniuses but that does not mean our lives are not meaningful! Very insightful there amigo!!!

@vincentconti3633 3 жыл бұрын

Cause we are small!! It's all good!

@louisng114 3 жыл бұрын

42:40 "To be continued" I see what you did there.

@sharpfang 3 жыл бұрын

Let's hope not, 'cause at the depicted progression fifth video from this one would be just under 33 seconds long.

@gabor6259 3 жыл бұрын

42:37 "Until next time remember, it's okay to be a little crazy"

@_abdul 3 жыл бұрын

@@gabor6259 Hey 👋 ma buddy from Science Asylum.

@vgernyc 3 жыл бұрын

kzfaq.info/get/bejne/ar93pKt7tZuUmGg.html

@ViratKohli-jj3wj 3 жыл бұрын

@@gabor6259 Nick Lucid

@Mathologer 3 жыл бұрын

Greetings from Melbourne. For a change I am posting this video at a reasonable time, 8:51 a.m. on a Sunday morning. We are still in lockdown around here, but things appear to be improving: 63 new cases. There is a very interesting footnote to what I am talking about today contained in the description of this video. Check it out :)

@Amateur0Visionary 3 жыл бұрын

Glad to hear it! Much love to you and yours!

@michaelgian2649 3 жыл бұрын

Saturday night in Rockport Texas. Reasonable time here too.

@user-ws7kp1yh9l 3 жыл бұрын

Love your vid

@windrush104 3 жыл бұрын

Mathologer Are in Melbourne. ?? From a Melbournian

@Mathologer 3 жыл бұрын

@@windrush104 Yes, I teach maths at Monash :)

@amazinggrace5692 3 жыл бұрын

One day I hope to answer your “did you see it?” with an equally enthusiastic “Yes!” 💕🐝

@Chuckie_Baby 3 жыл бұрын

Sometimes I almost see it but not until he says "Did you see it?".

@mohanappavoo4798 3 жыл бұрын

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@mohanappavoo4798 3 жыл бұрын

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@govindasharman425 3 жыл бұрын

I saw that coming

@arvindtech408 3 жыл бұрын

Cathi shaner best of luck

@eminekitapc3877 3 жыл бұрын

Have you ever wondered why his t-shirt says TAXI 1729? The number 1729 is known as the Hardy-Ramanujan number after a famous visit by Hardy to see Ramanujan at a hospital. This number is the number of the taxi Hardy used to visit him, and Ramanujan looked at the number of the taxi and said 'very interesting'. The great mathematician Hardy did not understand what Ramanujan was talking about and asked. Ramanujan, who kept his mind busy with only numbers, said that 1729 is the smallest number, which is the sum of the cubes of two positive numbers in two different forms.

@elcheapo9444 3 жыл бұрын

Indeed!

@AmarDamani 3 жыл бұрын

Knew this one, but a slightly different story...

@nataala_ 3 жыл бұрын

@@AmarDamani Please tell it!

@eminekitapc3877 3 жыл бұрын

@@AmarDamani Yes, there is also a slightly different version of this story. In Hardy's words: I remember once going to see him when he was ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one and that I hoped it was not an unfavorable omen. "No", he replied, "it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways." Is this the story you're talking about?

@RockBrentwood 3 жыл бұрын

The first time I heard the story, I *immediately* blurted out, in reply: "it's also the difference of the squares of two triangular numbers" ... the triangular numbers being (1, 2, 6, 10, ⋯) = (1·2/2, 2·3/2, 3·4/2, 4·5/2, ⋯) ... and in case you weren't paying attention, two of the solutions to the problem for house numbers are 12·17, 29·41. At the time, I *was* going to say in reply that it was the difference of the squares of two triangular numbers in *two* ways, but stopped short, because the other one is off by one.

@stevewhisnant 3 жыл бұрын

This is perhaps the best math video I've seen. Clever, well-explained, and elegant. Keep up the great work. Stay safe amid the Covid.

@michaelscheuermann6949 3 жыл бұрын

Rfrfrry3q TV r

@achyuththouta6957 3 жыл бұрын

Ramanujan was a genius

@velvetpaws999 3 жыл бұрын

Can anybody ever say anything again without referring to Covid? I AM safe, and have not felt unsafe a single second ever since this hubris started! So stop it already, will ya? Thanks!

@bisnisteknoutama3841 3 жыл бұрын

Disagree. There are many math videos out there much better than this.

@johnpearcey 3 жыл бұрын

@@velvetpaws999 Well said.

@oak_meadow9533 3 жыл бұрын

Thank you from the heart. You have such kindness, generosity, and humor in your lectures. I trained to be a mathematician but realized that I didn't have any real talent, so I became an Engineer ( all three). And tutored math in my free time.

@channel100tube 3 жыл бұрын

I love your Ramanujan inspired TAXI 1729 T-shirt

@Mathologer 3 жыл бұрын

Check out this wiki page en.wikipedia.org/wiki/Taxicab_number :)

@swarnimvajpai6373 3 жыл бұрын

It was in 'the man who knew infinity'

@omeragam8628 3 жыл бұрын

"...The Euclidean Algorithem, an ancient mathematical superweapon" perfect description!

@patstevens8970 3 жыл бұрын

Having it demonstrated from 23:36 onwards in the video along with the accompanying soundtrack - a moment of poetic beauty ...

@my-love404 3 жыл бұрын

An impossible problem

@gustavozubieta8767 3 жыл бұрын

Splendid 21st Century math honoring the great Ramanujan. He would have loved this digital age!!

@victorhermestorrestomara3050 3 жыл бұрын

I was watching one of your videos about infinite fractions and... WOW, NEW VIDEO, THAAAANKS

@Mathologer 3 жыл бұрын

That reminds me that I should really add some cards linking to those videos :)

@xCorvus7x 3 жыл бұрын

29:49 The width of the white rectangle is sqrt(2) - 1 . Its height is 1 - (sqrt(2) - 1) = 2 - sqrt(2) = sqrt(2) * (sqrt(2) - 1) . This height divided by the width is: sqrt(2) * (sqrt(2) - 1)/(sqrt(2) - 1) = sqrt(2) .

@sanferrera 3 жыл бұрын

Thank you!

@xCorvus7x 3 жыл бұрын

@@sanferrera You're welcome.

@PickleRickkkkkkk 3 жыл бұрын

WTFhappenedWITHyou factor out the √2 from the left side you get the right side

@greogryhouse8341 3 жыл бұрын

@@WTFhappenedWITHyou 2 = sqrt(2) * sqrt(2), therefore with factorising sqrt(2) *sqrt(2) - sqrt(2) = sqrt(2) * (sqrt(2) - 1)

@astrobullivant5908 3 жыл бұрын

If I could only have had 30 seconds in Ramanujan's brain

@frozenmoon998 3 жыл бұрын

He might not be your typical name, which you could give and everyone would recognize it, such as its with Newton. Despite that however, Ramanujan is a genius, who sadly didn't live for long and would probably be one of the most important mathematicians, should he have lived and published more papers, perhaps even be an advisor to some future mathematicians :)

@2sridhark 2 жыл бұрын

The notebooks he left is an area of research to this day.

@magicmulder 2 жыл бұрын

I read Hofstadter’s “Gödel Escher Bach” at age 13, that was the first time I remember him being mentioned.

@magicmulder 2 жыл бұрын

@@rjwh67220 I ended up becoming a mathematician, so its influence was profound. :)

@robertveith6383 2 жыл бұрын

No, he *was* a genius. He is not alive.

@brindatakley9858 2 жыл бұрын

"Not be your typical name"? What do you really mean?

@dheerdaksh 3 жыл бұрын

I am so happy to have access to such great content without any charge. I love mathematics so much and this satiates my curiosity! Looking forward to more of your amazing work ❤️

@anthonyiodice 3 жыл бұрын

I have almost no grasp of basic algebra. I watch these videos in complete aww of the innate problem solving potential of human beings. I feel like learning math is akin to finally being able to leave Plato’s allegory cave, in that math seems like a key to understanding the entire world around us.

@Mathologer 3 жыл бұрын

Well, pretty sure that the more of this kind of video you watch the more you'll understand :)

@AmadeuShinChan 2 жыл бұрын

Are you interested to study together?

@HiddenTerminal 3 жыл бұрын

Your infinite fractions/sum videos have been absolutely amazing. Please don't ever stop making videos, they are super clear and entertaining.

@user-sw3ro6hh3j 3 жыл бұрын

this is a great presentation. easy to understand and breaks down seemingly mysterious mathematical intuition. thank you!

@jamaluddin9158 3 жыл бұрын

Your videos are really calming to the mind. Pleasant music during algebra autopilot and then fascinating math explained in a natural way!

@user-xt6ee9sx4o Жыл бұрын

Your videos are amazing and very amusing! I don’t think I understand all that you present but I enjoy them a lot! These videos are like a brain “oil change” for me. Used to enjoy math when I was in school a century ago. I have gotten rusty now but thanks to videos like yours I can enjoy math again! 👍👌

@migfed 3 жыл бұрын

This video is really something special, the level of insight, beauty, deep concepts and even mathematics history is staggering. Thank you so so much!

@jeremytaylor3532 3 жыл бұрын

It's sad that Ramanujan did not achieve the Lucasian or Plumian Chair ( Although he would have to of graduated from Cambridge) It would have been nice to see his name on that list of Luminaries. Sometimes Incredible men are taken before they can make those contributions that would leapfrog our Society ahead. Possibly because as a group we are not worthy of what they could gift us with.

@dinofx35 Жыл бұрын

*had to have

@leif1075 Жыл бұрын

Why on earth do you say that? They were human just like the rest of us..Who says we can't do the same as he? I could never admit I'm not as gifted and smart as Ramanujan and couldn't make as great contributions..why elevate one person unduly?

@jeremytaylor3532 Жыл бұрын

@@leif1075 Well I can tell that you are neither gifted or smart from your single comment. But obviously pride and ignorance are your forte.

@gongbadcommunity 3 жыл бұрын

No words could explain the infinite joy I get while going on this journey with your way of telling this story. Thanks 🙏

@TrueMachine2 3 жыл бұрын

I really like how you brought this all down to the pictorial version of squares. Then to stop it from going on forever, adjusted it slightly... which in the end allowed the process to work out very well. I am not a math scientist like you, but use it every day in programming, bookkeeping, estimating, and formula... for automation and business. I was able to follow along, and what you did... made perfect sense in the end. On another note: when I was 17 years old, I took the number 7 to the power of 277, and calculated this by hand. The resulting length of paper ended up from floor to ceiling... or maybe more than 12 pages or so. What became very interesting, is somewhere down the pages... the answers, or the next calculated figure was a pattern. I could write out directly, as it developed a pattern that went right on down. So I could simply just write the number. Why did I go this... I don't know, but it was fun!!! Maybe I'm weird?

@johnmorrow4719 3 жыл бұрын

Wow. That is a show. An interview.

@stephenruby141 3 жыл бұрын

I love the geometric intuition you continue to provide in your videos. I can't wait to see what you have next with this series.

@black_jack_meghav 3 жыл бұрын

Mathologer and on Ramanujan , absolutely amazing. Highly appreciated work sir.

@HebaruSan 3 жыл бұрын

I paused and worked out as much of the problem as I could on paper. Then I unpaused and he covered everything I did in 5 seconds. :~(

@shilpisarker4344 3 жыл бұрын

If Sir Ramanujan was alive for more 50 years mathmetics could prosper more ,specially number theory.

@magicmulder 2 жыл бұрын

He’d have solved Riemann and Fermat rather quickly I presume.

@star_ms 2 жыл бұрын

Even more identities? 😨

@Eonilien 3 жыл бұрын

Your way of explaining things is just lovely: pleasant, clear, open to anyone who's curious and knows basic algebra. Great work once again!

@sebastiensoubiale6482 3 жыл бұрын

Mathologer, Just a note to thank you for all your videos, this is great work, so inspiring. Just the right level of compromise between rigour and popularisation to deal with such amazing topics! Thanks! Sebastien

@Effivera 3 жыл бұрын

This was one your best videos ever Mathologer; thank you. I'm curious if anyone has the answer to the puzzle about the Red Cross at 2:18? Cheers.

@Mathologer 3 жыл бұрын

Someone posted this solutions imgur.com/JfpClXR

@nescafezos4265 2 жыл бұрын

very nice! I tried to solve it for almost 40 mins with no success

@alisaiterkan 3 жыл бұрын

Dude, I teach this stuff at the college level and all I can say is you are the most incredible math educator I have ever seen. Hands down. I had heard somewhere (can't remember where) that Gauss used to consider intuition about proofs to be sort of like the ugly scaffolding around a large structures in restoration. Assuming this is true, it explains so much about why math is feared. What you are doing is the antithesis of that perspective and you are totally nailing it. Thank you.

@joetursi9573 7 ай бұрын

Yu don't refer t this man as"Dude!!"

@stevebeal73 3 жыл бұрын

I discovered this only today (October 20th) and thoroughly enjoyed it. Really looking forward to watching some more like this. Many thanks for producing it.

@malcolmkirk3343 2 жыл бұрын

Absolutely love your presentation style: challenging, yet fun, engaging, and clear explanations!

@RamanKumar-is7xb 3 жыл бұрын

I feel like a similar version of this problem is in NCERT Ex. 5.4 (Optional) Class X Mathematics. Who else have tried that problem using AP?

@Mathologer 3 жыл бұрын

Maybe share this problem with the rest of us ?

@nishadthakur 3 жыл бұрын

@@Mathologer The houses of a row are numbered consecutively from 1 to 49. Show that there is value of x such that the sum of the numbers of the houses preceding the house numbered x is equal to the sum of the numbers of the houses following it. Find this value of x.

@tcadityaa 3 жыл бұрын

Ya. I too thought the same...

@EebstertheGreat 3 жыл бұрын

@@nishadthakur If you already know there are 49 houses, this problem becomes much easier. You can just write and solve a single equation for x, though you do need to know or find a way to sum 1 + 2 + ... + k for any whole number k.

@kakalimukherjee3297 3 жыл бұрын

Yeah that's a hot question for the board examinations of the CBSE here in India 😅

@zanti4132 3 жыл бұрын

Also worth noting about this sequence (the first few terms are shown at 41:11) is that the odd-numbered terms produce all the Pythagorean triples in which the legs of the right triangle differ by one: 1/1: 1 = 0 + 1; 1² + 0² = 1² (trivial case to get started) 7/5: 7 = 3 + 4; 3² + 4² = 5² 41/29: 41 = 20 + 21; 20² + 21² = 29² 239/169: 239 = 119 + 120; 119² + 120² = 169² ...and so on. Every Pythagorean triple of the form x² + (x + 1)² = y² is hit.

@phoquenahol7245 Жыл бұрын

That's not a coincidence. If you do the challenge at 13:37, you will find that the nth partial sum s_n in terms of the (n-1)th partial sum s_(n-1) is ((s_n)+2)/((s_n)+1). Keep in mind that this is the continued fraction for sqrt(2), we will use that fact later. Expressing s_(n-1) as a fraction in lowest terms p/q, we get s_n = (p+2q)/(p+q) (which is actually the same rule described at your timestamp now that I look at it 😅) Edit: Or just skip to 39:30 for the relation. In case you haven't noticed, all of the numerators are odd (which makes sense, otherwise constructing a Pythagorean triple from it whose legs differ by one is clearly impossible). To be rigorous however, we first have to prove that the numerator p+2q is always odd which will be done by induction. Base case: Consider the second partial sum 1/(1+1/2). This simplifies to 3/2 and 3 is odd. The case for all n: Assume s_n = p/q. Then s_(n+1) = (p+2q)/(p+q). If p is odd, then clearly p+2q is odd, no matter the parity of q. Since in the base case, p=3, which is odd, p should be odd for all partial sums. This ensures that when we attempt to construct the Pythagorean triple from its corresponding partial sum, the legs are integers. Next, we express the actual elements of the Pythagorean triple in terms of p and q. The 2 legs are (p-1)/2 and (p+1)/2 and the hypotenuse is q. The sum of squares of the legs are ((p-1)^2+(p+1)^2)/4 = (p^2+1)/2 and by the Pythagorean theorem, is equal to q^2. Multiplying both sides by 2 and moving the 1, we end up with the Pell equation 2q^2-p^2 = 1. However, since p/q is the nth partial sum for sqrt(2) (I told you we would use it :D), p and q are indeed solutions of that Pell equation (check 17:00). I forget the proof though, sorry 😞. Please have mercy on me, I am just a grade 9 student with no social life. Edit: As an aside, you may have noticed that the Pell equation provided at 17:00 is actually 2q^2-p^2 = -1 and not positive 1. Actually, s_n only produces a Pythagorean triple if n is odd because p and q satisfy the other Pell equation for even n. This is the reason why there is no Pythagorean triple for 3/2, 17/12, 99/70 etc; the sum of the squares of the legs is actually 1 greater than the denominator squared. I could prove this to you, but I am sure you are tired of this rambling and I am getting tired of typing. Also 14:30 is exactly what I just said 😅.

@WarpFactor999 3 жыл бұрын

Burkard - having struggled with math all my life, you bring clarity and fresh air to an otherwise somewhat rarefied endeavor. Your efforts are most appreciated. Thank you kind sir!

@Green_Phosphorus 3 жыл бұрын

13:36 - Numerator is equal to 2x the denominator of the previous term plus the numerator of the previous term, denominator is equal to the numerator minus the denominator of the previous term. Maybe not the simplest rule but it’s the first one I saw, by looking at the sequence of partial fractions. I appreciate the little challenges included in these videos. Not many math KZfaq channels include them. Most of the time I don’t go for them, but whenever I do and find the solution, it’s rewarding 🙂

@silvernekode7526 2 жыл бұрын

Another slightly cleaner way to phrase this same pattern is: The denominator is equal to the sum of the previous term's denominator and numerator. The numerator is equal to the sum of the previous term's denominator and the current term's denominator.

@mindlesskris 3 жыл бұрын

Red Cross solution: Align the centre of one of the smaller crosses to the centre the big cross, matching their orientation. Rotate the smaller cross until its vertices touch the edges of the big cross. The 4 segments produced form the 2nd smaller cross.

@Mathologer 3 жыл бұрын

That's it :)

@therealsachin 3 жыл бұрын

@@Mathologer I am confused. This solution assumes that the smaller crosses are of the right size. But we don't know the size yet to begin with. So while rotating the smaller cross, what if it keeps freely rotating without touching the bigger cross?

@MarkVersteegh 3 жыл бұрын

@@therealsachin the area of the large cross should be twice the area of the small crosses, therefore the ratio of the lengths has to be 1 : √2. So the diagonal of a unit square in the small crosses equals the length of the edges of the squares in the large cross.

@davisdawson5047 3 жыл бұрын

@@MarkVersteegh I will pretend to understand that. Ah that's what I thought too.

@therealsachin 3 жыл бұрын

Hi@@MarkVersteegh, Yes, I got that... but that was not put as part of proof so I was wondering. I have a different proof based on that fact. I am still not able to wrap my head around this proof though. The proof I got: Side of square of larger cross = √2 * side of square of smaller cross. Then if we cut all the four outer squares of the large cross by their diagonal, we will only need 4 cuts to cut all of them. The center square wont' be cut yet, so we just use one remaining cut horizontally on it and we now have all the pieces required to rebuild the 2 smaller crosses. Link to solution image: www.linkpicture.com/q/Cross-Puzzle-Solution.png

@fathicoltd6774 Жыл бұрын

Algebra gets very interesting when it's described with geometry. I love it that way and probably Euclid's approach to the problem was derived from geometry as well.

@alishawamreh5752 3 жыл бұрын

Your persistent ability to make me restless until I wrap my head around these concepts really shows how good you are as a content creator. I don’t think I would have been able to grasp the infinite fraction without your animation/explanation using squares. It’s very inspiring - I’m excited for your next video!

@leofranklin84 3 жыл бұрын

You are probably the best math teacher in the world...the way u bring out the magic in math is mesmerising....one can get hooked on for hours

@mayabartolabac 3 жыл бұрын

I feel like Ramanujan was an alien that was sent to Earth to accelerate our knowledge in mathematics, and once he taught everything he could to the human race, he left Earth to teach another underdeveloped civilization.

@Mathologer 3 жыл бұрын

If you have not read it yet there is this very nice biography of Ramanujan by Roger Kanigel (I found the video pretty much unwatchable :)

@videosforyou567 3 жыл бұрын

You should read up about Indian science and Maths. 1) How Fibonacci was introduced to Indian mathematics 2)How maharishi kannada postulated (kinda) the atomic theory 3) How Schopenhauer had declared, “In the whole world there is no study so beneficial and so elevating as that of the Upanishads. It has been the solace of my life. It will be the solace of my death.” 4) How schrödinger named his dog Atman after getting inspired by Hindu texts..I've got endless stuff to write!

@hanniffydinn6019 3 жыл бұрын

The dude killed himself. Not every intelligent move! 🤯🤯🤯

@mangai3599 3 жыл бұрын

Well, you said Ramanujan was an Alien! In hinduism, we can say he was avatar of god who came here to teach the mankind! He was a really very brilliant great mathematician and it must been great for the other fellow mathematician who contemporary of Ramanujan. Well, we know that Ramanujan was highly self taught but there are many more examples of scientist that have appeared in the History that are self taught genius! The real ability of Ramanujan that made him brilliant and compared him will god or Alien that was his brilliant ability to play with mathematics!😁

@SoleaGalilei 3 жыл бұрын

What are you talking about? Ramanujan didn't kill himself.

@nafrost2787 3 жыл бұрын

36:29 there is always more to dig on every subject in math, this is a never ending quest.

@piwi2005 3 жыл бұрын

Cool ! So from Euclide's algorithm, we also get decomposition of products into squares: 38*16=2*16^2+2*6^+1*4^2+2*2^2 p*r0=d1*r0^2+d2*r1^2+d3*r2^2+...+dn*gcd^2 with gcd

@Mathologer 3 жыл бұрын

Yes !

@piwi2005 3 жыл бұрын

@@Mathologer :) I hope you'll do a video to explain what we can do with that ! Thanks so much for your amazing videos.

@juanluisclaure6485 3 жыл бұрын

i must comment your channel after watching this chapter, My boldness is feed from your phrase that repeat, "is this brillant,isnt?" and i must say as hardcore student of your teachings that yes it is brillant. Thanks for sharing some awesome math issues. Really make my life better. Gracias por tanto y saludos desde Bolivia.

@BardaKWolfgangTheDrug 3 жыл бұрын

Always quality content 💪💪 one of the best channels on YT 💕

@firefly618 3 жыл бұрын

24:42 is not only a beautiful palindrome timestamp, but in a flash it gave me a deep understanding of both the Euclidian algorithm and of continued fractions. Thank you!

@Mathologer 3 жыл бұрын

That's the reaction I was hoping for :)

@AdamSpanel 3 жыл бұрын

Wow, this was so satisfying, relaxing and just generally wonderful and elegant bit of math for a sunday evening. Thank you!

@proth1951 3 жыл бұрын

my apologies for asking an unnecessary question yesterday. I was watching this wonderful lesson on continuous fractions using my cell phone and was unable to navigate to your full explanation which included the credits and titles for the background music. Thanks for helping us readers get really excited and interested in furthering our math education well beyond what we learned in high school.

@xCorvus7x 3 жыл бұрын

35:09 The fractions we get by truncation when put into the left side of the Pell equation alternatingly yield 1 and -1 because we alternate between cutting off rectangles where the longer side aligns with the longer side of the original rectangle and rectangles where the longer side is orthogonal to the longer side of the original rectangle. If it aligns, the truncation means to make the denominator slightly smaller because we rescale the shorter side of the big rectangle to be reduced by the width of the truncated rectangle. This makes the truncated fraction bigger than the number we approximate and so the equation yields 1 . If it is orthogonal, the truncation means to make the numerator slightly smaller since we rescale the longer side of the big rectangle to be reduced by the width of the truncated rectangle. This makes the truncated fraction slightly smaller than the number we approximate and the equation yields -1 . Edit: In terms of 40:03 : L^2 - 2*S^2 = ±1 => (2*S + L)^2 - 2*(S + L)^2 = 4*S^2 + 4*S*L + L^2 - 2*S^2 - 4*S*L - 2*L^2 = 2*S^2 - L^2 = (-1)*(L^2 - 2*S^2) = ∓1 .

@christianneisler2962 3 жыл бұрын

13:43 the numerators follow the pattern a1=1, a2=3, a(n)=2*a(n-1)+a(n-2)

@christianneisler2962 3 жыл бұрын

the denonimators follow the same pattern sorry forgot to mention that

@christianneisler2962 3 жыл бұрын

except that b2=2 instead of 3

@louisvandermerwe8012 3 жыл бұрын

Breaking the video into chapters was a great idea. Each chapter was a gem on its own, complementing the whole video.

@t0mstone581 3 жыл бұрын

A new mathologer video always feels like a great birthday present. I love all of them!

@GuRuGeorge03 3 жыл бұрын

I was supposed to study english for an exam tomorrow, but this is just way too fascinating

@beautifulsmall 3 жыл бұрын

School maths should teach more like this , love the geometric drawing conceptualiseations.

@colinnewton5254 3 жыл бұрын

Fantastic. This is the first time I have watched this video and I understand it! ! ! Congratulations Mathologer, I look forward to the two hour extension you promised? ? ?

@accountname1047 3 жыл бұрын

You really are the GOAT of mathematics youtube. Fantastic as always!

@Bhatakti_Hawas 3 жыл бұрын

Yesterday I saw Stand-up Math's video on how to approximately calculate the perimeter of an ellipse. And lo and behold Ramanujan was in there And today, I meet Ramanujan once again 😀

@manjusarangi8536 3 жыл бұрын

Great fan of your work sir . I just love to see your videos The way you explain concepts is just amazing. Love from India sir

@dcterr1 3 жыл бұрын

Wonderful video highlighting the genius of Ramanujan and the power of continued fractions.

@Reynolt2k 3 жыл бұрын

First time i saw one of your videos. Absolutely love your style. I'm am engineer and typically use maths only insofar as necessary, but i found this video so engaging and paced just right.

@KillianDefaoite 3 жыл бұрын

Me when I see a mathologer video: "My *excitement* is immeasurable and my day is *made* ."

@quickyummy8120 3 жыл бұрын

Even if ur videos are long still it keeps us engaged. Good job 👍 appreciable ❤️love from india 🇮🇳 Ramanujan was great🙏

@mathwithjanine 3 жыл бұрын

Your videos are so fascinating! Looking forward to watching your next video!

@Sandy-rv9tv 2 жыл бұрын

Excellent video. Ramanujan, what a legend he was! Also the other friend visiting Ramanujan mentioned in the early part of the video - PC Mahalanobis - is also a great name, he is considered the Father of Indian Statistics. He is famous for Mahalonobis Distance

@pankajdave5591 3 жыл бұрын

Wonderful video Every maths lover must watch.

@eliyasne9695 3 жыл бұрын

This is a masterpiece!

@irvingg2342 3 жыл бұрын

Incredible job, as always! Your visual derivations make many of the familiar ideas so much more magical to me. I’d love to see you talk about partitions, modular forms, and theta functions at some point :)

@Mathologer 3 жыл бұрын

There may be something on partitions pretty soon. Sort of got a half-finished presentation on partitions open on my laptop at the moment. Having said that, these days I never plan ahead with these videos and just go "where the wind takes me" :)

@johnnygodoy8329 3 жыл бұрын

I just very recently ran into the Pell Equation for a problem, thanks for helping to make the conection to more maths. I think it's worth noting that since the recursion is linear, it can be calculated with powers of a matrix, which leaves a very short expression to calculate them in the same time order as an explicit formula but without roots (avoiding having to use a rounding function in a computer due to the numerical errors). If you wanted, you can diagonalize it to get the formula as well!

@pranavlimaye 3 жыл бұрын

36:57 "Outramanujan" is now my new favourite verb!

@2mat012 3 жыл бұрын

I liked that

@GlutesEnjoyer 3 жыл бұрын

This the the only _strand_-type math problem

@Mathologer 3 жыл бұрын

@babulalmarandi1243 3 жыл бұрын

Wow

@Jop_pop 3 жыл бұрын

Upset I had to scroll so far to find this

@user-me7hx8zf9y 3 жыл бұрын

@@Jop_pop Same

@brendawilliams8062 3 жыл бұрын

Thankyou for being the excellent teacher are. I solved my questions when I found I was working in designer space. Different opinions exist over lambda. So many of you helped me figure it out. Thankyou.

@nagas7722 3 жыл бұрын

Srinivasa Ramanujan was indeed genius.. not matter how many westerners say about aliens or whatever..

@greggjohnson5634 3 жыл бұрын

8 houses could still be a "long street" just depends how close the houses are 😆

@SoWe1 3 жыл бұрын

really liked this one!

@williedewit5223 3 жыл бұрын

Love these videos. Saying I understand half of what he says would be exaggerating :) But his love for Math is keeping me going . Well done!!

@timh.6872 3 жыл бұрын

I've recently gotten into euclidean constructions (compass + straightedge) while doing woodworking. I wouldn't necessarily use them to build a house, but for laying out cuts for visible jointery it's a lot easier and relaxing to grab a compass and start marking off ratios on the lumber itself and ignore the precise dimensions. The expression of the euclidean algorithm via snugged-up squares is really neat and a heck of a lot easier to do as a geometric construction than long division. Since it also uniquely expresses the simplified form of a fraction, it gives the largest squares that tile the rectangle. Not sure if I'll ever be able to use that, but I'll certainly keep it tucked in the back of my mind.

@themrflibbleuk 3 жыл бұрын

Yay! I think I need to invest in Mathologer T-Shirts!

@johnchessant3012 3 жыл бұрын

To me the fact that you can get from one convergent to the next is amazing. At first it looks like putting one more coefficient means you need to calculate the fraction again from scratch. But the picture makes the quick way so obvious!

@Mathologer 3 жыл бұрын

Yes, it's really an amazingly insightful way of looking at these infinite fraction :) Hardly ever taught though :(

@iabervon 3 жыл бұрын

Actually, if you look at the continued fraction, it's reasonably obvious that adding one, flipping it, and adding one again will push everything down a layer. The hard part is noticing that pushing it down is going to work, due to the numbers being the same all the way down.

@pharaohgarmar5611 3 жыл бұрын

Enjoyable and looking forward to the sequel!

@zeitgeist2720 3 жыл бұрын

Can’t believe I haven’t found this channel earlier. Thank you for this amazing content I can’t believe it’s free

@PapaFlammy69 3 жыл бұрын

*_/w magic_*

@Mathologer 3 жыл бұрын

@shantanunene4389 3 жыл бұрын

Yayy two of my favourite maths KZfaqrs

@PapaFlammy69 3 жыл бұрын

@@Mathologer (:

@PapaFlammy69 3 жыл бұрын

@@shantanunene4389 hehe :3

@darkseid856 3 жыл бұрын

@@PapaFlammy69 Papa flammy spotted .

@deanc9195 3 жыл бұрын

The way Mathologer pronounces ramanujan makes me so happy

@r.k.jangra1638 3 жыл бұрын

In continuation to my last comment 4 hours ago, below is my solution. I not watched the solution in video yet. I am going to watch that now. I am not sure, if my solution and solution in video will be same or not, as I haven't seen the video yet. Objective of this exercise by me was 'to feel inner-delight' that I felt while solving this myself; nothing else ;). I used my computer to do my calculations. I always interested in such mathematics. And undoubtedly, Ramanujan was a legend. So, after some R&D, I finally came up with below 2 equations, which gives next set of solution in terms of current known solution. This way we can get the series of infinite solutions Hnext = ceil( [3 + 2 * sqrt(2)] * Hcur ) Tnext = ceil( 2*[2 + sqrt(2)]* Hcur + Tcur ) In above equations: 1. Hcur represents person's house number and Tcur represents total number of houses in CURRENT solution. 2. Hnext represents person's house number and Tnext represents total number of houses respectively for NEXT solution in the series of infinite solutions. 3. Initial value of Hcur=Tcur=1 be taken. 4. 'ceil' function changes decimal number to nearest next integer number. E.g. ceil (2.334) => 3 Some other observations: - Last digit of 'Person's House#' is in pattern 6->5->4->9->0->1 and repeats again. - Similarly last digit of corresponding 'Total Houses' number is in pattern 8->9->8->1->0->1 and repeats again. - Also, there are patterns of last two digits as well. Solving above equations gives below. I calculated first 60 solution of series. (Solution#., Person's House#, Total Houses, Answer confirmed to be correct, # of digits in Person's house number, # of digits in Total houses number) 1. 6 8 True 1 1 2. 35 49 True 2 2 3. 204 288 True 3 3 4. 1189 1681 True 4 4 5. 6930 9800 True 4 4 6. 40391 57121 True 5 5 7. 235416 332928 True 6 6 8. 1372105 1940449 True 7 7 9. 7997214 11309768 True 7 8 10. 46611179 65918161 True 8 8 11. 271669860 384199200 True 9 9 12. 1583407981 2239277041 True 10 10 13. 9228778026 13051463048 True 10 11 14. 53789260175 76069501249 True 11 11 15. 313506783024 443365544448 True 12 12 16. 1827251437969 2584123765441 True 13 13 17. 10650001844790 15061377048200 True 14 14 18. 62072759630771 87784138523761 True 14 14 19. 361786555939836 511643454094368 True 15 15 20. 2108646576008245 2982076586042449 True 16 16 21. 12290092900109634 17380816062160328 True 17 17 22. 71631910824649559 101302819786919521 True 17 18 23. 417501372047787720 590436102659356800 True 18 18 24. 2433376321462076761 3441313796169221281 True 19 19 25. 14182756556724672846 20057446674355970888 True 20 20 26. 82663163018885960315 116903366249966604049 True 20 21 27. 481796221556591089044 681362750825443653408 True 21 21 28. 2808114166320660573949 3971273138702695316401 True 22 22 29. 16366888776367372354650 23146276081390728245000 True 23 23 30. 95393218491883573553951 134906383349641674153601 True 23 24 31. 555992422174934068969056 786292024016459316676608 True 24 24 32. 3240561314557720840260385 4582845760749114225906049 True 25 25 33. 18887375465171390972593254 26710782540478226038759688 True 26 26 34. 110083691476470624995299139 155681849482120242006652081 True 27 27 35. 641614773393652358999201580 907380314352243226001152800 True 27 27 36. 3739604948885443528999910341 5288600036631339114000264721 True 28 28 37. 21796014919919008815000260466 30824219905435791458000435528 True 29 29 38. 127036484570628609361001652455 179656719395983409634002348449 True 30 30 39. 740422892503852647351009654264 1047116096470464666346013655168 True 30 31 40. 4315500870452487274745056273129 6103039859426804588442079582561 True 31 31 41. 25152582330211071001119327984510 35571123060090362864306463840200 True 32 32 42. 146599993110813938731970911633931 207323698501115372597396703458641 True 33 33 43. 854447376334672561390706141819076 1208371067946601872720073756911648 True 33 34 44. 4980084264897221429612265939280525 7042902709178495863723045838011249 True 34 34 45. 29026058213048656016282889493864074 41049045187124373309618201271155848 True 35 35 46. 169176265013394714668085071023903919 239251368413567743993986161788923841 True 36 36 47. 986031531867319631992227536649559440 1394459165294282090654298769462387200 True 36 37 48. 5747012926190523077285280148873452721 8127503623352124799931806454985399361 True 37 37 49. 33496046025275818831719453356591156886 47370562574818466708936539960450008968 True 38 38 50. 195229263225464389913031439990673488595 276095871825558675453687433307714654449 True 39 39 51. 1137879533327510520646469186587449774684 1609204668378533586013188059885837917728 True 40 40 52. 6632047936739598733965783679534025159509 9379132138445642840625440926007312851921 True 40 40 53. 38654408087110081883148232890616701182370 54665588162295323457739457496158039193800 True 41 41 54. 225294400585920892564923613664166181934711 318614396835326297905811304050940922310881 True 42 42 55. 1313111995428415273506393449094380390425896 1857020792849662463977128366809487494671488 True 43 43 56. 7653377571984570748473437080902116160620665 10823510360262648485956958896805984045718049 True 43 44 57. 44607153436479009217334229036318316573298094 63084041368726228451764625014026416779636808 True 44 44 58. 259989543046889484555531937137007783279167899 367680737852094722224630791187352516632102801 True 45 45 59. 1515330104844857898115857393785728383101709300 2143000385743842104896020122110088683012980000 True 46 46 60. 8831991086022257904139612425577362515331087901 12490321576610957907151489941473179581445777201 True 46 47

@bryanbischof4351 3 жыл бұрын

Wow, this video had some particularly beautiful connections. Very very impressive results. The continued fractions and rectangles was new for me and much appreciated. I think I didn’t catch the measure of better approximation, but it seems like you might do it in more detail in a different video so I’ll try to check.

@falseprophet75 3 жыл бұрын

Perhaps the (legendary) fellow that upset the Pythagoreans so much by proving the irrationality of root 2 may not have suffered such a tragic fate if he had been able to demonstrate root 2 as an infinite continued fraction.

@Chad-qk1ig 2 жыл бұрын

The Pythagoreans also hated infinity

@theespatier4456 3 жыл бұрын

A Strand Is A Part Of A Rope Or Bond, While Stranded Means Being Washed Up On The Shore, And Being Stranded Is When You Can't Go Home.

@OMGclueless 3 жыл бұрын

The real question is whether Mathologer got the reference, or just gave this comment a heart because he's giving all the early comments on his video a heart...

@it6647 3 жыл бұрын

Death Stranding?

@Mathologer 3 жыл бұрын

@@OMGclueless Actually, I had to look it up :)

@JNCressey 3 жыл бұрын

@@OMGclueless, what's the reference?

@EebstertheGreat 3 жыл бұрын

@@JNCressey Apparently to a game called "Death Stranding."

@Adityarm.08 Жыл бұрын

The connection between continued fractions & euclidean algorithm was just mind blowing. Thank you.

@antoniomonteiro1203 3 жыл бұрын

I solved this problem some years ago by calculating a formula and copying down in Excel. In the first column it starts with 1. In the second column =SQRT(A1*A2/2). By copying down the formula (and the 1), on the first column you end up with the natural numbers. On the second column the resulting square root. Looking at the numbers it was immediately apparent which were the integer numbers out of the "forest" of fractional numbers. The ones smaller than 50 (1 , 6 and 35) and the one I wanted (204). On the left of the door number (when integer) is the total of houses on the road for each case.