WEBVTT 00:00:00.000 --> 00:00:02.240 Welcome back to the deep dive. So our mission 00:00:02.240 --> 00:00:04.519 today, it's a big one. We're going to take this 00:00:04.519 --> 00:00:07.459 the stack of sources that detail an almost impossible 00:00:07.459 --> 00:00:10.320 life. It really is. We're talking about a man 00:00:10.320 --> 00:00:13.380 who. Well, who arrived at the front lines of 00:00:13.380 --> 00:00:15.419 mathematics having never really studied the rulebook. 00:00:15.640 --> 00:00:18.519 Exactly. And our job is to extract the insights 00:00:18.519 --> 00:00:21.839 that explain not just what he did, but how. We 00:00:21.839 --> 00:00:25.019 are diving into the extraordinary life and, frankly, 00:00:25.179 --> 00:00:27.879 the shocking mathematical legacy of Srinivasa 00:00:27.879 --> 00:00:30.120 Ramanujan. Yeah, we're looking at a true mathematical 00:00:30.120 --> 00:00:33.020 anomaly. Srinivasa Ramanujan, born in India, 00:00:33.119 --> 00:00:36.500 1887. He was a figure in the early 20th century 00:00:36.500 --> 00:00:39.359 whose whole career, his entire life, It just 00:00:39.359 --> 00:00:41.840 defies every standard academic story we know. 00:00:41.960 --> 00:00:44.500 It does. The journey he took from, you know, 00:00:44.520 --> 00:00:46.399 extreme poverty in Madras all the way to the 00:00:46.399 --> 00:00:49.460 very rigid formal halls of Cambridge. It's just 00:00:49.460 --> 00:00:52.460 this powerful testament to raw, undiluted genius. 00:00:52.740 --> 00:00:54.479 OK, let's unpack that right away. Let's define 00:00:54.479 --> 00:00:56.020 the central paradox here because the sources 00:00:56.020 --> 00:00:58.399 are so clear on this. Remigian had almost no 00:00:58.399 --> 00:01:02.200 formal training in pure math, no access to modern 00:01:02.200 --> 00:01:05.439 European textbooks, no degree, no rigorous instruction 00:01:05.439 --> 00:01:09.120 in things like complex analysis. And yet. He 00:01:09.120 --> 00:01:11.739 made these huge contributions to some of the 00:01:11.739 --> 00:01:14.859 most difficult fields out there. Number theory, 00:01:15.060 --> 00:01:17.739 infinite series, continued fractions. Right. 00:01:17.819 --> 00:01:21.140 And he was solving problems that establish academics, 00:01:21.400 --> 00:01:23.620 you know, the best in the world. They considered 00:01:23.620 --> 00:01:26.400 them completely unsolvable. It's just it's an 00:01:26.400 --> 00:01:28.239 unprecedented achievement. And when you look 00:01:28.239 --> 00:01:31.040 at the sheer output, the numbers themselves are 00:01:31.040 --> 00:01:33.599 almost unbelievable. Let's get into that. Ramanujan 00:01:33.599 --> 00:01:36.879 independently compiled nearly 3 ,900 results. 00:01:37.180 --> 00:01:39.310 We're talking mostly. identities and equations, 00:01:39.590 --> 00:01:42.030 all recorded in his famous notebooks. And the 00:01:42.030 --> 00:01:44.030 really astonishing part. The astonishing part 00:01:44.030 --> 00:01:46.250 is that the vast majority of them over the decades 00:01:46.250 --> 00:01:48.989 have been proven correct. And so many of them 00:01:48.989 --> 00:01:51.569 were completely new, opening up research areas 00:01:51.569 --> 00:01:54.409 that are still defining modern math today. Can 00:01:54.409 --> 00:01:57.310 we just stop there for a second? 3 ,900. It's 00:01:57.310 --> 00:01:58.909 mind -boggling. I mean, you have to put that 00:01:58.909 --> 00:02:00.730 in perspective for someone listening. A typical 00:02:00.730 --> 00:02:03.230 PhD thesis in pure mathematics, what does that 00:02:03.230 --> 00:02:06.629 have? Five, maybe six major provable results. 00:02:06.769 --> 00:02:09.620 If you're lucky, yeah. And Ramanujan generated 00:02:09.620 --> 00:02:14.120 3 ,900 of them in isolation with no formal proofs 00:02:14.120 --> 00:02:16.680 attached. Exactly. And that context is precisely 00:02:16.680 --> 00:02:19.500 why his early attempts to get noticed failed. 00:02:19.879 --> 00:02:22.500 As Hans Heisling pointed out, he couldn't get 00:02:22.500 --> 00:02:24.460 the attention of leading mathematicians at first 00:02:24.460 --> 00:02:26.759 because his work was, and I'm quoting here, too 00:02:26.759 --> 00:02:29.520 novel, too unfamiliar, and additionally presented 00:02:29.520 --> 00:02:31.659 in unusual ways. They didn't know what they were 00:02:31.659 --> 00:02:33.699 looking at. Not at all. They'd look at his equations 00:02:33.699 --> 00:02:35.500 and couldn't even figure out what branch of math 00:02:35.500 --> 00:02:37.900 they belonged to, let alone start to verify them. 00:02:38.569 --> 00:02:41.330 His genius was so far outside the established 00:02:41.330 --> 00:02:43.629 framework that to them it probably just looked 00:02:43.629 --> 00:02:45.930 like nonsense. It's this classic clash, isn't 00:02:45.930 --> 00:02:49.210 it? Intuition versus rigor, East versus West. 00:02:50.110 --> 00:02:53.509 So our mission today is to explore two key ideas. 00:02:54.090 --> 00:02:57.349 First, how did the solitary genius find his voice 00:02:57.349 --> 00:02:59.610 through that extraordinary partnership with G 00:02:59.610 --> 00:03:02.090 .H. Hardy? A crucial relationship. And second, 00:03:02.310 --> 00:03:05.370 how do his insights continue to inspire such 00:03:05.370 --> 00:03:07.789 profound research? I mean, we're not talking 00:03:07.789 --> 00:03:09.870 about dusty historical facts here. We're talking 00:03:09.870 --> 00:03:13.250 about things, mere comments in his writings. 00:03:13.900 --> 00:03:16.259 that researchers were still uncovering and proving 00:03:16.259 --> 00:03:19.719 his profound number theory as late as 2012. His 00:03:19.719 --> 00:03:22.219 work is a perpetual mathematical goldmine. It 00:03:22.219 --> 00:03:24.080 really is. So let's go back to the beginning 00:03:24.080 --> 00:03:26.939 of this incredible story, the isolated prodigy. 00:03:27.620 --> 00:03:31.060 Srinivasa Ramanujan was born December 22, 1887, 00:03:31.280 --> 00:03:35.159 in Erode, India. He was from a traditional Tamil 00:03:35.159 --> 00:03:39.099 Brahmin Iyengar family in what was then the Madras 00:03:39.099 --> 00:03:42.120 presidency of British -ruled India. And his roots 00:03:42.120 --> 00:03:44.520 were very modest. His father was a clerk in a 00:03:44.520 --> 00:03:46.919 small shop. His mother was a housewife who also 00:03:46.919 --> 00:03:48.599 sang at a local temple. And she'd often take 00:03:48.599 --> 00:03:50.340 him with her, right? Which gave him this really 00:03:50.340 --> 00:03:52.699 early, strong connection to spirituality that 00:03:52.699 --> 00:03:54.759 we'll see come up again and again. Absolutely. 00:03:54.800 --> 00:03:57.860 And his childhood, I mean, it immediately signaled 00:03:57.860 --> 00:04:00.349 that this was no ordinary mind. The sources say 00:04:00.349 --> 00:04:02.750 that by age 11, he had completely exhausted the 00:04:02.750 --> 00:04:05.069 mathematical knowledge of two college students 00:04:05.069 --> 00:04:06.810 who were boarding at his family's home. Just 00:04:06.810 --> 00:04:08.669 to think about that. These are guys studying 00:04:08.669 --> 00:04:11.569 for higher exams, and this 11 -year -old kid 00:04:11.569 --> 00:04:14.409 is basically running them out of things to teach 00:04:14.409 --> 00:04:17.329 him. Until they had nothing left to give. It's 00:04:17.329 --> 00:04:20.129 an incredible developmental leap. Most kids that 00:04:20.129 --> 00:04:22.529 age are, what, struggling with fractions. And 00:04:22.529 --> 00:04:25.529 he's already pushing into university -level concepts. 00:04:25.790 --> 00:04:29.209 And it didn't stop. By age 13, he gets his hands 00:04:29.209 --> 00:04:32.009 on a book by S .L. Looney on advanced trigonometry. 00:04:32.230 --> 00:04:34.629 And this wasn't some simple textbook. Louie's 00:04:34.629 --> 00:04:36.889 book is a real challenge, even for university 00:04:36.889 --> 00:04:39.189 students at the time. But Ramanujan doesn't just 00:04:39.189 --> 00:04:42.230 master it. He starts discovering his own sophisticated 00:04:42.230 --> 00:04:44.949 theorems, completely independent of any curriculum. 00:04:45.250 --> 00:04:47.420 He was just a natural problem solver. He had 00:04:47.420 --> 00:04:49.459 this high aptitude for logistics, too, even as 00:04:49.459 --> 00:04:51.319 a teenager. Oh, the school scheduling problem. 00:04:51.620 --> 00:04:54.019 Exactly. The story goes that when he was 14, 00:04:54.259 --> 00:04:56.879 he helped his school solve this really complex 00:04:56.879 --> 00:05:00.139 scheduling challenge. How to assign 1 ,200 students 00:05:00.139 --> 00:05:02.779 to about 35 teachers. That's not just arithmetic. 00:05:02.860 --> 00:05:05.860 That's a complex optimization problem. It shows 00:05:05.860 --> 00:05:08.779 he had this innate, practical, applied intelligence 00:05:08.779 --> 00:05:11.759 long before those fields were even properly formalized. 00:05:11.920 --> 00:05:15.879 It does. But the really crucial text, the one 00:05:15.879 --> 00:05:18.600 that shaped his whole unique methodology, was 00:05:18.600 --> 00:05:21.040 a book by G .S. Carr. A Synopsis of Elementary 00:05:21.040 --> 00:05:23.660 Results in Pure and Applied Mathematics. That's 00:05:23.660 --> 00:05:26.500 the one. He got a copy in 1903 when he was 16. 00:05:26.800 --> 00:05:29.759 And it just... It fundamentally changed how he 00:05:29.759 --> 00:05:32.240 approached mathematics. So what was this book? 00:05:32.459 --> 00:05:35.060 It wasn't a normal textbook, was it? No, not 00:05:35.060 --> 00:05:37.560 at all. It's arguably the single most important 00:05:37.560 --> 00:05:40.060 document in his entire development. It was this 00:05:40.060 --> 00:05:43.339 massive two -volume collection of about 5 ,000 00:05:43.339 --> 00:05:46.170 theorems and results. Wow. But... And this is 00:05:46.170 --> 00:05:48.730 the key part. They were stated largely without 00:05:48.730 --> 00:05:50.810 any detailed proofs or derivations. It's just 00:05:50.810 --> 00:05:53.149 the answers. It's just the final answers. Carr 00:05:53.149 --> 00:05:55.670 designed it to be a memory aid for students cramming 00:05:55.670 --> 00:05:57.930 for exams. It gave you the destination without 00:05:57.930 --> 00:06:00.389 showing you the route. So imagine you're a 16 00:06:00.389 --> 00:06:02.629 -year -old prodigy, and you get a book with 5 00:06:02.629 --> 00:06:05.389 ,000 mathematical final answers, but no explanation 00:06:05.389 --> 00:06:08.110 of why they work. And because he studied it so 00:06:08.110 --> 00:06:11.009 intensely, line by line, he internalized this 00:06:11.009 --> 00:06:13.970 method where intuition and the final result were 00:06:13.970 --> 00:06:17.230 everything. The formal proof was, well, secondary. 00:06:17.610 --> 00:06:20.009 Or maybe just something you didn't need to write 00:06:20.009 --> 00:06:22.420 down. And that's the period of isolated discovery, 00:06:22.699 --> 00:06:24.899 right, where he has to invent the steps and the 00:06:24.899 --> 00:06:27.379 proofs himself. This is why he later struggled 00:06:27.379 --> 00:06:30.240 so much with the Western demand for rigor. Absolutely. 00:06:30.339 --> 00:06:33.220 And yet, even completely on his own, his mind 00:06:33.220 --> 00:06:36.000 was just exploding with discoveries. He independently 00:06:36.000 --> 00:06:38.800 developed algorithms for Bernoulli numbers. He 00:06:38.800 --> 00:06:41.120 calculated the Euler -Mascheroni constant to 00:06:41.120 --> 00:06:44.060 15 decimal places. That's just world -class analysis. 00:06:44.240 --> 00:06:46.759 Performed by a young man with no formal guidance 00:06:46.759 --> 00:06:51.120 whatsoever. Which brings us to the... The tragedy 00:06:51.120 --> 00:06:53.920 of his academic career, because despite this 00:06:53.920 --> 00:06:56.600 obvious genius, his formal performance was a 00:06:56.600 --> 00:06:59.420 complete disaster. A total disaster. Yeah. He 00:06:59.420 --> 00:07:00.980 got a scholarship to government arts college, 00:07:01.120 --> 00:07:04.000 but his obsession with math was so all consuming 00:07:04.000 --> 00:07:06.500 that he just he neglected everything else. He 00:07:06.500 --> 00:07:08.420 failed every other required subject. Especially 00:07:08.420 --> 00:07:11.139 English and physiology, the sources say. He literally 00:07:11.139 --> 00:07:13.019 couldn't force his brain to care about anything 00:07:13.019 --> 00:07:15.379 that wasn't mathematics. Yeah. So the scholarship 00:07:15.379 --> 00:07:18.740 was revoked instantly. Yeah. And that was followed 00:07:18.740 --> 00:07:21.920 by this long, dark. He failed his fellow of arts 00:07:21.920 --> 00:07:24.959 exam multiple times, left college without a degree. 00:07:25.220 --> 00:07:28.220 And this isn't just an academic setback. In that 00:07:28.220 --> 00:07:31.040 rigid colonial system, not having credentials 00:07:31.040 --> 00:07:34.000 meant extreme poverty. Real poverty. The sources 00:07:34.000 --> 00:07:37.480 describe him as often being on the brink of starvation. 00:07:37.879 --> 00:07:40.180 This wasn't a struggling student with a support 00:07:40.180 --> 00:07:43.139 system. This was genuine economic desperation. 00:07:43.779 --> 00:07:46.600 And it was made worse by his refusal to work 00:07:46.600 --> 00:07:49.459 in any field that wasn't math. it took until 00:07:49.459 --> 00:07:52.360 about 1910 he's in his early 20s newly married 00:07:52.360 --> 00:07:54.920 to his wife janaki That he begins this systematic 00:07:54.920 --> 00:07:57.639 search, not just for a job, but for mathematicians 00:07:57.639 --> 00:07:59.540 who could actually understand what he was doing. 00:07:59.839 --> 00:08:03.339 And that search leads him to V. Ramaswamy Ayer, 00:08:03.480 --> 00:08:05.800 the founder of the Indian Mathematical Society. 00:08:06.160 --> 00:08:09.300 And Ayer was just instantly struck by the power 00:08:09.300 --> 00:08:11.639 and originality in Ramanujan's notebooks. And 00:08:11.639 --> 00:08:14.459 Ayer made a crucial decision. A really humane 00:08:14.459 --> 00:08:16.839 and critical decision. He later said he had, 00:08:16.959 --> 00:08:19.680 quote, no mind to smother his genius by an appointment 00:08:19.680 --> 00:08:21.360 in the lowest rungs of the revenue department. 00:08:21.740 --> 00:08:23.660 In other words, he refused. refused to let this 00:08:23.660 --> 00:08:25.519 talent get buried in bureaucratic paperwork. 00:08:25.879 --> 00:08:29.019 Right. He became his crucial early advocate. 00:08:29.240 --> 00:08:31.319 He helped him get letters of introduction to 00:08:31.319 --> 00:08:33.659 European scholars and, importantly, helped him 00:08:33.659 --> 00:08:36.139 secure a small research scholarship, about 75 00:08:36.139 --> 00:08:38.220 rupees a month, at the University of Madras. 00:08:38.500 --> 00:08:41.740 That small bit of recognition and Ayer's refusal 00:08:41.740 --> 00:08:45.139 to just give him a menial job, it really saved 00:08:45.139 --> 00:08:47.940 him from being lost to clerical obscurity. It 00:08:47.940 --> 00:08:51.090 saved his genius. But even with that small grant, 00:08:51.269 --> 00:08:53.730 the financial pressure was still immense. The 00:08:53.730 --> 00:08:56.409 sources detail it clearly, especially after he 00:08:56.409 --> 00:08:59.070 married Janaki in 1909 and then had to recover 00:08:59.070 --> 00:09:02.210 from surgery for hydrosil testis. He needed a 00:09:02.210 --> 00:09:04.649 stable income. So he eventually secures a post 00:09:04.649 --> 00:09:06.909 as an accounting clerk at the Madras Port Trust. 00:09:07.149 --> 00:09:09.549 He's earning a very meager 30 rupees a month. 00:09:09.710 --> 00:09:12.169 The irony is just, it's incredible. The man who 00:09:12.169 --> 00:09:14.090 would redefine number theory is working as an 00:09:14.090 --> 00:09:15.789 accountant. But what's so telling is that he 00:09:15.789 --> 00:09:18.629 would finish his clerical work easily. very quickly, 00:09:18.750 --> 00:09:20.850 which left him with all this spare time for his 00:09:20.850 --> 00:09:23.590 actual research. And his colleagues and his boss, 00:09:23.750 --> 00:09:25.750 they supported him. They recognized his unique 00:09:25.750 --> 00:09:28.990 focus. That support was absolutely vital. It 00:09:28.990 --> 00:09:31.230 let him continue his true work, which brings 00:09:31.230 --> 00:09:33.889 us to the moment that changed absolutely everything. 00:09:34.330 --> 00:09:37.850 The letter to Cambridge. Before G .H. Hardy, 00:09:37.970 --> 00:09:40.570 though. Ramanujan's work went through what you 00:09:40.570 --> 00:09:42.610 could call a rejection filter. Right. He'd sent 00:09:42.610 --> 00:09:45.090 it out before. Yes. His manuscript pages were 00:09:45.090 --> 00:09:47.669 sent to several established British mathematicians, 00:09:47.669 --> 00:09:51.429 and they just failed to recognize what they were 00:09:51.429 --> 00:09:53.769 seeing. The sources cite a couple of examples. 00:09:54.129 --> 00:09:56.950 M .J .M. Hill, a prominent mathematician, sent 00:09:56.950 --> 00:09:59.330 the papers back with this incredibly condescending 00:09:59.330 --> 00:10:01.909 comment that they were riddled with holes and 00:10:01.909 --> 00:10:04.529 that Ramanujan lacked the necessary educational 00:10:04.529 --> 00:10:08.039 background. Wow. And even worse. H .F. Baker 00:10:08.039 --> 00:10:10.480 and E .W. Hobson, both very respected names. 00:10:10.620 --> 00:10:12.919 They just returned the work. No comment at all. 00:10:12.960 --> 00:10:14.519 It was so unconventional. They didn't even think 00:10:14.519 --> 00:10:16.539 it was worth a rejection letter. So three strikes. 00:10:16.659 --> 00:10:19.340 It wasn't until January 1913 that he wrote to 00:10:19.340 --> 00:10:22.539 G .H. Hardy at Cambridge. And Hardy was, well, 00:10:22.639 --> 00:10:25.279 arguably the most brilliant and rigorous mathematician 00:10:25.279 --> 00:10:28.039 in Britain at the time. And he receives these 00:10:28.039 --> 00:10:30.340 nine pages of complex handwritten mathematics 00:10:30.340 --> 00:10:34.139 from a completely unknown clerk in India. Hardy's 00:10:34.139 --> 00:10:36.620 initial suspicion must have been off the charts. 00:10:36.700 --> 00:10:39.340 Oh, absolutely. It was warranted. He got unsolicited 00:10:39.340 --> 00:10:41.600 manuscripts all the time, usually from cranks 00:10:41.600 --> 00:10:43.840 claiming they, you know, trisected the angle 00:10:43.840 --> 00:10:46.059 or something. His first reaction was that it 00:10:46.059 --> 00:10:48.379 was probably a fraud or the work of a lunatic. 00:10:48.519 --> 00:10:51.799 But that suspicion quickly changed into something 00:10:51.799 --> 00:10:54.610 else. He and his colleague, J .E. Littlewood, 00:10:54.750 --> 00:10:57.250 started looking at the work more closely. And 00:10:57.250 --> 00:10:59.710 Hardy later confessed that, yes, some of the 00:10:59.710 --> 00:11:02.870 theorems were recognizably advanced, which confirmed 00:11:02.870 --> 00:11:05.809 Ramanujan had genuine talent. But the others, 00:11:05.990 --> 00:11:09.210 the truly novel results, he said they defeated 00:11:09.210 --> 00:11:11.929 me completely. I had never seen anything in the 00:11:11.929 --> 00:11:14.490 least like them before. It was that sheer novelty, 00:11:14.730 --> 00:11:17.210 the fact that they had no traceable lineage in 00:11:17.210 --> 00:11:19.929 Western mathematics, that forced Hardy to reconsider. 00:11:20.309 --> 00:11:22.730 And this brings us to that famous quote. the 00:11:22.730 --> 00:11:24.909 one that basically validated Ramanujan's genius 00:11:24.909 --> 00:11:28.009 for all time. Hardy concluded that the theorems 00:11:28.009 --> 00:11:30.929 must be true because if they were not true, no 00:11:30.929 --> 00:11:33.009 one would have the imagination to invent them. 00:11:33.110 --> 00:11:36.700 That quote is just. It's pivotal. It's the ultimate 00:11:36.700 --> 00:11:40.600 recognition of pure intuition. The idea is so 00:11:40.600 --> 00:11:43.139 beautiful and strange that it can't be a mistake. 00:11:43.200 --> 00:11:46.000 It has to be a revelation. It took a genius like 00:11:46.000 --> 00:11:48.340 Hardy to recognize a genius that looked nothing 00:11:48.340 --> 00:11:50.759 like himself. And let's get into the specifics 00:11:50.759 --> 00:11:53.100 of what convinced him, because it really shows 00:11:53.100 --> 00:11:56.039 the technical gulf Ramanujan had bridged on his 00:11:56.039 --> 00:11:59.750 own. The sources mention a complex infinite integral 00:11:59.750 --> 00:12:02.289 formula. But it was the infinite series for pi 00:12:02.289 --> 00:12:04.710 that really blew Hardy away, wasn't it? It was. 00:12:05.070 --> 00:12:07.470 For the listener, an infinite series is just 00:12:07.470 --> 00:12:10.049 a way to express a number by adding up an infinite 00:12:10.049 --> 00:12:13.070 sequence of smaller and smaller terms. Reminiscence 00:12:13.070 --> 00:12:16.289 series for, specifically, two -piler was derived 00:12:16.289 --> 00:12:18.309 from something called a hypergeometric series. 00:12:18.470 --> 00:12:20.440 Yeah. It was completely new to Hardy. And he 00:12:20.440 --> 00:12:22.500 said it was much more intriguing than even the 00:12:22.500 --> 00:12:24.940 work of giants like Gauss and Legendre. Yes. 00:12:25.240 --> 00:12:27.679 And it's a masterpiece of technical complexity. 00:12:27.960 --> 00:12:30.720 But what makes it so groundbreaking is its convergence 00:12:30.720 --> 00:12:33.360 rate, how fast it gets you to the answer. Right. 00:12:33.440 --> 00:12:34.940 Let's slow down on that. What does that actually 00:12:34.940 --> 00:12:37.679 mean, extraordinarily rapid convergence? Well, 00:12:37.740 --> 00:12:41.019 if you use the older standard methods, you need 00:12:41.019 --> 00:12:44.240 thousands of terms to get, say, a dozen decimal 00:12:44.240 --> 00:12:47.250 places of pi. But with Ramanujan's formula, if 00:12:47.250 --> 00:12:49.350 you just take the first two terms of the sum. 00:12:49.509 --> 00:12:51.990 Just two. Just the first two. You get a value 00:12:51.990 --> 00:12:53.950 that's already correct to 14 decimal places. 00:12:54.309 --> 00:12:57.450 Wow. That phenomenal efficiency is the basis 00:12:57.450 --> 00:12:59.889 of some of the fastest algorithms we use today 00:12:59.889 --> 00:13:02.990 to calculate pi. It shows he wasn't just generating 00:13:02.990 --> 00:13:05.629 true ideas. He was generating ideas that were 00:13:05.629 --> 00:13:08.490 computationally superior. So after this, Hardy 00:13:08.490 --> 00:13:10.450 immediately starts trying to get Ramanujan to 00:13:10.450 --> 00:13:12.730 Cambridge. But then he hits this huge cultural 00:13:12.730 --> 00:13:16.259 and religious barrier. Right. Initially, Ramanujan 00:13:16.259 --> 00:13:19.159 refused to go. Why? He cited his Orthodox Brahmin 00:13:19.159 --> 00:13:22.399 upbringing. There was intense parental and societal 00:13:22.399 --> 00:13:25.179 opposition to him traveling overseas to a foreign 00:13:25.179 --> 00:13:27.820 land. It was a massive religiously fraught decision. 00:13:28.039 --> 00:13:30.320 You could lose caste. You'd be forced to break 00:13:30.320 --> 00:13:32.419 strict dietary laws. So what was the turning 00:13:32.419 --> 00:13:34.940 point? The turning point is one of the most remarkable 00:13:34.940 --> 00:13:38.279 anecdotes in the whole history of science. It 00:13:38.279 --> 00:13:40.639 took E .H. Neville, a colleague of Hardy's who 00:13:40.639 --> 00:13:42.659 was lecturing in Madras, to help convince him. 00:13:42.970 --> 00:13:45.710 but the final decision came from his family the 00:13:45.710 --> 00:13:49.159 dream the dream The sources say Ramanujan's mother 00:13:49.159 --> 00:13:52.139 had this vivid, decisive dream where the family 00:13:52.139 --> 00:13:55.679 goddess Namagiri Thayer commanded her to let 00:13:55.679 --> 00:13:58.539 her son go to England to fulfill his life's purpose. 00:13:58.799 --> 00:14:01.960 I love that. A divine mandate that just overrides 00:14:01.960 --> 00:14:04.320 these deeply ingrained social rules. So this 00:14:04.320 --> 00:14:06.399 wasn't just a decision. It was a spiritual turning 00:14:06.399 --> 00:14:09.200 point for the whole family. It was. So in March 00:14:09.200 --> 00:14:12.600 1914, Ramanujan boards the SS Navassa, leaving 00:14:12.600 --> 00:14:15.179 his wife behind in India. And he sets sail for 00:14:15.179 --> 00:14:17.110 a life he couldn't possibly have imagined. A 00:14:17.110 --> 00:14:20.250 life that would define his genius, but also tragically 00:14:20.250 --> 00:14:22.490 break his body. So Ramanujan arrives in Cambridge 00:14:22.490 --> 00:14:25.370 and this incredible collaboration begins. And 00:14:25.370 --> 00:14:28.169 this is where the tension, but also the real 00:14:28.169 --> 00:14:30.409 genius, really comes to the fore. It's a study 00:14:30.409 --> 00:14:33.309 in contrasts. You have G .H. Hardy, who is the 00:14:33.309 --> 00:14:35.789 personification of the Western academic establishment. 00:14:36.169 --> 00:14:39.009 He's an atheist, an apostle of mathematical rigor, 00:14:39.250 --> 00:14:42.250 classically educated. For him, it was all about 00:14:42.250 --> 00:14:44.789 proof, proof, and more proof. And then you have 00:14:44.789 --> 00:14:47.909 Ramanujan. Deeply religious, relying almost entirely 00:14:47.909 --> 00:14:50.110 on intuition and completely lacking any formal 00:14:50.110 --> 00:14:52.149 grounding in the systematic methods of the day. 00:14:52.230 --> 00:14:55.669 Hardy himself was just astonished to find out 00:14:55.669 --> 00:14:57.929 that Ramanujan had what the vaguest idea of what 00:14:57.929 --> 00:15:00.529 a function of a complex variable was. And that 00:15:00.529 --> 00:15:02.549 he'd never even heard of fundamental cornerstones 00:15:02.549 --> 00:15:06.200 of modern analysis like Cauchy's theorem. Okay, 00:15:06.240 --> 00:15:08.620 but how is that even possible? How can a mathematician 00:15:08.620 --> 00:15:11.580 be world -class in number theory, a field that 00:15:11.580 --> 00:15:14.679 leans so heavily on complex analysis, but have 00:15:14.679 --> 00:15:16.840 never heard of one of its most basic theorems? 00:15:17.039 --> 00:15:19.460 It just tells you that his mind worked in a fundamentally 00:15:19.460 --> 00:15:22.259 different way. He was discovering these complex 00:15:22.259 --> 00:15:25.240 analytical results through a process that Hardy 00:15:25.240 --> 00:15:28.340 described as a mingled argument, intuition, and 00:15:28.340 --> 00:15:30.899 induction. A process that, frankly, Ramanujan 00:15:30.899 --> 00:15:33.299 himself couldn't really explain coherently. So 00:15:33.299 --> 00:15:35.960 Hardy is faced with this unbelievable intellectual 00:15:35.960 --> 00:15:39.419 challenge. He has to mentor this guy, teach him 00:15:39.419 --> 00:15:42.500 the systematic rules of formal proof, fill these 00:15:42.500 --> 00:15:45.320 enormous gaps in his education. But he has to 00:15:45.320 --> 00:15:47.600 do it without crushing the very source of that 00:15:47.600 --> 00:15:50.659 extraordinary intuition. It's an incredibly delicate 00:15:50.659 --> 00:15:53.299 balancing act. He had to treat him less like 00:15:53.299 --> 00:15:55.639 a student who needed to be fixed and more like 00:15:55.639 --> 00:15:58.419 an oracle who sometimes needed translation. Exactly. 00:15:58.659 --> 00:16:01.120 It required tremendous patience and respect from 00:16:01.120 --> 00:16:03.740 Hardy. He was trying to bring Ramanujan up to 00:16:03.740 --> 00:16:07.480 speed on like 150 years of European mathematics, 00:16:07.799 --> 00:16:10.840 while Ramanujan was simultaneously leaping 50 00:16:10.840 --> 00:16:13.580 years into the future with entirely new theorems. 00:16:13.639 --> 00:16:15.740 And the key struggle was just translating that 00:16:15.740 --> 00:16:18.659 instantaneous, intuitive mathematical truth into 00:16:18.659 --> 00:16:22.220 the language of rigorous, verifiable proof that 00:16:22.220 --> 00:16:24.379 the academic world demanded. And all this is 00:16:24.379 --> 00:16:26.299 happening during the worst possible time. World 00:16:26.299 --> 00:16:30.149 War I. From 1914 to 1918, which just compounded 00:16:30.149 --> 00:16:32.370 his physical struggle, the silent tragedy of 00:16:32.370 --> 00:16:34.950 his time in England. Ramanujan was a strict Brahmin 00:16:34.950 --> 00:16:37.629 vegetarian. And this is a point that often gets 00:16:37.629 --> 00:16:40.149 overlooked, but it's absolutely critical. Maintaining 00:16:40.149 --> 00:16:42.830 that diet, it wasn't just about not eating meat. 00:16:43.110 --> 00:16:45.690 No, it was about abstaining from eggs, fish, 00:16:45.950 --> 00:16:48.309 often certain root vegetables, and sometimes 00:16:48.309 --> 00:16:51.409 even requiring self -preparation to ensure ritual 00:16:51.409 --> 00:16:54.740 purity. This was nearly impossible in Edwardian, 00:16:54.840 --> 00:16:58.019 and especially wartime England. And wartime rationing 00:16:58.019 --> 00:17:01.000 made it exponentially harder. Access to specialized 00:17:01.000 --> 00:17:03.519 Indian ingredients was non -existent. He was 00:17:03.519 --> 00:17:05.859 forced to cook all his own food, which took up 00:17:05.859 --> 00:17:08.480 hours of his time. He became deeply isolated, 00:17:08.799 --> 00:17:11.400 and that was compounded by the cold, the climate, 00:17:11.500 --> 00:17:14.039 the relentless intellectual pressure. His health 00:17:14.039 --> 00:17:16.650 just... deteriorates he's eventually diagnosed 00:17:16.650 --> 00:17:18.970 with tuberculosis coupled with severe vitamin 00:17:18.970 --> 00:17:21.450 deficiency which was a direct result of his diet 00:17:21.450 --> 00:17:24.349 and his isolation and yet despite this overwhelming 00:17:24.349 --> 00:17:27.349 physical and mental distress the academic establishment 00:17:27.349 --> 00:17:29.430 was finally forced to acknowledge his genius 00:17:30.059 --> 00:17:32.240 We see that through this rapid succession of 00:17:32.240 --> 00:17:35.059 honors. In March 1916, he gets a Bachelor of 00:17:35.059 --> 00:17:37.380 Arts by research degree for his massive paper 00:17:37.380 --> 00:17:40.339 on highly composite numbers. A paper Hardy later 00:17:40.339 --> 00:17:42.779 said displayed Ramanujan's extraordinary mastery 00:17:42.779 --> 00:17:45.920 over the algebra of inequalities. And then come 00:17:45.920 --> 00:17:49.819 the historic elections in 1918. In May, he's 00:17:49.819 --> 00:17:52.380 elected a fellow of the Royal Society, the FRS, 00:17:52.420 --> 00:17:55.079 a monumental achievement. He was only the second 00:17:55.079 --> 00:17:57.480 Indian ever admitted. And then in October of 00:17:57.480 --> 00:17:59.799 that same year. He becomes the first Indian to 00:17:59.799 --> 00:18:02.200 be elected a fellow of Trinity College, Cambridge. 00:18:02.460 --> 00:18:05.819 That swift series of high honors all within about 00:18:05.819 --> 00:18:08.799 seven months was absolutely crucial. It cemented 00:18:08.799 --> 00:18:12.140 his legacy. And maybe more importantly, it completely 00:18:12.140 --> 00:18:15.160 validated Hardy's initial, highly controversial 00:18:15.160 --> 00:18:18.240 belief in this isolated clerk from Madras. It 00:18:18.240 --> 00:18:20.440 proved that even if his methods were unconventional, 00:18:20.640 --> 00:18:23.519 the depth of his work was simply undeniable. 00:18:23.960 --> 00:18:25.759 So let's turn to the mathematics itself, the 00:18:25.759 --> 00:18:28.700 architecture of his genius and that distinction 00:18:28.700 --> 00:18:31.660 between his insight and the formal proof. That 00:18:31.660 --> 00:18:34.119 remains the core theme. He was just generating 00:18:34.119 --> 00:18:36.180 this abundance of formulae that were completely 00:18:36.180 --> 00:18:38.839 novel. I mean, he was opening up entire new fields 00:18:38.839 --> 00:18:40.640 of research before anyone else even realized 00:18:40.640 --> 00:18:43.119 those fields existed. Let's go back to some of 00:18:43.119 --> 00:18:45.099 the specific examples to really understand the 00:18:45.099 --> 00:18:47.380 depth here. We talked about his famous infinite 00:18:47.380 --> 00:18:50.259 series for Tarkas. Which is a highly technical 00:18:50.259 --> 00:18:52.019 formula based on something called the negative 00:18:52.019 --> 00:18:54.220 fundamental discriminant dollar dollars equals 00:18:54.220 --> 00:18:57.980 known as just 232. It's a complex concept from 00:18:57.980 --> 00:19:00.519 number theory. And we noted its speed, its convergence. 00:19:00.900 --> 00:19:03.140 But let's just pause on what that means for us 00:19:03.140 --> 00:19:06.440 today for modern computation. Because that series 00:19:06.440 --> 00:19:08.880 is so efficient, it's basically the ancestor 00:19:08.880 --> 00:19:11.599 of algorithms that are being used right now by 00:19:11.599 --> 00:19:13.799 researchers and engineers who need extremely 00:19:13.799 --> 00:19:17.160 precise values for pi. His intuitive algebraic 00:19:17.160 --> 00:19:19.680 structure provided the most practical path forward, 00:19:19.799 --> 00:19:22.259 even a century later. Okay, now let's switch 00:19:22.259 --> 00:19:24.619 gears completely. Let's go from that cold computational 00:19:24.619 --> 00:19:29.049 rigor to pure instantaneous insight. Which brings 00:19:29.049 --> 00:19:32.490 us to the famous taxi cab number anecdote. 1729. 00:19:32.890 --> 00:19:35.829 1729. This story, it takes place when Hardy is 00:19:35.829 --> 00:19:37.630 visiting Ramanujan in the hospital at Putney, 00:19:37.750 --> 00:19:40.069 and it just perfectly captures his unique relationship 00:19:40.069 --> 00:19:42.269 with numbers. It does. Hardy, you know, trying 00:19:42.269 --> 00:19:44.230 to make small talk, mentions he arrived in a 00:19:44.230 --> 00:19:47.089 cab with the number 1729, and he remarks that 00:19:47.089 --> 00:19:49.269 it seemed rather a dull one. It's such a beautifully 00:19:49.269 --> 00:19:52.089 reserved British understatement. But Ramanujan, 00:19:52.109 --> 00:19:55.369 despite being severely ill, he instantly corrects 00:19:55.369 --> 00:19:57.819 him. He says, no, it is a very interesting number. 00:19:57.920 --> 00:20:00.119 It is the smallest number expressible as the 00:20:00.119 --> 00:20:02.680 sum of two cubes in two different ways. That's 00:20:02.680 --> 00:20:05.980 just that immediate recall and computation. The 00:20:05.980 --> 00:20:08.259 ability to see the deep properties of a four 00:20:08.259 --> 00:20:10.779 digit number instantly is astounding. And the 00:20:10.779 --> 00:20:17.279 solution is $1729 equals 13 plus $129 equals 00:20:17.279 --> 00:20:22.059 93 plus 103 to 3. It shows that numbers weren't 00:20:22.059 --> 00:20:24.500 just abstract concepts to him. They were personal. 00:20:24.990 --> 00:20:27.470 As Hardy later recalled his collaborator Littlewood 00:20:27.470 --> 00:20:30.910 saying, every positive integer was one of Ramanujan's 00:20:30.910 --> 00:20:33.589 personal friends. He wasn't calculating. He was 00:20:33.589 --> 00:20:35.670 just recognizing a friend's hidden talent. And 00:20:35.670 --> 00:20:37.890 this instantaneous insight wasn't limited to 00:20:37.890 --> 00:20:40.069 small numbers. It extended to solving entire 00:20:40.069 --> 00:20:42.490 classes of difficult problems, like the one posed 00:20:42.490 --> 00:20:45.250 by his Indian colleague, P .C. Mahalanobis, the 00:20:45.250 --> 00:20:47.349 houses problem. Right. The problem is a clever 00:20:47.349 --> 00:20:49.250 one. Imagine a street with houses numbered 1, 00:20:49.410 --> 00:20:52.000 2, 3. up to some number nullar. And there's a 00:20:52.000 --> 00:20:54.720 specific house, let's call it $6, where the sum 00:20:54.720 --> 00:20:56.539 of the house numbers to the left of it equals 00:20:56.539 --> 00:20:58.299 the sum of the house numbers to the right of 00:20:58.299 --> 00:21:00.740 it. Mahalanobis had worked out a specific case 00:21:00.740 --> 00:21:03.839 and asked Ramanujan for the solution. Most mathematicians 00:21:03.839 --> 00:21:06.259 would spend hours setting up the equations. But 00:21:06.259 --> 00:21:09.529 Ramanujan's response was just... astounding he 00:21:09.529 --> 00:21:12.410 didn't just give the specific solution he immediately 00:21:12.410 --> 00:21:14.930 gave the answer as a continued fraction which 00:21:14.930 --> 00:21:17.650 solved the entire class of problems the infinite 00:21:17.650 --> 00:21:19.930 set of possibilities mahalanobis asked him how 00:21:19.930 --> 00:21:22.289 he did it and ramojin just said it is simple 00:21:22.289 --> 00:21:25.190 the minute i heard the problem i knew that the 00:21:25.190 --> 00:21:27.829 answer was a continued fraction which continued 00:21:27.829 --> 00:21:29.930 fraction i asked myself then the answer came 00:21:29.930 --> 00:21:32.390 to my mind He saw the structure, not the calculation. 00:21:32.710 --> 00:21:35.789 That's the essence of his genius. And this focus 00:21:35.789 --> 00:21:38.730 on integers led to what is probably his greatest 00:21:38.730 --> 00:21:42.430 collaboration with Hardy in 1918, the study of 00:21:42.430 --> 00:21:44.730 the partition function, P -S -A -9. Okay, so 00:21:44.730 --> 00:21:47.029 the partition function, P -A -N -A, it asks a 00:21:47.029 --> 00:21:49.349 simple question. In how many different ways can 00:21:49.349 --> 00:21:51.809 you break up an integer, noller, into a sum of 00:21:51.809 --> 00:21:53.650 positive integers? So for the number four, for 00:21:53.650 --> 00:21:55.450 example. The number four can be partitioned in 00:21:55.450 --> 00:21:58.470 five ways. You can have 4, 3 plus 1, 2 plus 2, 00:21:58.490 --> 00:22:00.829 plus 1 plus 1, or 1 plus 1 plus 1 plus 1. It 00:22:00.829 --> 00:22:02.990 seems simple, but as the number gets bigger... 00:22:02.990 --> 00:22:06.069 The numbers just explode. The number of partitions 00:22:06.069 --> 00:22:09.549 for 200 is a number that has 14 digits. It's 00:22:09.549 --> 00:22:11.529 computationally impossible to list them all. 00:22:11.710 --> 00:22:14.509 So how did they solve it? Their work together 00:22:14.509 --> 00:22:17.109 yielded a non -convergent asymptotic series, 00:22:17.349 --> 00:22:20.390 which is a highly complex formula that lets you 00:22:20.390 --> 00:22:23.650 calculate to act exactly. And crucially... This 00:22:23.650 --> 00:22:26.170 collaboration pioneered what became known as 00:22:26.170 --> 00:22:29.049 the powerful circle method. Okay, so what is 00:22:29.049 --> 00:22:30.990 the circle method in simple terms? It sounds 00:22:30.990 --> 00:22:33.250 very abstract. It is abstract, but we can make 00:22:33.250 --> 00:22:35.369 it accessible. Think of a complex number not 00:22:35.369 --> 00:22:37.609 as a point on a line, but as a point on a two 00:22:37.609 --> 00:22:39.890 -dimensional plane. Right. The circle method 00:22:39.890 --> 00:22:41.670 is a technique that uses complex integration, 00:22:42.029 --> 00:22:44.769 basically calculating an area, but on this complex 00:22:44.769 --> 00:22:48.380 plane, by integrating over a circle. By carefully 00:22:48.380 --> 00:22:50.420 dividing that circle into what they called major 00:22:50.420 --> 00:22:53.099 and minor arcs, they could isolate the contribution 00:22:53.099 --> 00:22:55.859 of specific mathematical properties. It was a 00:22:55.859 --> 00:22:57.920 revolutionary way of thinking about how to count 00:22:57.920 --> 00:23:00.440 extremely difficult things. Using advanced geometry 00:23:00.440 --> 00:23:02.960 to solve what seems like a simple counting problem. 00:23:03.339 --> 00:23:06.140 Exactly. And even in the last year of his life, 00:23:06.220 --> 00:23:09.019 when he was frail and suffering, he made one 00:23:09.019 --> 00:23:12.359 final, crucial discovery that remained a mystery 00:23:12.359 --> 00:23:16.190 for decades. Mach theta functions. Yes, this 00:23:16.190 --> 00:23:18.589 is maybe the ultimate example of his insights 00:23:18.589 --> 00:23:20.829 being ahead of their time. He generated these 00:23:20.829 --> 00:23:23.589 functions, and they were related to modular forms, 00:23:23.730 --> 00:23:25.730 these mathematical structures with incredible 00:23:25.730 --> 00:23:28.410 symmetry, but they didn't quite fit the definitions 00:23:28.410 --> 00:23:31.430 of the time. For years, mathematicians didn't 00:23:31.430 --> 00:23:32.930 know what to do with them. They were like puzzle 00:23:32.930 --> 00:23:35.730 pieces from a future game. So what's their significance 00:23:35.730 --> 00:23:38.009 today? Where did they fit in? Well, it wasn't 00:23:38.009 --> 00:23:40.730 until modern mathematics caught up that we fully 00:23:40.730 --> 00:23:43.559 understood them. They are now known as the holomorphic 00:23:43.559 --> 00:23:46.480 parts of harmonic weak mass forms. Which means? 00:23:46.759 --> 00:23:49.200 It means they are specific parts of a much larger, 00:23:49.279 --> 00:23:52.019 highly symmetric structure that wasn't even fully 00:23:52.019 --> 00:23:55.079 defined until the 21st century. It just proves 00:23:55.079 --> 00:23:57.819 that even on his deathbed, Ramanujan was generating 00:23:57.819 --> 00:24:00.359 insights that were literally 70 or 80 years ahead 00:24:00.359 --> 00:24:03.539 of the field's ability to classify or even understand 00:24:03.539 --> 00:24:06.039 them. When we talk about Ramanujan's immense 00:24:06.039 --> 00:24:10.220 output, those almost 3 ,900 results, we have 00:24:10.220 --> 00:24:12.319 to go back to the physical record of his work, 00:24:12.519 --> 00:24:15.640 the famous notebooks. And they were filled almost 00:24:15.640 --> 00:24:19.160 entirely with final results, virtually no derivations 00:24:19.160 --> 00:24:21.759 or proofs recorded anywhere. And this is a huge 00:24:21.759 --> 00:24:24.099 point of fascination for mathematicians. This 00:24:24.099 --> 00:24:27.160 lack of proof led to the early and incorrect 00:24:27.160 --> 00:24:29.759 assumption that he just received the results 00:24:29.759 --> 00:24:32.440 through divine insight and couldn't actually 00:24:32.440 --> 00:24:35.720 prove them. But that's not the case. No. Brucie 00:24:35.720 --> 00:24:37.700 Berndt, the mathematician who has spent decades 00:24:37.700 --> 00:24:40.619 meticulously studying these notebooks, he asserts 00:24:40.619 --> 00:24:42.640 that Ramanujan most certainly was able to prove 00:24:42.640 --> 00:24:45.940 the vast majority of his results. So why the 00:24:45.940 --> 00:24:48.380 radical omission? Why only record the destination 00:24:48.380 --> 00:24:51.059 and not the route? Well, the practical reasons 00:24:51.059 --> 00:24:53.000 are really compelling, and they speak volumes 00:24:53.000 --> 00:24:55.539 about his impoverished circumstances. First... 00:24:55.799 --> 00:24:57.819 Caper was extremely expensive in India at the 00:24:57.819 --> 00:24:59.500 time. He couldn't afford to waste it. He couldn't 00:24:59.500 --> 00:25:01.880 afford it. So he adopted a common practice for 00:25:01.880 --> 00:25:04.640 math students in the Madras presidency. He worked 00:25:04.640 --> 00:25:07.240 out the full proofs and all the complex algebra 00:25:07.240 --> 00:25:10.299 on a reusable slate. Which you can just... wipe 00:25:10.299 --> 00:25:12.440 clean for the next problem. Right. So imagine 00:25:12.440 --> 00:25:15.160 the physical labor involved. He's not just working 00:25:15.160 --> 00:25:18.599 out 3 ,900 results. He's calculating, writing, 00:25:18.779 --> 00:25:20.980 erasing, and refining these monumental themes 00:25:20.980 --> 00:25:23.920 on a slate, and then committing only the final 00:25:23.920 --> 00:25:27.000 distilled identity to the expensive paper notebook. 00:25:27.279 --> 00:25:29.720 That contrast between the physical struggle for 00:25:29.720 --> 00:25:31.960 resources and the intellectual sophistication 00:25:31.960 --> 00:25:35.359 is just, it's staggering. And second, the format 00:25:35.359 --> 00:25:38.319 was also heavily influenced by Carr's synopsis, 00:25:38.319 --> 00:25:40.559 the book that started it all for him, which just 00:25:40.559 --> 00:25:43.160 stated results without proofs. He was likely 00:25:43.160 --> 00:25:45.059 using the notebooks as a personal inventory, 00:25:45.339 --> 00:25:47.740 needing only the final equations as reference 00:25:47.740 --> 00:25:50.529 points. And speaking of records, we have to mention 00:25:50.529 --> 00:25:53.769 the rediscovery in 1976 of a fourth notebook. 00:25:54.009 --> 00:25:56.910 A lost notebook. 87 unorganized pages containing 00:25:56.910 --> 00:25:59.470 discoveries from his very last year of life. 00:25:59.589 --> 00:26:02.029 And its discovery caused a firestorm of excitement 00:26:02.029 --> 00:26:04.910 among mathematicians. It's fueled decades of 00:26:04.910 --> 00:26:07.430 work that's still going on today. It just emphasizes 00:26:07.430 --> 00:26:10.670 that his output was so vast, so rich, that even 00:26:10.670 --> 00:26:13.130 the unorganized messy notes he wrote in the final 00:26:13.130 --> 00:26:15.650 months of his life contained foundational new 00:26:15.650 --> 00:26:18.740 mathematics. And now we have to address the spiritual 00:26:18.740 --> 00:26:21.740 dimension because he himself linked it inextricably 00:26:21.740 --> 00:26:25.000 to his work. Ramanujan was a rigorously orthodox 00:26:25.000 --> 00:26:27.720 Hindu. And he credited his mathematical acumen 00:26:27.720 --> 00:26:30.299 directly to his family goddess, Namagiri Thayar. 00:26:30.460 --> 00:26:33.160 For him, mathematics wasn't just abstract logic. 00:26:33.380 --> 00:26:36.279 It was a way of accessing profound eternal truths. 00:26:37.000 --> 00:26:39.559 He often said he looked to the goddess for inspiration. 00:26:39.920 --> 00:26:42.819 He even described this vivid dream similar to 00:26:42.819 --> 00:26:44.420 the one that convinced his mother to let him 00:26:44.420 --> 00:26:47.680 travel. He saw a red screen formed by flowing 00:26:47.680 --> 00:26:51.059 blood. And on this screen, a hand began to write 00:26:51.059 --> 00:26:54.440 complex elliptic integrals that just stuck instantly 00:26:54.440 --> 00:26:56.700 in his mind. And as soon as he woke up, he wrote 00:26:56.700 --> 00:26:59.079 them down. It all leads to one of his most powerful 00:26:59.079 --> 00:27:01.880 and famous quotes. An equation for me has no 00:27:01.880 --> 00:27:04.079 meaning unless it expresses a thought of God. 00:27:04.599 --> 00:27:06.660 And while Hardy, the staunch atheist, really 00:27:06.660 --> 00:27:08.940 struggled to reconcile this mystical view with 00:27:08.940 --> 00:27:11.339 academic rigor, there's no question that for 00:27:11.339 --> 00:27:13.680 Ramanujan, the deep truths of mathematics and 00:27:13.680 --> 00:27:16.460 divine insight were one in the same. It gave 00:27:16.460 --> 00:27:18.720 him this unwavering confidence in his results. 00:27:18.980 --> 00:27:21.259 It gives you a balanced understanding of his 00:27:21.259 --> 00:27:23.519 work. His intuition wasn't just a psychological 00:27:23.519 --> 00:27:26.480 phenomenon. For him, it was a deeply personal 00:27:26.480 --> 00:27:28.859 spiritual connection that allowed him to bypass 00:27:28.859 --> 00:27:31.640 the standard process and arrive directly at these 00:27:31.640 --> 00:27:34.359 fundamental realities. But despite this staggering 00:27:34.359 --> 00:27:36.640 flow of insights he was providing right up to 00:27:36.640 --> 00:27:39.500 the end, his failing health really dictated his 00:27:39.500 --> 00:27:42.740 final days. The sources note this immense strain 00:27:42.740 --> 00:27:45.359 of his life in England, which led to a devastating 00:27:45.359 --> 00:27:48.539 physical and mental collapse. Including a suicide 00:27:48.539 --> 00:27:52.400 attempt in late 1917 or early 1918, he tried 00:27:52.400 --> 00:27:54.299 to jump onto the tracks of a London underground 00:27:54.299 --> 00:27:56.980 station. An attempt that Hardy intervened to 00:27:56.980 --> 00:27:59.430 prevent? Which led to his release from arrest. 00:27:59.789 --> 00:28:02.329 His illness, combined with the dietary strain 00:28:02.329 --> 00:28:04.690 and the miserable climate, it all compelled him 00:28:04.690 --> 00:28:08.109 to return to India in 1919. He was severely weakened, 00:28:08.269 --> 00:28:10.589 but even back home, he kept working. He died 00:28:10.589 --> 00:28:14.069 the following year, April 26, 1920, at the tragic 00:28:14.069 --> 00:28:16.730 age of 32. And it's just heart -wrenching to 00:28:16.730 --> 00:28:19.230 read that even in severe pain in his final days, 00:28:19.450 --> 00:28:21.950 his wife recounts that he continued doing his 00:28:21.950 --> 00:28:24.710 mathematics, filling sheet after sheet with numbers. 00:28:25.230 --> 00:28:27.529 And what makes his early death even more tragic 00:28:27.529 --> 00:28:30.170 is the modern analysis of his medical condition. 00:28:31.230 --> 00:28:34.569 A 1994 review of his medical records suggested 00:28:34.569 --> 00:28:37.190 his illness was actually more consistent with 00:28:37.190 --> 00:28:40.049 hepatic amoebiasis. Which is a serious complication 00:28:40.049 --> 00:28:42.130 from untreated dysentery, something that was 00:28:42.130 --> 00:28:45.210 rampant in Madras at the time. Not the tuberculosis 00:28:45.210 --> 00:28:47.470 he was originally diagnosed with. And if we connect 00:28:47.470 --> 00:28:49.809 that to the bigger picture, the tragedy just 00:28:49.809 --> 00:28:52.869 deepens because amoebiasis was treatable. It 00:28:52.869 --> 00:28:55.470 was a treatable and often curable disease, especially 00:28:55.470 --> 00:28:58.119 by the time he left England. If he had received 00:28:58.119 --> 00:29:00.339 an accurate diagnosis and the right medical care, 00:29:00.539 --> 00:29:03.319 it is highly probable he would have lived. The 00:29:03.319 --> 00:29:05.880 fact that a treatable illness, compounded by 00:29:05.880 --> 00:29:08.319 the strain of living abroad in wartime, might 00:29:08.319 --> 00:29:10.900 have been the cause. It's arguably the single 00:29:10.900 --> 00:29:13.079 greatest tragedy in modern mathematical history. 00:29:13.160 --> 00:29:15.980 It is. Yet his work transcended his short life. 00:29:16.240 --> 00:29:18.640 It laid the groundwork for the next century of 00:29:18.640 --> 00:29:21.759 math. A major focus of that posthumous work is 00:29:21.759 --> 00:29:24.119 the Ramanujan conjecture. This focuses on the 00:29:24.119 --> 00:29:26.039 size of something called the tau function, which 00:29:26.039 --> 00:29:28.940 comes from his work on modular forms. And the 00:29:28.940 --> 00:29:31.519 Ramanujan conjecture became absolutely central 00:29:31.519 --> 00:29:34.539 to 20th century mathematics. It was finally proven 00:29:34.539 --> 00:29:37.700 in 1973 as a consequence of Pierre de Lyon's 00:29:37.700 --> 00:29:40.240 proof of the Weyl conjectures. And that work 00:29:40.240 --> 00:29:42.220 earned de Lyon the Fields Medal, the highest 00:29:42.220 --> 00:29:45.680 honor in math, in 1978. But the connection goes 00:29:45.680 --> 00:29:48.200 even deeper. This entire theoretical framework, 00:29:48.559 --> 00:29:50.960 which sprang directly from Ramanujan's insight, 00:29:51.200 --> 00:29:53.920 was later essential to the work needed to prove 00:29:53.920 --> 00:29:56.640 Fermat's last theorem. Maybe the most famous 00:29:56.640 --> 00:29:58.799 and difficult problem in mathematical history. 00:29:59.180 --> 00:30:02.339 So think about that intellectual flow. A conjecture 00:30:02.339 --> 00:30:04.880 made by a self -taught clerk, based on intuition 00:30:04.880 --> 00:30:08.000 in the early 1900s, provided the structural blueprint 00:30:08.000 --> 00:30:10.599 needed almost a century later to solve a problem 00:30:10.599 --> 00:30:12.740 that had stumped mathematicians since the 17th 00:30:12.740 --> 00:30:15.400 century. That's a legacy that defines the entire 00:30:15.400 --> 00:30:18.099 field. And this overwhelming contribution led 00:30:18.099 --> 00:30:20.819 directly to Hardy's ultimate, iconic assessment 00:30:20.819 --> 00:30:23.720 of him. Hardy famously devised this scale of 00:30:23.720 --> 00:30:26.960 pure mathematical talent from 0 to 100. He rated 00:30:26.960 --> 00:30:29.880 himself at a modest 25, his great collaborator 00:30:29.880 --> 00:30:32.920 Littlewood at 30, and David Hilbert, one of the 00:30:32.920 --> 00:30:35.380 foundational titans of 20th century math, at 00:30:35.380 --> 00:30:37.579 80. And Ramanujan. Hardy gave him a score of 00:30:37.579 --> 00:30:40.910 100. A perfect score, placing him above his own 00:30:40.910 --> 00:30:43.309 greatest contemporaries. You'd have to pause 00:30:43.309 --> 00:30:46.029 and reflect on the sheer magnitude of that, placing 00:30:46.029 --> 00:30:48.809 a self -taught figure above a giant like Hilbert. 00:30:49.009 --> 00:30:52.250 It speaks volumes about the raw, explosive power 00:30:52.250 --> 00:30:55.390 of his genius. But Hardy's eulogy, which was 00:30:55.390 --> 00:30:58.230 published in 1940, it was ultimately balanced. 00:30:58.890 --> 00:31:01.690 He noted both the staggering depth and the glaring 00:31:01.690 --> 00:31:04.509 limitations imposed by his lack of formal education. 00:31:04.849 --> 00:31:07.430 He said Ramanujan's insight was quite amazing 00:31:07.430 --> 00:31:09.789 and altogether beyond anything I've met with 00:31:09.789 --> 00:31:12.390 in any European mathematician. He called him 00:31:12.390 --> 00:31:14.910 the greatest formalist of his time, meaning the 00:31:14.910 --> 00:31:17.710 best at manipulating complex expressions. But 00:31:17.710 --> 00:31:19.829 he also noted that his lack of knowledge was 00:31:19.829 --> 00:31:22.839 as startling as its profanity. For instance, 00:31:23.059 --> 00:31:25.119 Ramanujan had to rediscover many known results 00:31:25.119 --> 00:31:26.940 because he just hadn't read the modern texts. 00:31:27.079 --> 00:31:29.359 His achievements were greatest in algebra, specifically 00:31:29.359 --> 00:31:31.720 hypergeometric series and continued fractions. 00:31:32.099 --> 00:31:34.680 But despite those limitations, Hardy's final 00:31:34.680 --> 00:31:36.900 words on his collaborator were so prophetic. 00:31:37.180 --> 00:31:39.779 He believed that what Ramanujan had accomplished 00:31:39.779 --> 00:31:43.359 was wonderful enough. When the researches which 00:31:43.359 --> 00:31:45.940 his work has suggested have been completed, it 00:31:45.940 --> 00:31:48.000 will probably seem a good deal more wonderful 00:31:48.000 --> 00:31:50.380 than it does today. And that prediction has absolutely 00:31:50.380 --> 00:31:53.549 proven true. Today, Ramanujan's influence extends 00:31:53.549 --> 00:31:56.910 far beyond academic papers. He's rightly recognized 00:31:56.910 --> 00:32:00.670 as a global cultural icon. India celebrates his 00:32:00.670 --> 00:32:03.609 birthday, December 22nd, as National Mathematics 00:32:03.609 --> 00:32:06.309 Day. And the institutional honors are plentiful. 00:32:06.730 --> 00:32:09.250 There's the Sastra Ramanujan Prize, which gives 00:32:09.250 --> 00:32:11.950 $10 ,000 annually to mathematicians under 32 00:32:11.950 --> 00:32:14.990 working in areas he influenced. There's even 00:32:14.990 --> 00:32:17.670 the Ramanujan IT City, a huge special economic 00:32:17.670 --> 00:32:20.450 zone in Chennai. Cementing his image as a national 00:32:20.450 --> 00:32:22.349 figure of intellectual prowess and innovation 00:32:22.349 --> 00:32:24.970 and his life story is just, it's tailor -made 00:32:24.970 --> 00:32:27.690 for film. Oh, absolutely. The 2015 film The Man 00:32:27.690 --> 00:32:29.630 Who Knew Infinity brought his struggles and his 00:32:29.630 --> 00:32:32.190 partnership with Hardy to a massive international 00:32:32.190 --> 00:32:35.410 audience. And even in broader pop culture, his 00:32:35.410 --> 00:32:38.089 name has become shorthand for intuitive, effortless 00:32:38.089 --> 00:32:41.029 genius. He was famously mentioned in the 1997 00:32:41.029 --> 00:32:44.369 film Good Will Hunting. Where Professor Lambeau 00:32:44.369 --> 00:32:46.269 compares the main character's raw mathematical 00:32:46.269 --> 00:32:49.950 brilliance to Ramanujan's. And he inspired the 00:32:49.950 --> 00:32:52.690 character Amita Ramanujan in the TV series Numbers, 00:32:52.690 --> 00:32:55.670 who was also steeped in number theory. So what 00:32:55.670 --> 00:32:58.190 does this all mean for us? For the learner trying 00:32:58.190 --> 00:33:01.319 to grasp the magnitude of his life? If there's 00:33:01.319 --> 00:33:03.359 one summary takeaway from this deep dive, for 00:33:03.359 --> 00:33:05.539 me, is that Ramanujan proved that mathematical 00:33:05.539 --> 00:33:08.160 rigor, while absolutely essential for validation, 00:33:08.539 --> 00:33:11.059 can sometimes follow intuition and inspiration. 00:33:11.420 --> 00:33:13.779 It's a powerful reminder that pure talent can 00:33:13.779 --> 00:33:16.500 emerge, unstructured and profound, from the most 00:33:16.500 --> 00:33:19.519 isolated circumstances. Indeed. And the study 00:33:19.519 --> 00:33:21.740 of his work continues to yield surprises even 00:33:21.740 --> 00:33:24.539 now. Bruce Burnt noted that nearly all of the 00:33:24.539 --> 00:33:26.460 thousands of conjectures he made have been proven 00:33:26.460 --> 00:33:28.710 correct. often leading to brand new insights. 00:33:28.910 --> 00:33:30.910 Which brings us to a final provocative thought, 00:33:31.069 --> 00:33:33.289 building on the profound depth of his unfinished 00:33:33.289 --> 00:33:36.430 legacy. Even the casual passing comments in his 00:33:36.430 --> 00:33:38.970 notebooks, where he'd remark about certain formulas 00:33:38.970 --> 00:33:42.349 having simple properties or similar outputs to 00:33:42.349 --> 00:33:45.470 others, have turned out to be profound and subtle 00:33:45.470 --> 00:33:48.109 number theory results that remained completely 00:33:48.109 --> 00:33:50.809 unsuspected until almost a century after his 00:33:50.809 --> 00:33:52.970 death. So you have to ask... What other profound 00:33:52.970 --> 00:33:55.410 mathematical secrets, entire new fields of inquiry, 00:33:55.609 --> 00:33:57.970 might still be hidden within those unexamined 00:33:57.970 --> 00:34:00.150 notes, just waiting for us to catch up to the 00:34:00.150 --> 00:34:02.450 mind of a genius who lived a century too soon? 00:34:02.609 --> 00:34:05.410 The exploration is... far far from over. Thank 00:34:05.410 --> 00:34:07.529 you for joining us for this deep dive into the 00:34:07.529 --> 00:34:09.789 extraordinary life and shocking mathematical 00:34:09.789 --> 00:34:11.889 legacy of Srinivasa Ramanujan.