4 ms·

How did he come up with it?

deleted 4y ago
[deleted]
version_five 4y ago
I don't know, other than he was very smart. But a potential way that the article hints at is that the factorial is related to the gamma function which I believe is the solution to a differential equation... and so there may be some insight to be had in looking at the relevant differential equation and trying to approximate its solution. Edit after googling: the gamma function not the solution to a differential equation, in fact there is a proof that it cant be [*] but it's formulated as an integral, whose approximation could lead to odd looking equations like the one in the article https://en.m.wikipedia.org/wiki/H%C3%B6lder%27s_theorem https://en.m.wikipedia.org/wiki/H%C3%B6lder%27s_theorem
- rq1 4y ago
  Algebraic differential equation*
vouaobrasil 4y ago
Ramanujan had a miraculous facility for just seeing expressions without much formal derivation. He worked with the British mathematician G.H. Hardy and it is clear from Hardy's description and Ramanujan's biography that Ramanujan did not have much insight into his own process or following clearly defined steps. He must have thus had a very natural insight into number theory that was mostly instinctual.
- steppi 4y ago
  Early the following year, K. S. Srinivasan, a student at Madras Christian College who'd known Ramanujan back in Kumbakonam, dropped by to see him at Summer House. "Ramanujan," he said, "they call you a genius." Hardly a genius, replied Ramanujan, "Look at my elbow. That will tell you the story." It was rough, dirty, and black. Working from his large slate, he found the quick flip between writing hand and erasing elbow a lot faster, when he was caught up in the throes of his work, than reaching for a rag. "My elbow is making a genius of me," he said. Why, Srinivasan asked, didn't he just use paper? Can't afford it, replied Ramanujan. He was getting money from Ramachandra Rao. But that only went so far. Paper? He'd need four reams of it a month. From Robert Kanigel's The Man Who Knew Infinity [0]. My understanding is that Ramanujan did most of his derivations on slate and tended to preserve only the results on paper. His insight was tremendous, but it was backed up by a tremendous amount of elbow work. [0] https://en.wikipedia.org/wiki/The_Man_Who_Knew_Infinity_(book) https://en.wikipedia.org/wiki/The_Man_Who_Knew_Infinity_(boo...
  - UpstandingUser 4y ago
    This is one of the big tragedies of Ramanujan. We have all these miraculous mathematics from him but we don't really know how he got there. A lot of this stuff we already knew, but he seems to have found it all out independently. No idea how though, all because he was too poor to buy paper. Jeez.
    - nerdponx 4y ago
      I had no idea he did all his work on a big slate. His working memory must have been astoundingly good.
      - tshanmu 4y ago
        Maybe thats what made him what he was - working in an ephemeral medium forcing his working memory to adapt in ways that are conducive to doing mathematics? For a comparison on the other end our spatial memory structures in the brain have changed since widespread use of gps/mapping apps, apparently.
        linspace 4y ago
        Short term memory is very important for problem solving. Euler continued to work and publish mathematics after becoming blind
- deleted 4y ago
  [deleted]
- Kranar 4y ago
  I think that's part of the myth of Ramanujan, that he could just see these things and pull them seemingly out of thin air, and in fact Ramanujan himself has said that math comes to him in visions from a Hindu goddess, but a closer inspection demystifies a lot of this. What's true is that Ramanujan spent an absolute significant amount of time working on math problems of all kinds, and the vast majority of his work was simply incorrect. In fact one thing Hardy appreciated and found most extraordinary about Ramanujan was not what Ramanujan managed to get right, but the ways in which he managed to get things wrong since that kind of revealed some unique insight into how Ramanujan went about solving problems. Of course that is never what is showcased about him, instead we get infamous stories such as how Ramanujan seemingly figured out in a matter of seconds that the number 1729 is the smallest number expressible as the sum of two cubes in two different ways. What the story leaves out is that Ramanujan did not just figure this out on the fly but had been working on trying to prove Fermat's Last Theorem for years and as part of that work he had come across the number 1729 since it's related to elliptic curves (the basis of which would eventually lead to a proof of Fermat's Last Theorem). At any rate, none of this is intended to diminish Ramanujan in any way, on the contrary my hope is to celebrate the very hard work and dedication that Ramanujan put towards mathematics. He is not someone who just divined correct solutions to complex math problems out of thin air but instead he spent countless hours for decades on end devoted to his passion and persevered through much trial and error. The only things we see of his work are the things he got right, and to that end it may seem like his accomplishments are magic; the truth is that they were anything but.
  - kumarm 4y ago
    Thank you for one most rational explanation I have ever seen about Ramanujan's work.
  - qart 4y ago
    > the vast majority of his work was simply incorrect I had heard this before, and I believed it. But I got convinced that that's not true by the likes of George E. Andrews and Ken Ono. They say that most of his work is correct. All of this is still hearsay as far as I'm concerned. I'm not qualified to verify his math.
  - 4y ago
EpiMath 4y ago
There was a paper a few years ago in the American Mathematical Monthly ( sorry... can't remember the author, I think it was a Russian mathematician ) that gave some interesting heuristics for why this form is natural to consider, where the 1/30 comes from ( and considerations of a couple of alternatives to the "30" ), and the kind of intuition/thinking that Ramanujan may have used. When you only see the final form of the equation like this, it looks very mysterious and impossible that someone could find it ( and to be fair, there are other results from Ramanujan that are definitely in that class! ). Probably a google search could find the paper, it was delightful to read.

deleted 4y ago

[deleted]

I don't know, other than he was very smart. But a potential way that the article hints at is that the factorial is related to the gamma function which I believe is the solution to a differential equation... and so there may be some insight to be had in looking at the relevant differential equation and trying to approximate its solution. Edit after googling: the gamma function not the solution to a differential equation, in fact there is a proof that it cant be [*] but it's formulated as an integral, whose approximation could lead to odd looking equations like the one in the article https://en.m.wikipedia.org/wiki/H%C3%B6lder%27s_theorem https://en.m.wikipedia.org/wiki/H%C3%B6lder%27s_theorem

rq1 4y ago
Algebraic differential equation*

vouaobrasil 4y ago

Ramanujan had a miraculous facility for just seeing expressions without much formal derivation. He worked with the British mathematician G.H. Hardy and it is clear from Hardy's description and Ramanujan's biography that Ramanujan did not have much insight into his own process or following clearly defined steps. He must have thus had a very natural insight into number theory that was mostly instinctual.

steppi 4y ago
Early the following year, K. S. Srinivasan, a student at Madras Christian College who'd known Ramanujan back in Kumbakonam, dropped by to see him at Summer House. "Ramanujan," he said, "they call you a genius." Hardly a genius, replied Ramanujan, "Look at my elbow. That will tell you the story." It was rough, dirty, and black. Working from his large slate, he found the quick flip between writing hand and erasing elbow a lot faster, when he was caught up in the throes of his work, than reaching for a rag. "My elbow is making a genius of me," he said. Why, Srinivasan asked, didn't he just use paper? Can't afford it, replied Ramanujan. He was getting money from Ramachandra Rao. But that only went so far. Paper? He'd need four reams of it a month. From Robert Kanigel's The Man Who Knew Infinity [0]. My understanding is that Ramanujan did most of his derivations on slate and tended to preserve only the results on paper. His insight was tremendous, but it was backed up by a tremendous amount of elbow work. [0] https://en.wikipedia.org/wiki/The_Man_Who_Knew_Infinity_(book) https://en.wikipedia.org/wiki/The_Man_Who_Knew_Infinity_(boo...
- UpstandingUser 4y ago
  This is one of the big tragedies of Ramanujan. We have all these miraculous mathematics from him but we don't really know how he got there. A lot of this stuff we already knew, but he seems to have found it all out independently. No idea how though, all because he was too poor to buy paper. Jeez.
  - nerdponx 4y ago
    I had no idea he did all his work on a big slate. His working memory must have been astoundingly good.
    - tshanmu 4y ago
      Maybe thats what made him what he was - working in an ephemeral medium forcing his working memory to adapt in ways that are conducive to doing mathematics? For a comparison on the other end our spatial memory structures in the brain have changed since widespread use of gps/mapping apps, apparently.
      - linspace 4y ago
        Short term memory is very important for problem solving. Euler continued to work and publish mathematics after becoming blind
deleted 4y ago
[deleted]
Kranar 4y ago
I think that's part of the myth of Ramanujan, that he could just see these things and pull them seemingly out of thin air, and in fact Ramanujan himself has said that math comes to him in visions from a Hindu goddess, but a closer inspection demystifies a lot of this. What's true is that Ramanujan spent an absolute significant amount of time working on math problems of all kinds, and the vast majority of his work was simply incorrect. In fact one thing Hardy appreciated and found most extraordinary about Ramanujan was not what Ramanujan managed to get right, but the ways in which he managed to get things wrong since that kind of revealed some unique insight into how Ramanujan went about solving problems. Of course that is never what is showcased about him, instead we get infamous stories such as how Ramanujan seemingly figured out in a matter of seconds that the number 1729 is the smallest number expressible as the sum of two cubes in two different ways. What the story leaves out is that Ramanujan did not just figure this out on the fly but had been working on trying to prove Fermat's Last Theorem for years and as part of that work he had come across the number 1729 since it's related to elliptic curves (the basis of which would eventually lead to a proof of Fermat's Last Theorem). At any rate, none of this is intended to diminish Ramanujan in any way, on the contrary my hope is to celebrate the very hard work and dedication that Ramanujan put towards mathematics. He is not someone who just divined correct solutions to complex math problems out of thin air but instead he spent countless hours for decades on end devoted to his passion and persevered through much trial and error. The only things we see of his work are the things he got right, and to that end it may seem like his accomplishments are magic; the truth is that they were anything but.
- kumarm 4y ago
  Thank you for one most rational explanation I have ever seen about Ramanujan's work.
- qart 4y ago
  > the vast majority of his work was simply incorrect I had heard this before, and I believed it. But I got convinced that that's not true by the likes of George E. Andrews and Ken Ono. They say that most of his work is correct. All of this is still hearsay as far as I'm concerned. I'm not qualified to verify his math.
- 4y ago

EpiMath 4y ago

There was a paper a few years ago in the American Mathematical Monthly ( sorry... can't remember the author, I think it was a Russian mathematician ) that gave some interesting heuristics for why this form is natural to consider, where the 1/30 comes from ( and considerations of a couple of alternatives to the "30" ), and the kind of intuition/thinking that Ramanujan may have used. When you only see the final form of the equation like this, it looks very mysterious and impossible that someone could find it ( and to be fair, there are other results from Ramanujan that are definitely in that class! ). Probably a google search could find the paper, it was delightful to read.