The legacy of Srinivasa Ramanujan(thehindu.com)
thehindu.com
The legacy of Srinivasa Ramanujan
http://www.thehindu.com/sci-tech/science/article2746988.ece
7 comments
I remember once going to see him [Ramanujan] 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.
Courtesy of G.H.Hardy
Courtesy of G.H.Hardy
Did he just figured that out on the spot when his ill or knew it from before?
I think Kanigel's book talks about this. It seems he had worked through first few thousand numbers and knew all of them quite well. Thus instead of a spark of genius, this seems to be a fruit of years worth of hard work and an excellent memory.
Few thousand? There are only 12 third powers below 1729: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331, 1728.
Spotting that 729 + 1000 almost is equal to 1728 is not hard, either. Everybody who has computed that list will have noticed it. Translating that observation into a concise, interesting description is not hard, either, but it does require creativity. That, he had way more than most. I wonder what he would have said about the pair 3^5 and 7^3 (243 and 343)
Spotting that 729 + 1000 almost is equal to 1728 is not hard, either. Everybody who has computed that list will have noticed it. Translating that observation into a concise, interesting description is not hard, either, but it does require creativity. That, he had way more than most. I wonder what he would have said about the pair 3^5 and 7^3 (243 and 343)
The first few thousand integers, and knew their properties quite well; that seems to be what is meant.
He was a genius,Yet shy person ,Its said that he also gave properties of many other numbers which were not documented by anyone, But only Hardy knew them and would tell it to others.
I have a feeling that even he would not be able to tell you the answer to that question if he were alive!
robert-kanigel's book, "the man who knew infinity" is an excellent chronicle into his life, and cambridge's culture during 20th century. it was hardy who created a rising scale of mathematical abilities as follows:
g.h.hardy - 25
littlewood - 30
hilbert - 80
ramanujan - 100
You might also like to read this interview of Kanigel on his book and how he researched on Ramanujan.
http://www.thehindu.com/opinion/interview/article2747541.ece
PS: I was half inclined to share this interview (instead of the article, this is also quite interesting)
http://www.thehindu.com/opinion/interview/article2747541.ece
PS: I was half inclined to share this interview (instead of the article, this is also quite interesting)
wow ! it definitely is pretty interesting. thanks for sharing.
Whenever I think about Ramanujan, I cannot help but wonder how many Ramanujans are out there that are just not visible and struggling with poverty.
Ramanujan's second letter to Hardy [1]:
"Dear Sir, I am very much gratified on perusing your letter of the 8th February 1913. I was expecting a reply from you similar to the one which a Mathematics Professor at London wrote asking me to study carefully Bromwich's Infinite Series and not fall into the pitfalls of divergent series. … I told him that the sum of an infinite number of terms of the series: 1 + 2 + 3 + 4 + ... = −1/12 under my theory. If I tell you this you will at once point out to me the lunatic asylum as my goal. I dilate on this simply to convince you that you will not be able to follow my methods of proof if I indicate the lines on which I proceed in a single letter. ..."
[1] http://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4
"Dear Sir, I am very much gratified on perusing your letter of the 8th February 1913. I was expecting a reply from you similar to the one which a Mathematics Professor at London wrote asking me to study carefully Bromwich's Infinite Series and not fall into the pitfalls of divergent series. … I told him that the sum of an infinite number of terms of the series: 1 + 2 + 3 + 4 + ... = −1/12 under my theory. If I tell you this you will at once point out to me the lunatic asylum as my goal. I dilate on this simply to convince you that you will not be able to follow my methods of proof if I indicate the lines on which I proceed in a single letter. ..."
[1] http://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4
Maybe it is just me, but overstating anyone's impact to this extent is disrespectful to that person.
"....wealth of ideas that have transformed and reshaped 20th century mathematics"
"His work has had a fundamental role in the development of 20th century mathematics"
Maybe someone who has delved deeper into math can shine some light here.
"....wealth of ideas that have transformed and reshaped 20th century mathematics"
"His work has had a fundamental role in the development of 20th century mathematics"
Maybe someone who has delved deeper into math can shine some light here.
I suppose Ramanujan doesn't seem important to the layperson, because his major contributions were both very advanced and difficult to explain -- compare Gödel's second incompleteness theorem, that no axiomatic system representing Robinson arithmetic can prove its own consistency: that one is much simpler than a result dealing with mock theta functions -- and they were also in pure mathematics rather than applied mathematics, and the main application of Ramanujan's work has been in some areas of quantum mechanics and string theory, which are themselves a bit beyond a layperson. The partition function does pop up in computing from time to time, though.
I've heard of a division of mathematicians into so-called "theory-builders" and "problem-solvers" -- Ramanujan and Hilbert are firmly in the first class, while Erdös, Wiles and Gödel fall into the second. Ramanujan was transformative because he changed the questions that were asked, as well as answering them.
If you look at pure mathematics, most in the field will agree that Ramanujan was a -- if not the -- preeminent mathematician of the 20th century. In applied mathematics, that title probably goes to John von Neumann.
I've heard of a division of mathematicians into so-called "theory-builders" and "problem-solvers" -- Ramanujan and Hilbert are firmly in the first class, while Erdös, Wiles and Gödel fall into the second. Ramanujan was transformative because he changed the questions that were asked, as well as answering them.
If you look at pure mathematics, most in the field will agree that Ramanujan was a -- if not the -- preeminent mathematician of the 20th century. In applied mathematics, that title probably goes to John von Neumann.
Not a mathematician, but some references I found after goolging a bit:
[1] "The meaning of Ramanujam now and for the future", George E. Andrews. http://www.math.psu.edu/andrews/pdf/274.pdf
[2] "The Ramanujam Journal", http://www.springer.com/mathematics/numbers/journal/11139
Quote from the conclusion of [1]:
In his Presidential address to the London Mathematical Society in 1936, G. N. Watson spoke movingly of his emotional response to Ramanujan's achievements. I close by quoting his last few paragraphs [54; p. 80]:
"The study of Ramanujan's work and of the problems to which it gives rise inevitably recalls to mind Lame's remark that, when reading Hermite's papers on modular functions, \on a la chair de poule." I would express my own attitude with more prolixity by saying that such a formula as:
<snipped formula>
gives me a thrill which is indistinguishable from the thrill which I feel when I enter the Sagrestia Nuova of the Capelle Medicee and see before me the austere beauty of the four statues representing Day, Night, Evening, and Dawn which Michelangelo has set over the tombs of Guiliano de Medici and Lorenzo de Medici. Ramanujan's discovery of the mock theta functions makes it obvious that his skill and ingenuity did not desert him at the oncoming of his untimely end. As much as any of his earlier work, the mock theta functions are an achievement sucient to cause his name to be held in lasting remembrance"
To me, it's interesting that a community where fairly easy to build web-applications are called transformative and disruptive also has two out of thirteen comments critical of Ramanujam's impact.
[1] "The meaning of Ramanujam now and for the future", George E. Andrews. http://www.math.psu.edu/andrews/pdf/274.pdf
[2] "The Ramanujam Journal", http://www.springer.com/mathematics/numbers/journal/11139
Quote from the conclusion of [1]:
In his Presidential address to the London Mathematical Society in 1936, G. N. Watson spoke movingly of his emotional response to Ramanujan's achievements. I close by quoting his last few paragraphs [54; p. 80]:
"The study of Ramanujan's work and of the problems to which it gives rise inevitably recalls to mind Lame's remark that, when reading Hermite's papers on modular functions, \on a la chair de poule." I would express my own attitude with more prolixity by saying that such a formula as:
<snipped formula>
gives me a thrill which is indistinguishable from the thrill which I feel when I enter the Sagrestia Nuova of the Capelle Medicee and see before me the austere beauty of the four statues representing Day, Night, Evening, and Dawn which Michelangelo has set over the tombs of Guiliano de Medici and Lorenzo de Medici. Ramanujan's discovery of the mock theta functions makes it obvious that his skill and ingenuity did not desert him at the oncoming of his untimely end. As much as any of his earlier work, the mock theta functions are an achievement sucient to cause his name to be held in lasting remembrance"
To me, it's interesting that a community where fairly easy to build web-applications are called transformative and disruptive also has two out of thirteen comments critical of Ramanujam's impact.
Being critical of the article is not the same as being critical of Ramanujan's impact. So, there isn't a need to down vote.
I am genuinely interested in knowing what Ramanujan did to "transform and reshape" 20th century mathematics.
I am genuinely interested in knowing what Ramanujan did to "transform and reshape" 20th century mathematics.
I was upset by this statement which assumes a priori that the article must be overstating his impact.
Maybe it is just me, but overstating anyone's impact to this extent is disrespectful to that person.
I don't claim that the article is not overstating Ramanujam's impact, but you seemed to be arguing that there's no way this could be true, so the article must be overstating his impact. Given that even you don't claim to have the background required to objectively evaluate Ramanujam's impact, this seemed to be a prejudiced comment.
In retrospect, I might have been a bit trigger-happy with the downvote. Sorry about that.
Maybe it is just me, but overstating anyone's impact to this extent is disrespectful to that person.
I don't claim that the article is not overstating Ramanujam's impact, but you seemed to be arguing that there's no way this could be true, so the article must be overstating his impact. Given that even you don't claim to have the background required to objectively evaluate Ramanujam's impact, this seemed to be a prejudiced comment.
In retrospect, I might have been a bit trigger-happy with the downvote. Sorry about that.
Incidentally that was my reaction too after reading the article. No doubt he contributed a lot to the areas he was working in, but you have to consider that his areas were limited. He primarily dealt with Number Theory and analysis.
To my mind, the body of work looking into the foundations of mathematics (Russell, Whitehead, Godel etc) which also started around the same time has played a more fundamental role since it changed out very basic ideas of what can and cannot be achieved within the bounds of mathematics.
To my mind, the body of work looking into the foundations of mathematics (Russell, Whitehead, Godel etc) which also started around the same time has played a more fundamental role since it changed out very basic ideas of what can and cannot be achieved within the bounds of mathematics.
Perhaps as a lay observer, We can take your stand. But i would assume practitioners of the art would have a stronger grasp on subtlety and nuance.
Some art looks boring while others look vibrant. To the more experienced- the same paintings speak of control and ability while the other evokes childish garishness.
Some art looks boring while others look vibrant. To the more experienced- the same paintings speak of control and ability while the other evokes childish garishness.
In public key cryptography ,we use ramanujan's theory about
prime numbers to calculate large prime numbers.The fastest algorithm known is based on his work.
http://en.wikipedia.org/wiki/Ramanujan_prime
[deleted]
I believe there are at least several previouslies for this guy.
mcl(1)
"At the age of 12, he borrowed from a friend a copy of Loney's book on Plane Trigonometry, published by Cambridge University Press in 1894"...
"It is not a remarkable book, and Ramanujan's use of it to propel himself to the centre stage of 20th century mathematics, has made the book remarkable. It was largely used by students of Carr who were preparing for the entrance examination in mathematics at Cambridge University. Ramanujan used the book to master all of 18th and 19th century mathematics. He set about to demonstrate each of the assertions of the book, using only his slate to do the calculations. He would jot down the formula to be proved, and then erase it with his elbow, and then continue to jot down some more formulas. In this way, he worked through the entire book. People used to speak of his “bruised elbow.” Sadly, he took Carr's book as a model for mathematical writing and left behind his famous notebooks containing many formulas but practically no proofs."