Kanigel, Robert;
The man who knew infinity: A life of the genius Ramanujan
Little Brown & Co London 1991 / Rupa 1993
ISBN 0671750615
topics: | biography | math | india
A narrative that thrives on the extraordinariness of Srinivasan Ramanujan. A self-taught mathematician who never went to college, Ramanujan discovered for himself a number of recondite theorems in number theory, and went on to state major theorems involving theta functions and other domains at the intersection of algebra and arithmetic. The notebooks he kept during his short life, full of mathematical scribblings, are still being studied as source of possible new theorems in mathematics. Kanigel manages to enter the mind of a traditional Tamil brahmin boy, as he grows up in poverty, and the challenges he faces when finally his talents are recognized by Hardy and he is invited to Cambridge. This was to be one of the most mathematically productive encounters in history. The two men could hardly have been more different. P. Sundaresan and R.Padmavijayam write, in an article on Ramanujan: Ramanujan arrived at Trinity Col¬lege, Cambridge, in April 1914, a few months before World War I began. There is no record of his first meet¬ing with Hardy. But no two men could be more different. Hardy, then 37, was lean and handsome, passionately fond of cricket, a skeptic and rationalist who, as one of his friends put it, “con¬sidered God his personal enemy.” Chubby Ramanujan, on the other hand had no interest in sports and was a devout Hindu who saw the divine everywhere. “An equation,” he once said, “has no meaning unless it expresses a thought of God.” Ramanujan was completely out of place in England during the war years. He fell ill and became seriously depressed. In 1918, he became suicidal and jumped in front of an underground train. Fortunately, the driver was able to stop the train. Hardy managed to convince the police not to prosecute Ramanujan.
This remains the most stimulating biography of one of the undisputed geniuses of 20th c. mathematics. We still keep wondering about this mad genius, how he came to arise from the valley of the kaveri in sough india. Part of the reason the book is so mesmerizing of course, is this wonder. As Cambridge mathematician Bela Bollobas says of him: I believe Hardy was not the only mathematician who could have [done the part in the collaboration with Ramanujan]. Probably Mordell could have done it. Polya could have done it. I'm sure there are quite a few people who could have played Hardy's role. But Ramanujan's role in that particular partnership I don't think could have been played at any time by anybody else. - p. 253.
Series that converge to PI, p.209:
pi/4 = 1 - 1/3 + 1/5 - 1/7 + 1/9 .... [Gregory, or the Kerala
Mathematicians a century before him. can be obtained by plugging
x=1 into the Leibniz series for tan^(-1)x. The error after the nth term
of this series is larger than 1/(2n) so this sum converges so slowly
that 300 terms are not sufficient to calculate pi correctly to two
decimal places!]
1/4. (pi-3) = 1/2.3.4 - 1/4.5.6 + 1/6.7.8 - ... [Anon]
pi/2 = 2/1 x 2/3 x 4/3 x 4/5 x 6/5 x 6/7 x ... [Wallis]
2/pi = 1 -5(1/2)³ + 9(1.3/2.4)³ -13(1.3.5/2.4.6)³ . . . [Bauer, Ramanujan]
(p.167)
2/pi = probability that needle will fall on a line, on infinite
grid of lines, spaced apart by needle length.
[one of Ramanujan's formulae was found to be very quick to converge]
[
1/4. pi. sqrt(2) = 1 + 1/3 - 1/5 - 1/7 + 1/9 + 1/11 - ...
1/6. pi² = 1 + 1/4 + 1/9 + 1/16 + 1/25 + ...
1/8. pi² = 1 + 1/9 + 1/16 + 1/25 + ... 1/((2k-1)²)
pi/2 = 1 + 1/3 (1+ 2/5 (1+ 3/7 (1+ 4/9 (1+ ...))))
= 1 + 1/3 + (1.2)/(3.5) + (1.2.3)/(3.5.7) + ...
[Formula used by Newton, after Euler's convergence improvement]
Bailey-Borwein-Plouffe (BBP formula): digit-extraction algorithm in base 16:
pi = SIGMA{ [4/(8n+1)-2/(8n+4)-1/(8n+5)-1/(8n+6)] (1/16)^n }
More rapidly converging BBP-type formula (F. Bellard):
pi = 1/(26)sum_(n==0)^infty((-1)^n)/(2^(10n)) (-(2⁵)/(4n+1)-1/(4n+3)+(28)/(10n+1)-(26)/(10n+3)-(2²)/(10n+5)-(2²)/(10n+7)+1/(10n+9)).
http://mathworld.wolfram.com/PiFormulas.html ]
(from http://mathworld.wolfram.com/RamanujanConstant.html ) Ramanujan constant: e^(pi.sqrt(163) ) (irrational number, very nearly integer ) = 262 537 412 640 768 743.999 999 999 999 2 Ramanujan (1913-1914) gave few rather spectacular examples of almost integers (such as e^(pi.sqrt(58)) ), he did not actually mention e^(pi.sqrt(163)). In fact, Hermite (1859) observed this property of 163 long before Ramanujan's work. The name "Ramanujan's constant" was coined by Simon Plouffe and derives from an April Fool's joke played by Martin Gardner (Apr. 1975) on the readers of Scientific American. In his column, Gardner claimed that e^(pi.sqrt(163)) was exactly an integer, and that Ramanujan had conjectured this in his 1914 paper. Gardner admitted his hoax a few months later (Gardner, July 1975). ] --- It is very proper that in England, a good share of the produce of the earth should be appropriated to support certain families in affluence, to produce senators, sages and heroes . . . The leisure, independence and high ideals which the enjoyment of this rent affords has enabled them to raise Britain to pinnacles of glory. Long may they enjoy it. But in India that haughty spirit, independence and deep thought which the possession of great wealth sometimes gives ought to be suppressed. They are directly averse to our power and interest. The nature of things, the past experience of all governments, renders it unnecessary to enlarge on this subject. We do not want generals, statesmen and legislators; we want industrious husbandmen. If we wanted restless and ambitious spirits there are enough of them in Malabar to supply the whole peninsula. William Thackeray, Report on Canara, Malabar, and Ceded Districts, 1807. --- The Indian character has seldom been wanting in examples of what may be called passive virtues. Patience, personal attachment, gentleness and such like have always been prominent. . . In all departments of life the Hindus require a vigorous individuality, a determination to succeed and to sacrifice everything in the attempt. - Hindu editorial 1889 - p. 67-8 -- There can be no doubt that boots and trousers with the European coat constitute the most convenient dress for moving about quickly. The oriental dress is suited to a life of leisure, indolence, and slow locomotion, whereas the Western costume indicates an actie and self-confident life. - Hindu editorial, late 1890's, p.95 ---
Once Ramanujan was cooking in the kitchen, and P.C. Mahalanobis sat in the
drawing room reading a puzzle in the newspaper:
The house was in a long street, numbered on this side one, two,
three. and so on, and that all the numbers on one side of him added
up exactly the same as all the numbers on the other side of him.
Funny thing that! He said he knew there were more than fifty houses
on that side of the street, but not so many as five hundred . . .
Through trial and error, Mahalanobis (who would go on to found the Indian
Statistical Institute and become a Fellow of he Royal Society) had figured
it out in a few minutes. Ramanujan figured it out, too, but with a twist.
"Please take down the solution," he said -- and proceeded to dictate a
continued fraction, a fraction whose denominator consist of a number plus a
continued fraction, a / (1 + b/(1+ c/. . . )). As stated the problem had
but one solution -- house number 204 on a street of 288 houses; 1 + 2 +
. . . 203 = 205 + . . . 288. But without the 50 to 500 contraint there
were other solutions. For example, on an eight-house street, no. 6 would
be the answer. Ramanujan's continued fraction comprised within a single
expression all the correct answers.
Mahalanobis was astounded. How, he asked Ramanujan, had he done
it?
"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." - p. 215
[Now, how's that for an explanation ! No wonder Hardy says:]
His ideas as to what constituted a mathematical proof were of the most
shadowy description. - Hardy on SR, p. 216
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. - Hardy -- Littlewood (of Ramanujan): Every positive integer was one of his personal friends. - --- I believe Hardy was not the only mathematician who could have [done the part in the collaboration with Ramanujan]. Probably Mordell could have done it. Polya could have done it. I'm sure there are quite a few people who could have played Hardy's role. But Ramanujan's role in that particular partnership I don't think could have been played at any time by anybody else. - Cambridge mathematician Bela Bollobas, p. 253.
Rigor, Littlewood would observe, "is not of first-rate importance in analysis beyond the undergraduate state, and can be supplied, given a real idea, by any competent professional." . . . "Mathematics has been advanced most by those who are distinguished more for intuition than for rigorous methods of proof," the German mathematician Felix Klein once noted. . . . Years later, [Hardy] would contrive an informal scale of mathematical ability on which he assigned himself a 25 and Littlewood a 30. To David Hilbert, the most eminent mathematician of his day, he assigned an 80. To Ramanujan, he gave 100.
Back home, Indians recalled, people would come up to you, sit down,
start talking, and in five minutes know all about you -- whether you
were married, had children, where you were from, whatkind of work you
did. One story told of a swimmer whose cries for help sent everyone
rushing to his aid. Everyone, that is, save the lone Englishman, who
sat where he was, apparently unmoved. "Oh," he replied when asked
later, "were we introduced?" 243
Cambridge boasted its own brand of aloofness. A book aimed at Indian
students in England: "Even college porters went about "without the least
concern about the newcomer and with an air of indifference." ... It was a
wall erected around one's feelings, a great silence of emotions. In
Cambridge, the emphasis was on ideas, events, things, work, games --
anything, it seemed, but the deeply personal. 243
[could this be the male approach?]
---
A woman was complaining that the problem with the working classes was
that they failed to bathe enough, sometimes not even once a week.
Seeing disgust writ large on Ramanujan's face, she moved to reassure
him that the Englishmen {\em he} met were sure to bathe daily. "You
mean," he asked, you bathe only {\em once} a day?" 244
One day in January or February of 1918, it was at a station somewhere in this network, then smaller and newer, that Ramanujan threw himself onto the tracks in front of an approaching train. What happened next would be easy enough to read as a miracle. A guard spotted him do it and pulled a switch, bringing the train screeching to a stop a few feet in front of him. Ramanujan was alive, though bloodied enough to leave his shins deeply scarred. He was arrested and hauled off to Scotland Yard. Called to the scene, Hardy, marshaling all his charm and academic stature, made a show of how there, before the police, stood the great Mr. Srinivasa Ramanujan, a Fellow of the Royal Society, and how a Fellow of the Royal Society simply could not be arrested. In fact, Ramanujan was not an F.R.S. He would hardly have been immune from arrest in any case, and the police were not fooled for a minute. But they investigated, learned Ramanujan was indeed reputed to be an eminent mathematician, and decided to let him go. "We in Scotland Yard did not want to spoil [his] life," the officer in charge of the case said later. 294
today in India Ramanujan could not get even a lectureship in a rural collge because he had no degree. Much less could he get a post through the Union Public Service Commission. This fact is a disgrace to India. I am aware that he was offered a chair in India after_ becoming a Fellow of the Royal Society. But it is scandalous that India's great men should have to wait for foreign recognition. If Ramanujan's work had been recognized in India as early as it was in England, he might never have emigrated and might be alive today. We can cast the blame for Ramanujan's non-recognition on the British Raj. We cannot do so when similar cases occur today. . . - JBS Haldane, 1960's [p.353] [Why were Ramanujan's research reports from 1914 lost while in Indian care? Why was it left to American, more than Indian, mathematicians to restore Ramanujan's reputation? . . . How many registrars in this country today, or for that matter how many vice chancellors of today, 100 years after Ramanujan was born, would give a failed pre-university student a research scholarship of what is now equivalent of Rs. 2000 or Rs 2500?" - S. Ramaseshan, in a Ramanujan centennial event. p.356 [R. Ramachandra Rao, district collector of Nellore, and an amateur mathematician, supported SR with a stipend of R 25 a month. ]