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Life of ramanujan Life of ramanujan Presentation Transcript

  • The Man who knew InfinitySrinivasa Ramanujan Iyengar(Best known as S. Ramanujan)(22 Dec 1887 - 26 April 1920)
  • June 1, 2013 Ramanujan Educational Institutions - Palwal (India) 2Life of Ramanujan• This presentation is about life & achievements offamous mathematicians S. Ramanujan & isorganized as below:• Genius Mathematician (3)• Child Prodigy (4-5)• Struggle / Life Downs & Ups (6-13)• Prof. G.H. Hardy (14-16)• Hardy Ramanujan Taxicab Numbers (17-18)• The Lost Notebook (19-22)• Misperception about Ramanujan (23-24)• Personal Life (25-29)• God To him (30-33)• Quotes about Ramanujan (34-42)• Ramanujan Eponyms (43)• Lessons Learnt / Unlearnt from Life of Ramanujan (44-48)• References / Further Readings (49)• Ramanujan song (50)
  • June 1, 2013 Ramanujan Educational Institutions - Palwal (India) 3The Genius Mathematician• S. Ramanujan hailed as an all-time greatmathematician, like Euler, Gauss or Jacobi, forhis natural genius, has left behind 4000 originaltheorems, despite his lack of formal educationand a short life-span.• Probably Ramanujan’s life has no parallel in thehistory of human thought. He was a autodidactmathematical genius not only of the twentiethcentury but for all time to come.
  • June 1, 2013 Ramanujan Educational Institutions - Palwal (India) 4Child Prodigy• Learned college-level mathematics by age 11,and generated his own theorems in numbertheory and Bernoulli numbers by age 13(including independently re-discovering Eulersidentity).• While in school, he was gifted GeorgeSchoobridge Carr’s Synopsis. This book listedwithout proofs 4865 formulae in algebra,trigonometry, analytical geometry and calculus.Ramanujan not only proved for himself each ofthese formulae but also derived many newresults and recorded them.
  • June 1, 2013 Ramanujan Educational Institutions - Palwal (India) 5Child ProdigyA thought of a 7 year old Prodigy• Teacher: n / n =1, for every integer n.• Ramanujan: “Is zero divided by zero is alsoone?”• Ramanujan’s Answer: “Zero divided by zero maybe anything. The zero of the numerator may beseveral times the zero of the denominator andvice versa”.(7 year old prodigy was thinking of limits and limiting processes.)
  • June 1, 2013 Ramanujan Educational Institutions - Palwal (India) 6Life Down & Ups• After high school, Ramanujan passed acompetitive examination in English andMathematics and secured the Scholarship. Hejoined the Government Arts college, in the F.A.(First Examination in Arts, now a days BA I year)class at the age of 18 years.• Due to his love & preoccupation withMathematics, he could not pass in English andSanskrit and hence was not promoted to theSenior F.A. Class.
  • June 1, 2013 Ramanujan Educational Institutions - Palwal (India) 7• He lost his Scholarship & forced to discontinuehis studies due to this and the poverty of hisfamily.• However, he appeared privately for the F.A.examination but failed to pass the universityexaminations and this marked the end of hisformal education.• After having failed in his F.A. class at College, heran from pillar to post in search of a benefactorfor 6 (from 18 years of age to 24 years) years.Life Down & Ups
  • June 1, 2013 Ramanujan Educational Institutions - Palwal (India) 8Life Down & Ups• He tried for tutoring assignments to earnlivelihood but failed. His love and dedication formath continued.• He lived in absolute poverty. Once he said toone of his friends, “when food is problem, howcan I find money for paper? I may require fourreams of paper every month.”• With the help of a recommendation letter from amathematician, Ramanujan secured a clericalpost in the Accounts Section of the Madras PortTrust at an age of 24 years.
  • June 1, 2013 Ramanujan Educational Institutions - Palwal (India) 9Life Down & Ups• Meanwhile in 1911 at an age of 23 years, the firstfull length (15 page) research paper ofRamanujan, entitled: “Some properties ofBernoulli Numbers”, appeared in the Journal ofthe Indian Mathematical Society. This broughthim to the forefront of international mathematics.• After going through a paper by Britishmathematician Hardy, about some theorems onprime numbers, where he claimed that no definiteexpression has been found as yet, Ramanujanderived the expression which very nearlyapproximated to the real result.
  • June 1, 2013 Ramanujan Educational Institutions - Palwal (India) 10Life Down & Ups• Ramanujan communicated the derived results toHardy through a letter, who after gettingconvinced that Ramanujan is a natural genius,persuaded university authorities to invite him toCambridge.• The orchestrated efforts of his admirers, resultedin inducing the Madras University, in offering himthe first research scholarship of the University inMay 1913; then in offering him a scholarship of250 pounds a year for five years &100 poundsfor passage by ship and for initial outfit to go toEngland in 1914 at an age of 26 years.
  • June 1, 2013 Ramanujan Educational Institutions - Palwal (India) 11Life Down & Ups• Ramanujan initially refused to go to Englandpossibly due to his caste prejudice. But aftermentoring from Neville, Ramanujan finally setsail for England at a age of 26 years in 1914.• Prior to Ramanujan departure to England, Prof.Littlehailes & Mr. Arthur Davies arranged with theUniversity for £ 60 (out of Scholarship amount of£ 250 ) per year to be sent to his parents inIndia. Thus, he fulfilled his responsibilities as theeldest son of the family.
  • June 1, 2013 Ramanujan Educational Institutions - Palwal (India) 12Life Down & Ups• At Cambridge, Ramanujan worked with Littlewoodand Hardy, mathematicians of repute of that time.• Ramanujan was awarded the B.A. degree byresearch (this degree was later renamed as PhD)in March 1916 at an age of 28 years, for his workon Highly Composite Numbers.• He was elected a Fellow of the Royal Society ofLondon in February 1918 at an age of 30 years.He was the second Indian to become a Fellow ofthe Royal Society (1stone was in 1841, around 80years prior) & one of the youngest Fellows in theentire history of the Royal Society.
  • June 1, 2013 Ramanujan Educational Institutions - Palwal (India) 13Life Down & Ups• He was elected to a Trinity College Fellowship (aprize fellowship worth 250 pounds a year, whichhe was not destined to avail of.) He was the firstIndian to be elected a Fellow of Trinity College,Cambridge.• During his five year stay in Cambridge,Ramanujan published twenty one researchpapers containing theorems.
  • June 1, 2013 Ramanujan Educational Institutions - Palwal (India) 14Prof. G. H. Hardy• The role of G.H. Hardy (1877 - 1947), Professorof Mathematics at Trinity College, Cambridge, inthe life and career of Ramanujan isimmeasurable, peerless and beyond praise.• Convinced that Ramanujan was a naturalgenius, Hardy made up his mind thatRamanujan should be brought to Cambridge.
  • June 1, 2013 Ramanujan Educational Institutions - Palwal (India) 15Prof. G. H. Hardy• The eccentric British mathematician G.H. Hardyis known for his achievements in number theoryand mathematical analysis. But he is perhapseven better known for his adoption andmentoring of the self-taught mathematicalgenius, Ramanujan.• Before writing to Hardy, Ramanujan had writtento two well-known Cambridge mathematicians.But both of them had expressed their inability tohelp Ramanujan.
  • June 1, 2013 Ramanujan Educational Institutions - Palwal (India) 16Prof. G. H. Hardy• “Ramanujan was”, Hardy wrote, “my discovery. Idid not invent him, like other great men, heinvented himself, but I was the first reallycompetent person who had the chance to seesome of his work, and I can still remember withsatisfaction that I could recognize at once whattreasure I had found.”• Some differences were also developed laterbetween Ramanujan & Hardy due to workingstyle.
  • June 1, 2013 Ramanujan Educational Institutions - Palwal (India) 17Hardy-Ramanujan "taxicab numbers"• A common anecdote about Ramanujan relateshow Hardy arrived at Ramanujans house in acab numbered 1729, a number he claimed to betotally uninteresting. Ramanujan is said to havestated on the spot that, on the contrary, it wasactually a very interesting numbermathematically, being the smallest numberrepresentable in two different ways as a sum oftwo cubes. Such numbers are now sometimesreferred to as "taxicab numbers".
  • June 1, 2013 Ramanujan Educational Institutions - Palwal (India) 18Hardy-Ramanujan "taxicab numbers"
  • June 1, 2013 Ramanujan Educational Institutions - Palwal (India) 19The Lost Notebook• Ramanujan recorded the bulk of his results in fournotebooks of loose leaf paper. First 3 notebookscontained his work before leaving for England.• Those 3 notebooks were published as a two-volume set in 1957 by the Tata Institute ofFundamental Research (TIFR), Mumbai, India.• The fourth notebook with 87 unorganized pages,the so-called "lost notebook", was sent to Hardy byhis wife after his death. The notebook remaineduntraced nearly for 40 years.
  • June 1, 2013 Ramanujan Educational Institutions - Palwal (India) 20The Lost Notebook• The “Lost Notebook” contained results he noted,during the last year of his life, after his return toIndia.• This “Lost Notebook” was found by Prof. GeorgeE. Andrews of the Pennsylvania State Universityin 1976 in the trash.• The “Lost Notebook” contained some 600theorems, which Ramanujan called as “mock”theta functions.
  • June 1, 2013 Ramanujan Educational Institutions - Palwal (India) 21The Lost Notebook• The discovery of the “Lost” Notebook ofRamanujan contributed to a resurgence ofinterest in the life and work of Ramanujan.• Its facsimile edition was brought out by NarosaPublishing House in 1987, on the occasion ofRamanujans birth centenary nearly after 68years of his death.
  • June 1, 2013 Ramanujan Educational Institutions - Palwal (India) 22The Lost Notebook• The results in his notebooks inspired numerouspapers by later mathematicians trying to provewhat he had found.• He stated results that were both original andhighly unconventional, and these have inspireda vast amount of further research.• The many conjectures of Ramanujan opened upnew areas of research in mathematics.
  • June 1, 2013 Ramanujan Educational Institutions - Palwal (India) 23Misperception• Ramanujan recorded the bulk of his results in fournotebooks of loose leaf paper. These results weremostly written up without any derivations. Sincepaper was very expensive, Ramanujan would domost of his work and perhaps his proofs on slate,and then transfer just the results to paper. This isprobably the origin of the misperception thatRamanujan was unable to prove his results andsimply thought up the final result directly.
  • June 1, 2013 Ramanujan Educational Institutions - Palwal (India) 24Misperception• Professor Bruce C. Berndt of University ofIllinois, who worked extensively on Ramanujannotebooks, during a lecture at IIT Madras in2011 stated that, over the last 40 years, asnearly all of Ramanujans theorems have beenproven right.• Also mathematicians agreed on the point that itwas not possible for someone to imagine thoseresults without solving / proving.
  • June 1, 2013 Ramanujan Educational Institutions - Palwal (India) 25Personal Life• Born to Kuppuswamy Srinivasa Iyengar &Komalathammal of Kumbakonam on Thursday,Dec 22, 1887, at 6 pm as their first child inErode (Tamil Nadu, India).• His nick name was Chinnaswamy, meaning LittleLord.• At the age of five, Ramanujan had a bout ofsmall pox, which left some permanent scars onhis face.
  • June 1, 2013 Ramanujan Educational Institutions - Palwal (India) 26Personal Life• His mother use to sing at a local temple, fromwhom he learnt religious traditions, which remainedpart of his daily life till the end. He had a deep faithin God and remained vegetarian throughout his life.• At the age of 21 in 1909, Ramanujan was marriedto Janaki, then a 10 year old girl.• Ramanujan developed a rare disease, which couldbe treated with a surgical operation only. His familydidn’t have the money for the operation, but in1910,a doctor volunteered to do the surgery for free.
  • June 1, 2013 Ramanujan Educational Institutions - Palwal (India) 27Personal Life• Ramanujan suffered illnesses before and afterhis marriage and before his departure toEngland.• His health was reasonably good during the firstthree years of his stay in Cambridge.• The period of Ramanujan’s stay in Englandalmost overlapped with World War.• Non availability of vegetarian food was one offactor behind his ill health.
  • June 1, 2013 Ramanujan Educational Institutions - Palwal (India) 28Personal Life• From May 1917, he had been in and out ofhospital for duration of 2-3 weeks to 5-6 monthsuntil his departure to India in early 1919 (Last 2years at Cambridge) due to suffering fromTuberculosis.• Ramanujan was persuaded by Hardy to return toIndia with the hope that he would recover soonand return to Trinity College.• After completing nearly five years at Cambridge,when Ramanujan appeared to have recoveredsufficiently, he left England on 27th February 1919for India at an age of 31.
  • June 1, 2013 Ramanujan Educational Institutions - Palwal (India) 29Personal Life• Unfortunately, his precarious health did notimprove, on his return to India.• Despite of the best medical attention from thedoctors, his untimely end came on 26th April1920, at Chetput, Madras, when Ramanujanwas 32 years old.• He was survived by wife Janaki, who lived untilher death in 1994 in Chennai alongwith herfoster son.
  • June 1, 2013 Ramanujan Educational Institutions - Palwal (India) 30God to Him• Ramanujan often quoted “An equation for mehas no meaning unless it expresses a thought ofGod.”• For example, 2n-1 will denote the primordial Godand several divinities. When n is zero theexpression denotes zero, there is nothing;when n is 1, the expression denotes unity, theInfinite God. When n is 2, the expressiondenotes Trinity; when n is 3, the expressiondenotes 7, the Saptha Rishis and so on.
  • June 1, 2013 Ramanujan Educational Institutions - Palwal (India) 31God to Him• Ramanujan was eager to work out a theory ofreality which would be based on the fundamentalconcepts of “zero”, “infinity” and the set of finitenumbers.• He spoke of “zero” as the symbol of the absolute(Nirguna-Brahmam) of the extreme monisticschool of Hindu philosophy, that is, the reality towhich no qualities can be attributed, whichcannot be defined or described by words, andwhich is completely beyond the reach of thehuman mind.
  • June 1, 2013 Ramanujan Educational Institutions - Palwal (India) 32God to Him• According to Ramanujan, the appropriatesymbol was the number “zero”, which is theabsolute negation of all attributes.• He looked on the number “infinity” as the totalityof all possibilities, which was capable ofbecoming manifest in reality and which wasinexhaustible.
  • June 1, 2013 Ramanujan Educational Institutions - Palwal (India) 33God to Him• According to Ramanujan, the product of infinityand zero would supply the whole set of finitenumbers. Each act of creation, could besymbolized as a particular product of infinity andzero, and from each such product would emergea particular individual of which the appropriatesymbol was a particular finite number.(As narrated by Prof. Prasantha Chandra Mahalanobius, the renownedStatistician, contemporary & good friend of Ramanujan atCambridge)
  • June 1, 2013 Ramanujan Educational Institutions - Palwal (India) 34Quotes about Ramanujan• “Try to imagine the quality of Ramanujans mind,one which drove him to work unceasingly whiledeathly ill, and one great enough to grow deeperwhile his body became weaker. I stand in awe ofhis accomplishments; understanding is beyondme. We would admire any mathematician whoselifes work was half of what Ramanujan found inthe last year of his life while he was dying.Prof. Richard AskeyUniversity of Wisconsin-Madison
  • June 1, 2013 Ramanujan Educational Institutions - Palwal (India) 35Quotes about Ramanujan• Hardys personal ratings of mathematicians:Suppose that we rate mathematicians on thebasis of pure talent on a scale from 0 to 100,Hardy gave himself a score of 25, Littlewood 30,Hilbert 80 and Ramanujan 100.Bruce C BerndtDistinguished Research Professor of Mathematics, University of IllinoisBest known for his work explicating the discoveries of Ramanujan. Heis a coordinating editor of The Ramanujan Journal.
  • June 1, 2013 Ramanujan Educational Institutions - Palwal (India) 36Quotes about Ramanujan• Sheer intuitive brilliance coupled to long, hardhours on his slate made up for most of hiseducational lapse. This ‘poor and solitary Hindupitting his brains against the accumulatedwisdom of Europe’ as Hardy called him, hadrediscovered a century of mathematics andmade new discoveries that would captivatemathematicians for next century.Robert KanigelAuthor of “The Man who Knew Infinity: A Life of the Genius Ramanujan”
  • June 1, 2013 Ramanujan Educational Institutions - Palwal (India) 37Quotes about Ramanujan• “I think it is fair to say that almost all themathematicians who reached distinction duringthe three or four decades following Ramanujanwere directly or indirectly inspired by hisexample.” Even those who do not know aboutRamanujan’s work are bound to be fascinatedby his life.S. ChandrasekharThe Indian born astrophysicist, who got Nobel Prize in 1983
  • June 1, 2013 Ramanujan Educational Institutions - Palwal (India) 38Quotes about Ramanujan• “I have to form myself, as I have never reallyformed before, and try to help you to form, someof the reasoned estimate of the most romanticfigure in the recent history of mathematics, aman whose career seems full of paradoxes andcontradictions, who defies all cannons by whichwe are accustomed to judge one another andabout whom all of us will probably agree in onejudgment only, that he was in some sense a verygreat mathematician.”Prof. Hardy
  • June 1, 2013 Ramanujan Educational Institutions - Palwal (India) 39Quotes about Ramanujan• “I found Hardy and Littlewood in a state of wildexcitement because they believe, they havediscovered a second Newton, a Hindu Clerk inMadras … He wrote to Hardy telling of someresults he has got, which Hardy thinks quitewonderful.”Bertrand Arthur William RussellBritish philosopher& mathematician, Noble prize holder
  • June 1, 2013 Ramanujan Educational Institutions - Palwal (India) 40Quotes about Ramanujan• “Ramanujan’s conjectures formulated and theirlater generalization, have come to play a morecentral role in the mathematics of today, servingas a kind of focus for the attention of quite alarge group of the best mathematicians of ourtime.• The estimates of Ramanujan’s nature inmathematics certainly have been growing overthe years. There is no doubt about that.Prof. Atle SelbergThe Norwegian mathematician one of the great number theorists of thiscentury
  • June 1, 2013 Ramanujan Educational Institutions - Palwal (India) 41Quotes about Ramanujan• S. Ramanujan, discovered by the Cambridgemathematician Hardy, whose great mathematicalfindings were beginning to be appreciated from1915 to 1919. His achievements were to be fullyunderstood much later, well after his untimelydeath in 1920. For example, his work onthe highly composite numbers started a wholenew line of investigations in the theory of suchnumbers."Prof. Jayant Narlikar, Indian astrophysicistIn his book Scientific Edge
  • June 1, 2013 Ramanujan Educational Institutions - Palwal (India) 42Quotes about Ramanujan• “The fact that Ramanujan’s early years werespent in a scientifically sterile atmosphere, thathis life in India was not without hardships thatunder circumstances that appeared to mostIndians as nothing short of miraculous, he hadgone to Cambridge, supported by eminentmathematicians, and had returned to India withvery assurance that he would be considered, intime as one of the most original mathematiciansof the century”.S. ChandrasekharThe Indian born astrophysicist, who got Nobel Prize in 1983
  • June 1, 2013 Ramanujan Educational Institutions - Palwal (India) 43EponymRamanujan is eponym of mathematical topics listed below:• Ramanujan Magic Square• Brocard-Ramanujan Diophantine equation• Dougall-Ramanujan identity• Hardy-Ramanujan number• Landau-Ramanujan constant• Ramanujans congruences• Ramanujan-Nagell equation• Ramanujan-Peterssen conjecture• Ramanujan-Skolems theorem• Ramanujan-Soldner constant• Ramanujan summation• Ramanujan theta function• Ramanujan graph• Ramanujans tau function• Ramanujans ternary quadratic form• Ramanujan prime• Ramanujans constant• Ramanujans sum• Rogers-Ramanujan identity
  • June 1, 2013 Ramanujan Educational Institutions - Palwal (India) 44Lessons Learnt / Unlearnt fromRamanujan’s Life• Fact, that despite the hardship faced byRamanujan, he rose to such a scientific standingand reputation that no Indian has enjoyedbefore, should be enough for young Indians tocomprehend that if they are deserving and canwork hard, they can perhaps soar the way whatRamanujan had.
  • June 1, 2013 Ramanujan Educational Institutions - Palwal (India) 45Lessons Learnt / Unlearnt fromRamanujan’s Life• Even today in India Ramanujan can not get alectureship in a school / college because he hadno degree. Many researchers / Universities willpursue studies / researches on his work but hewill have to struggle to get even a teaching job.
  • June 1, 2013 Ramanujan Educational Institutions - Palwal (India) 46Lessons Learnt / Unlearnt fromRamanujan’s Life• Even after more than 90 years of the death ofRamanujan the situation is not very different asfar the rigidity of the education system isconcerned. Today also a ‘Ramanujan’ has toclear all traditional subjects’ exams to get adegree irrespective of being genius in one ormore different subjects.
  • June 1, 2013 Ramanujan Educational Institutions - Palwal (India) 47Lessons Learnt / Unlearnt fromRamanujan’s Life• He was offered a chair in India after becoming aFellow of the Royal Society. But it is disgracefulthat India’s talent has to wait for foreignrecognition to get acceptance in India or elseimmigrate to other places. Similarly many otherscientists had no other option but to immigratefor opportunities & recognition. Many of thosewon international recognitions including nobleprizes. The process of this brain drain is stillcontinuing even after 60 years of independence.
  • June 1, 2013 Ramanujan Educational Institutions - Palwal (India) 48Lessons Learnt / Unlearnt fromRamanujan’s Life• The most important lesson that we should drawfrom Ramanujan’s life about the educationalsystem is that systems / provision should bemade to support specially gifted children withvery strong interests in one direction, at allstages of the educational system.
  • June 1, 2013 Ramanujan Educational Institutions - Palwal (India) 49References / Further Readings• http://www.imsc.res.in/~rao/ramanujan/newnow/main.htm• http://en.wikipedia.org/wiki/Srinivasa_Ramanujan• "Collected Papers of Srinivasa Ramanujan" Cambridge University Press(1927)• Ramanujan: Twelve Lectures on the Subjects Suggested by His Life andWork by G. H. Hardy, Chelsea Publishing Co, New York, 1940.• Srinivasa Ramanujan E.H. Neville, English Talk, Air April 22, 1941• The Man Who Knew Infinity : A Life of the Genius Ramanujan by R. Kanigel,Abacus Books, London, 1992.• Ramanujan’s Notebooks (Part I&II) by B.C. Berndt Springer, New York,1985-1989.• Ramanujan:The Man and the Mathematician by S.R. Ranganathan, AsiaPublishing House, Bombay, 1967.• Srinivasa Ramanujan : A Mathematical Genius by K. Srinivasa Rao; EastWest Books (Madras) Pvt. Ltd. 1998.• Srinivasa Ramanujan, Suresh Ram, National Book Trust India, 1989.• Ganit Jagater Bismay Ramanujan by Satyabachi Sar, Gyan BichitraPrakashani, Agartala, 2000. (Bangla)
  • June 1, 2013 Ramanujan Educational Institutions - Palwal (India) 50Ramanujan Song• Ramanujan Song: biographical rock epic about the life ofRamanujan.Words and Music by Mark Engelberghttp://archive.org/details/Ramanujan