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He is in fact, I am certain, one of the greatest mathematicians according to any criteria, let it be because he had no formal education, or because of the 3000 odd identities and theorems he came up with.anyway i don’t want to compare but i am just trying to say the striking things i saw in his life.Anyway I would first show the marklist Ramanujan acquired in the 1st year examinations.Roughly speaking, for these things,Ramanujan’s name is seen everywhere around the world, even if some might disagree.•Magic Square•Brocard – Ramanujan Diophatine equation•Dougall – Ramanujan identity•Hardy – Ramanujan number•Landau – Ramanujan constant•Ramanujan’s congruences•Ramanujan – Nagell equation•Ramanujan – Peterssen conjecture•Ramanujan – Skolem’s theorem•Ramanujan – Soldner constant•Ramanujan summation•Ramanujan theta function•Ramanujan graph•Ramanujan’s tau function•Ramanujan’s ternary quadratic form•Ramanujan’s prime•Ramanujan’s costant•Ramanujan’s sum•Rogers – Ramanujan’s identityNow, let us see a quote of an English Mathematician“Srinivasa Ramanujan was a mathematician so great that his name transcends jealousies, the one superlatively great mathematician whom India has produced in the last thousand years.”He continued thus: “His leaps of intuition confound mathematicians even today, a century after his death. His papers are still plumbed for their secrets. His theorems are being applied in areas- polymer chemistry, computers, astrophysics, molecular physics, even (it has been recently suggested) cancer – scarcely imaginable during his lifetime. And always the nagging question: What might have been, had he been discovered a few years earlier, or lived a few years longer?”Now just see Ramanujan’s childhood prodigy:Teacher: n/n = 1. Any number divided by itself is one. If there are 3 apples and there are three students, each one will get one apple. Likewise if there are 1000 children and 1000 pens, each will get one pen.Ramanujan: What about 0/0? If there are 0 apples and 0 students, will each still get one?Teacher got perplexed!Ramanujan’s Explanation: 0/0 can be anything, the zero in the numerator could be many times 0 in the denominator, and vice versa.Just before the age of 10, in November 1897, he passed his primary examinations in English, Tamil, geography and arithmeticWith his scores, he stood first in the district. That year, Ramanujan entered Town Higher Secondary School where he encountered formal mathematics for the first time.By age 11, he had exhausted the mathematical knowledge of two college students who were lodgers at his house.He was later lent a book on advanced trigonometry written by S. L. LoneyHe completely mastered this book by the age of 13 and discovered sophisticated theorems on his own.Now Ramanujan was shown how to solve cubic equations in 1902 and he went on to find his own method. Its like this:It is easy to solve simple equation of the first degree, e.g., 3a = 15. And we are taught how to solve second degree equations with the power of x as 2.Ramanujan found his own method in solving not only cubic equations but also equations of fourth degree.Next year not knowing that quintic equations, or equations with power of x as 5, cannot be solved, he tried and failed in his attempt.In 1903 when he was16, Ramanujan came across the book by G. S. Carr on A Synopsis of Elementary Results in Pure and Applied Mathematics, a collection of 4865 formula and theorems without proofThe book is generally acknowledged as a key element in awakening the genius of RamanujanThe next year, he had independently developed and investigated the Bernoulli numbers and had calculated Euler's constant up to 15 decimal placesWhen he graduated from Town Higher Secondary School in 1904, Ramanujan was awarded the K. Ranganatha Rao prize for mathematics as an outstanding student who deserved scores higher than the maximum possible marksHe received a scholarship to study at Government Arts College, Kumbakonam, However, Ramanujan could not focus on any other subjects and failed most of them, losing his scholarship in the processHe later enrolled at Pachaiyappa' College in Madras. He again excelled in mathematics but performed poorly in other subjectsRamanujan failed his Fine Arts degree exam in December 1906 and again a year laterWithout a degree, he left college and continued to pursue independent research in mathematics. At this point of his life, he lived in extreme poverty and was suffering from starvation.Deplorable Condition of Ramanujan is expressed in his own words:“When food is the problem, how can I find money for paper? I may require four reams of paper every month.”On 14 July 1909, Ramanujan was married to a nine-year old girl, Janaki Ammal (21 March 1899 - 13 April 1994)After the marriage, Ramanujan developed a hydrocele problemsHis family did not have the money for the operation, but in January 1910, a doctor volunteered to do the surgery for freeAfter his successful surgery, Ramanujan searched for a jobHe stayed at friends' houses while hewent door to door around the city of Chennai looking for a clerical positionTo make some money, he tutored some students at Presidency College who were preparing for their examRamanujan met deputy collector V. Ramaswamy Aiyer, who had recently founded the Indian Mathematical SocietyRamanujan, wishing for a job at the revenue department where Ramaswamy Aiyer worked, showed him his mathematics notebooksAs Ramaswamy Aiyer later recalled:“I had no mind to smother his genius by an appointment in the lowest level as clerk in the revenue department.”Ramaswamy Aiyer sent Ramanujan, with letters of introduction, to his mathematician friends.Some of these friends looked at his work and gave him letters of introduction to R. Ramachandra Rao, the district collector of Nellore and the secretary of the Indian Mathematical SocietyRamachandra Rao was impressed by Ramanujan's research but doubted that it was actually his own work !Ramanujan's friend, C. V. Rajagopalachari, persisted with Ramachandra Rao and tried to clear any doubts over Ramanujan's academic integrityRao listened as Ramanujan discussed elliptic integrals, hypergeometric series, and his theory of divergent series, through which Rao was convinced of Ramanujan's mathematical brilliance . When Rao asked him what he wanted, Ramanujan replied that he needed some work and financial supportRamanujan continued his mathematical research with Rao's financial aid taking care of his daily needsWith the help of Ramaswamy Aiyer, Ramanujan had his work published in the Journal of Indian Mathematical SocietyOne of the first problems he posed in the journal was to evaluate:He waited for a solution to be offered in three issues, over six months, but failed to receive any. At the end, Ramanujan supplied the solution to the problem himselfHe formulated an equation that could be used to solve the infinitely nested radicals problem. Using this equation, the answer to the question posed in the Journal was simply 3In early 1912 he got a job in the Madras Accountant Generals office with a salary of Rs 20 per month.Later he applied for a position under the Chief Accountant of the Madras Port TrustHe was Accepted as a Class III, Grade IV accounting clerk making 30 rupees per monthHe used to Spend spare time doing Mathematical ResearchIn the spring of 1913, Narayana Iyer and Ramachandra Rao tried to present Ramanujan's work to British mathematiciansOne mathematician, M. J. M. Hill of University College London, commented that although Ramanujan had "a taste for mathematics, and some ability", he lacked the educational background and foundation needed to be accepted by mathematiciansOn 16 January 1913, Ramanujan wrote to G. H. HardyComing from an unknown mathematician, the nine pages of mathematics made Hardy initially view Ramanujan's manuscripts as a possible "fraud“ !Hardy recognized some of Ramanujan's formulae but others "seemed scarcely possible to believe"G.H. Hardy was an academician at Cambridge UniversityHe was a prominent English mathematician, known for his achievements in number theory and mathematical analysis.Later on Ramanujan wrote to G.H.HardyHardy recognised some of his formulae but other “seemed scarcely possible to believe”. Some of them were –Initially, G. H. Hardy thought that the works of Ramanujan were fraud because most of them were impossible to believe.But eventually ,he was convinced and interested in his talent.This is one approximation formula of Pi mentioned in Ramanujan’s letters:Hardy was also impressed by some of Ramanujan's other work relating to infinite series:This second one was new to Hardy, and was derived from a class of functions called hypergeometric series which had first been researched by L. Euler and Carl F. Gauss.After he saw Ramanujan's theorems on continued fractions on the last page of the manuscripts, Hardy commented that the "[theorems] defeated me completely; I had never seen anything like them before”He figured that Ramanujan's theorems "must be true”Hardy asked a colleague, J. E. Littlewood, to take a look at the papersLittlewood was amazed by the mathematical genius of RamanujanRamanujan’s notebook referring calculus and number theory:Ramanujan boarded the S.S.Nevasa on 17 March 1914 and arrived in London on 14th AprilRamanujan began working with Hardy and LittlewoodHardy received 120 theorems from him in 1st 2 letters but there were many more results in his notebookRamanujan spent nearly 5 years in CambridgeRamanujan was awarded the B.A degree by Research in March 1916 at an age of 28 years for his work on Highly Composite Numbers.He was elected a Fellow of the Royal Society of London in February 1918 at an age of 30 years.He was the second Indian to become FRS.( First one was in 1841).He was elected to a Trinity College Fellowship as the FIRST INDIAN.During his five years stay in Cambridge he published twenty one research papers containing theorems.A few words regarding the 1729, Ramanujan NumberHardy arrived in a cab numbered 1729He commented that the number was uninteresting or dull.Instantly Ramanujan claimed that it was the smallest natural number which can be written as sum of cubes in 2 ways1729 = sum of cubes of 12 and 1/ sum of cubes of 10 and 9.Actually only this much is available in the popular version of the story.But Ramanujan had worked extensively on this number and made some simple reuslts along with other startling contributions.1729 = 7 x 13 x 19 product of primes in A.P1729 divisible by its sum of digits.1729 = 19 x 911729 is a sandwich number or HARSHAD number."Ramanujan was using 1729 and elliptic curves to develop formulas for a K3 surface," Ono says. "Mathematicians today still struggle to manipulate and calculate with K3 surfaces. So it comes as a major surprise that Ramanujan had this intuition all along."Ono had worked with K3 surfaces before and he also realized that Ramanujan had found a K3 surface, long before they were officially identified and named by mathematician André Weil during the 1950s.Just as K2 is an extraordinarily difficult mountain to climb, the process of generalizing elliptic curves to find a K3 surface is considered an exceedingly difficult math problem.And in Ramanujan’s writing he was relying on this number 1729 in order to arrive at some combination of numbers which could prove that Fermat’s last conjecture could be counter exampled.there are some popular misconceptions regarding ramanujan:Ramanujan recorded the bulk of his results in four notebooks of loose leaf paper (About 4000 theorems)These results written up without any derivations.Since paper was very expensive, He would do most of his work (derivations) on SLATE and transfer just the results to paper.Hence the perception that he was unable to prove his results and simply thought up the final result directly is NOT CORRECTProfessor Bruce C.Berndt of University of Illinois, who worked on Ramanujan note books, stated that “Over the last 40 years, nearly all of Ramanujan’s theorems have been proven right”.Also Mathematicians agreed unanimously on the point that it was not possible for someone to imagine those results without solving / proving.I think I will complete this answer tomorrow, because I feel sleepy: Good Night!Edited in Later:I am extremely sorry for not turning up yesterday to finish the answer I started, because I had gone for an outing to Hoggenakkal in Tamil Nadu.I think I would say something more about the GENIUS before I complete.Well, once G. H. Hardy rated his contemporary mathematicians based on pure talent.Hardy rated himself a score of 25 out of 100,J.E. Littlewood 30, David Hilbert 80 andRamanujan 100 !Hardy also said that Ramanujan’s solutions were "arrived at by a process of mingled argument, intuition, and induction, of which he was entirely unable to give any coherent account”Ramanujan’s genius was recognized by TN Government andNow, Tamil Nadu celebrates 22 December as ‘State IT Day’A Stamp was released by the Govt. in 196222nd December started to be celebrated as Ramanujan Day in Govt Arts College, Kumbakonam. Now on 22nd December 2011, Then prime minister Manmohan Singh said that the 125th birth year of Ramanujan will be celebrated as National Mathematics Year and from that year onwards, December 22 is National Mathematics Day.There is a National Symposium On Mathematical Methods and Applications on his name (NSMMA)And there is SASTRA Ramanujan Prize which is given under the auspices of National Mathematics Society and the society for Physics.Let me tell something about the Hardwork of Ramanujan:Once P.C. Mahalanobis, the founder of Indian Statistical Institute visited Ramanujan while in Cambridge and said to him: “ Ramanju, these English Mathematicians say that you are a Genius, A real incomparable Genius.Immediately, showing his thickly black elbow Ramanujan replied, dear friend, everything owes to this elbow.Shocked by the answer, P.C. Asked: How Can it be so?????Ramanujan replied with a smile: “During my childhood days, while using a slate for calculations, repeated erasing used to leave remnants of chalk in it, then I stopped using duster for rubbing.”“This meant that every few minutes I had to rub my slate using my elbow, it means I owe everything to this elbow.”Regarding the spiritual dimension of Ramanujan’s life, all will agree that he was a sort of a mystic, and in fact, Ramanujan was a person with a somewhat shy and quiet dispositionHe was absolutedly a dignified man with pleasant mannersRamanujan credited his success to his family Goddess, Namagiri of Namakkalin fact, He claimed to receive visions of scrolls of complex mathematical content unfolding before his eyes. And we have no idea to contradict his words.And this could be in one way regarded as his Dictom"An equation for me has no meaning, unless it represents a thought of God.”We get amazed the more we know about Ramanujan’s spiritual understanding of many mathematical concepts, I will brief just one.For example, 2n – 1 will denote the primordial GOD.When n is zero, the expression denotes ZERO.He spoke of “ZERO” as the symbol of the absolute (Nirguna – Brahmam) of the extreme monistic school of philosophy)The reality to which no qualities can be attributed,of which no qualities can be there.When n is 1, it denotes UNITY, the Infinite GOD.When n is 2, it denotes TRINITY.When n is 3, it denotes SAPTHA RISHIS and so on.Crazy isn’t it, but all such craziness constituted Ramanujan.He looked “infinity” as the totality of all possibilities which was capable of becoming manifest in reality and which was inexhaustible.According to Ramanujan, The product of infinity and zero would supply the whole set of finite numbers.Each act of creation, could be symbolized as a particular product of infinity and zero, and from each product would emerge a particular individual of which the appropriate symbol was a particular finite number.If you want to go through the life of Srinivasa Ramanujan in its fullness, I humbly refer to you my guide, the book which opened my eyes towards realizing the pearl of Indian Mathematics, and that is:“The man who knew infinity: A life of the Genius Ramanujan”It was written by Robert Kanigel.In that book Kanigel claims some very amazing facts about Ramanujan.Sheer intuitive brilliance coupled to long, hard hours on his slate made up for most of his educational lapse.This ‘poor and solitary Hindu pitting his brains against the accumulated wisdom of Europe’ as Hardy called him, had rediscovered a century of mathematics and made new discoveries that would captivate mathematicians for next century.S.Chandrasekhar, Indian Astrophysicist, Nobel laureate 1983, told thus:“I think it is fair to say that almost all the mathematicians who reached distinction during the three or four decades following Ramanujan were directly or indirectly inspired by his example.Even those who do not know about Ramanujan’s work are bound to be fascinated by his life.”“The fact that Ramanujan’s early years were spent in a scientifically sterile atmosphere, that his life in India was not without hardships that under circumstances that appeared to most Indians as nothing short of miraculous. He had gone to Cambridge, supported by eminent mathematicians, and had returned to India with very assurance that he would be considered, in time as one of the most original mathematicians of the century.The words of Hardy himelf speak volumes of Ramanujan:“I have to form myself, as I have never really formed before and try to help you to form, some of the reasoned estimate of the most romantic figure in the recent history of mathematics, a man whose career seems full of paradoxes and contradictions, who defies all cannons by which we are accustomed to judge one another andabout whom all of us will probably agree in one judgement only, that he was in some sense a very great mathematician.”Bertrand arthur william russell, British philosopher & mathematician, Nobel laureate and almost contemporary to Ramanujan, stated thus:“I found Hardy and Littlewood in a state of wild excitement because they believe, they have discovered a second Newton, a Hindu Clerk in Madras… He wrote to Hardy telling of some results he has got, which Hardy thinks quite wonderful.”The life of Ramanujan is actually a textbook from which many things could be conceived. Despite the hardship faced by Ramanujan, he rose to such a scientific standing and reputation no Indian has ever enjoyed.It should be enough for youngsters like us to comprehend that if we can work hard with indomitable determination, sheer perseverance and sincere commitment, we too can perhaps soar the way like Srinivasa Ramanujan.Even today in India, Ramanujan cannot get a lectureship in a school / college because he had no degree.Many researchers / Universities will pursue studies / researches on his work but he will have to struggle to get even a teaching job.Even after more than 90 years of the death of Ramanujan, the situation is not very different as far the rigidity of the education system is concerned. Today also a ‘Ramanujan’ has to clear all traditional subjects’ exams to get a degree irrespective of being genius in one or more different subjects.He was offered a chair in India only after becoming a Fellow of the Royal Society.But it is disgraceful that India’s talent has to wait for foreign recognition to get acceptance in India or else immigrate to other places.Many of those won international recognition including noble prizes had no other option but to migrate for opportunities & recognition.(Ex. Karmerkar)The process of this brain drain is still continuing.Here is a pic of Ramanujan with his colleagues in Cambridge University.Talking about certain contributions of Ramanujan which shook me off my feet.As we all know we use the notation P(n) to represent the number of partitions of an integer n. Thus P(4) = 5, similarly, P(7) = 15.I don’t need to explain that If we were to start enumerating the partitions for larger numbers, even for small numbers such as 10 we start seeing that there is a combinatorial explosion! To illustrate this consider P(30) = 5604 and P(50) = 204226 and so on. (btw, partitions can be visualized by Young tableau!).A similar search was on for asymptotic formulae for the partition number P(n) and because of the combinatorial explosion an accurate formula was considered difficult. Ramanujan believed that he could come up with an accurate formula even though it was considered extremely hard, and he came close.One work of Ramanujan (done with G. H. Hardy) is his formula for the number of partitions of a positive integer n, the famous Hardy-Ramanujan Asymptotic Formula for the partition problem. The formula has been used in statistical physics and is also used (first by Niels Bohr) to calculate quantum partition functions of atomic nuclei.The formula he proposed gives a very close value to that of the true value, and it is a mouth-watering feat considering its very pattern less nature.I had written another answer in quora regarding how Ramanujan provided a rapidly converging series as the value of Pi. I will just copy and paste it here.For a long time, the series used for finding the value of Pi was given by the Leibniz-Gregory Series.π = (4/1) - (4/3) + (4/5) - (4/7) + (4/9) - (4/11) + (4/13) - (4/15) ...But in order to give the value of Pi correctly upto 5 decimal places, this series required around 500000 terms.Now, in the Indian tradition, another formula was given by Nilakantha, a mathematician of Kerala School of Mathematics who lived couple of centuries before Leibniz and the series converged much rapidly.π = 3 + 4/(2*3*4) - 4/(4*5*6) + 4/(6*7*8) - 4/(8*9*10) + 4/(10*11*12) - 4/(12*13*14) ...And in order to give the value of Pi upto 5 decimal places, this series required only 6 terms. And thats a great thing but which failed to catch the eye of westerners until the nineteenth century.Now, take into consideration all these and what Ramanujan did. Ramanujan simply penned down an infinite series, looking so horrendous, which would be equal to the reciprocal of Pi.And this is the most rapidly converging series ever given for the value of Pi and the algorithm based on this have actually been used in computers.Now the most beautiful factor. In order to have the value of Pi upto 6 decimal places the infinite series of Ramanujan needed only ONE SINGLE TERM.And you take the second term and there you have suddenly the value of Pi upto 11 terms in your hands.I think it speaks something Great, and Ramanujan was indeed Great!!!Ramanujan has done extensive works in finding out highly composite numbers, and he has written down a long list of similar numbers which had more factors than any of the previous number.The highest highly composite number listed by Ramanujan is 6746328388800Having 10080 factorsHe received his degree from the university (later named Ph.D) for his work of highly composite numbers.I would just say another thing which caught my eye and unleashed an array of thoughts.Ramanujan while sick and dying in India, mentioned some very peculiarly behaving functions which mimicked the original moldular functions.The mock theta functions remained a mystery for most part of the last century and only the Great Ono made inroads towards their reality.In fact, no one at the time understood what Ramanujan was talking about.It wasn’t until 2002, through the work of Sander Zwegers, that we had a description of the functions that Ramanujan was writing about in 1920,' Ono said.Ono and his colleagues drew on modern mathematical tools that had not been developed before Ramanujan’s death to prove this theory was correct.Ramanujan actually wrote those functions claiming that he saw it in a scroll in the hands of A Goddess.Anyway now they are used to calculate the entropy of Black Holes ( A concept which developed years after his death.)Ono’s team was stunned to find the function could be used today.'No one was talking about black holes back in the 1920s when Ramanujan first came up with mock modular forms, and yet, his work may unlock secrets about them,' Ono says.Ramanujan’s Intuition Stands OUT!I think, just for a fun I would show the Mock Theta FunctionsNow I think I shoudl mention atleast something about the impact of Ramanujan’s work on statistical physics.For example imagine studying the statistics of a gas made of electrons confined to 2D. You could do something complicated like model the exact positions and momenta of many of electrons along with the force between them. Or you can simplify by imagining that the electrons can only occupy positions on a discrete triangular lattice, and instead of a repulsive force you can make the simple approximation that two electrons aren't allowed to be next to each other.The result is the Hard hexagon model and some work of Ramanujan's appears when you try to model it. Even if it's not physically realistic, these models share characteristics with more realistic physical models and give useful insight.In fact a whole bunch of different identities related to Ramanujan's work can appear when you study these kinds of simple physical models, especially 2-dimensional models. Eg. Hard Hexagon ModelI think I will conclude with a simple assumption of Ramanujan, I think it deserves mention:The mock theta functions which we mentioned earlier looked unlike any known modular forms, but he stated that their outputs would be very similar to those of modular forms when computed for the roots of 1, such as the square root -1. Characteristically, Ramanujan offered neither proof nor explanation for this conclusion.It was only 10 years ago that mathematicians formally defined this other set of functions, now called mock modular forms. But still no one fathomed what Ramanujan meant by saying the two types of function produced similar outputs for roots of 1.Ono and his colleagues have exactly computed one of Ramanujan’s mock modular forms for values very close to -1. They discovered that the outputs rapidly balloon to vast, 100-digit negative numbers, while the corresponding modular form balloons in the positive direction.Ono’s team found that if you add the corresponding outputs together, the total approaches 4, a relatively small number. In other words, the difference in the value of the two functions, ignoring their signs, is tiny when computed for -1, just as Ramanujan said. Incredible Intuition !I am just adding some pictures I came across.his notebooks, the last three,His handwritings and works mentioned without calculation:I think I can say nothing more, but if at all someone asks me, I would say if I know!By the way, I have actally spoken nothing regarding the complex mathematical contributions of this great mathematician,even without that I think you are thrilled and that is why, even if the statement is wrong in itself.“ Ramanujan is the greatest Mathematician of all time, atleast I believe so.”