The mathemagical genius of Srinivasa Ramanujan – Commemorating 125 years of infinite excitement

Srinivasa Ramanujan was the strangest man in all of mathematics, probably in the entire history of science.  He has been compared to a bursting supernova, illuminating the darkest, most profound corners of mathematics, before being tragically struck down by tuberculosis at the age of 33, like Riemann before him.

- Michio Kaku

G H Hardy (Ramanujan’s mentor) contrived 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 the day, he assigned an 80.  To Ramanujan he gave 100.

- Quoted by Robert Kanigel, Ramanujan’s biographer

Preamble

This year is the 125th birth anniversary of the great Indian mathematician Srinivasa Ramanujan.  The nation is celebrating it as the Year of Mathematics.  Also, his birth date, December 22, has been declared as the National Mathematics Day in apparent similarity to February 28 that is being observed since 1986 as the National Science Day in tribute to the Nobel Prize winning contribution of physicist C V Raman [See my earlier blog post: 45) Commemorating a historical achievement – Raman his effect and the National Science Day (Feb 12)]

Ramanujan’s advent on the mathematical stage 125 years ago was not only like the explosion of a supernova as Michio Kaku put it but also nearly as rare in a field that boasts the likes of Archimedes, Newton, Euler and Gauss.  Considering that the last great supernova in our (Milkyway) galaxy happened as far back as 1604, mathematics is perhaps fortunate with a much higher frequency of such exploding talent as exemplified by Ramanujan and his likes.  His appearance would have been dazzling enough anywhere in the developed world, but the fact that it happened in a little known place in an obscure and subjugated country, not particularly visible in the intellectual firmament, was infinitely more so.

 

Though supernovae flare up and fade away rapidly, they leave behind them remnants of the event that are of lasting significance and keep astronomers busy for decades and even centuries as is the case with the famous Crab Nebula, the remnant of a supernova that erupted in 1054 AD.  Similar is the case with infrequently erupting mathematical geniuses who keep lesser mathematicians busy studying, analyzing, and building upon their contributions for decades to come.  The story of Srinivasa Ramanujan of India is a classic example of this.

A star is born

The birth of Ramanujan on 22 Dec 1887 to a very poor and highly orthodox Hindu Brahmin family in a sleepy town in South India was as insignificant an event as anything can be.  Like the overwhelming majority of stars in the night sky, this one was invisible and would have remained so but for the circumstances that made it explode into a bright supernova soon and spend itself in the process.  His father, a sales assistant in a textile shop in Kumbakonam, was a virtual non-entity throughout his life while his mother, a domineering woman, had a very significant influence on his upbringing and future.  She nursed him through an attack of the dreaded small pox that devastated the region when he was just two.  He completed his primary education with flying colors when ten, standing first in the district, and joined the Town High School in Kumbakonam where he exhibited exceptional brilliance in Mathematics.  He was lent a book on advanced trigonometry by S L Loney which he not only mastered quickly but also made many sophisticated discoveries on his own.  He continued to excel in academic studies and received a much needed scholarship to pursue further education at the government college in Kumbakonam.

 

Soon came his sensational tryst with destiny in the form of a well-known book on Mathematics – A Synopsis of Elementary Results in Pure and Applied Mathematics, by the English mathematician G S Carr.  It produced such a magical and obsessive influence on the young man that he found himself immersed in it most of his wakeful hours, to the exclusion of other academic pursuits.  He not only mastered its contents in a matter of months, but went on to go far beyond their scope through discoveries of his own that Carr could never have imagined.  This was perhaps the first spark of the supernova that would soon draw compelling attention in the mathematical firmament.

Agony amidst ecstasy

Carr’s Synopsis had ignited such a burst of fiercely single-minded and unassisted intellectual activity in him that he lost interest in everything else.  He put in only a physical presence in his classes and was always lost in mathematical thought without paying any attention to the lessons.  This started reflecting in his academic performance in other subjects to such an extent that he actually failed in most of them.  An inevitable consequence was the loss of his scholarship and a life of extreme poverty and dependence on friends and relatives for livelihood.  He would earn a little by giving tuitions in mathematics to higher class students, but not many were attracted to this because his world of mathematics was very different from theirs.

 

Unable to bear with the difficulties of life confronting him, Ramanujan sought an escape from it by just running away from home to far off Vishakhapatnam apparently in search of a change in his fortunes, but was soon located and brought back home.  He then shifted to Madras and joined Pachaiyappa’s College where he again excelled in mathematics and performed so poorly in other subjects that he failed once again and had to give up formal studies.  But this drove him deeper into mathematics and into a world of his own where his creative genius flourished even as he found himself on the verge of starvation many a time.

Ramanujan’s mother decided that a solution to his misfortunes lay in his marriage, to a nine-year old girl, consistent with the practice of child marriages widely prevalent in Hindu society.  His circumstances grew if anything even worse and he went about looking for any sort of job in Madras to keep his body and soul together.  He would carry his mathematical notebooks with him and introduce himself as someone with an original attainment in the subject which he needed to pursue with a modest job to provide essential livelihood.  He was such a simple, honest and likeable person that people went out of the way to help him in various ways and was soon successful in his modest quest.

Light at the end of the tunnel

Ramanujan was soon able to enlist the sympathies and support of several well placed people most of whom were also mathematicians, though involved in more lucrative professions.  Initially he got introduced to deputy collector Mr V Ramaswamy Iyer who had recently founded the Indian Mathematical Society. The latter was quickly struck by the extraordinary mathematical findings contained in Ramanujan’s notebooks and he didn’t want to offer the young man a measly job in his office; so he sent him looking elsewhere for better openings with letters of introduction focusing on his mathematical prowess.  With tremendous difficulty Ramanujan eventually met Mr R Ramachandra Rao the district collector of Nellore and the secretary of the Indian Mathematical Society.  Ramachandra Rao initially found it very difficult to believe that whatever he saw in the notebooks was Ramanujan’s original work, but soon found overwhelming circumstantial evidence to support the claim.  He was so impressed that he dug into his own personal resources to provide some regular financial support for the young prodigy until he got a suitable job.

 

Ramanujan was soon to end up as a ‘class III, grade IV accounts clerk’ at a salary of rupees 30 per month in the Madras Port Trust office and was happy with it considering that he had no college degree and the job afforded sufficient spare time for his mathematical pursuits.  More significantly, his immediate boss was Mr S Narayana Iyer who was the treasurer of the Indian Mathematical Association and an admirer.  Also, the head of the Port Trust itself was Sir Francis Spring who had also been tremendously impressed with his mathematical researches.  Such contacts enabled him to get several of his research findings published in the Journal of the Indian Mathematical Society, enhancing his reputation as a mathematical genius of exceptional merit.  Earlier, one of the professional mathematicians who had encouraged him greatly and appreciated his work was P V Seshu Iyer, a Professor at Pachaiyappa’s College where Ramanujan had actually failed.

In search of recognition

It soon dawned on friends and well-wishers of Ramanujan that his raw talent could not grow to its full potential in Indian conditions and needed the vastly superior intellectual environment that existed in Europe, preferably England for understandable reasons.  Samples of his work were presented to a few eminent British mathematicians for assessment and professional support, but met with little or no response.  Undaunted, Ramanujan was advised to contact Professor G H Hardy at Cambridge, one of the most renowned contemporary mathematicians.  He did so in January 1913, enclosing nine pages of his results as a ‘sample’.  His covering letter said in part; “…I have not trodden through the… university course, but am striking out a new path for myself.  I have made a special investigation of divergent series in general and the results I get are termed by the local mathematicians as ‘startling’…  Being poor, if you are convinced that there is anything of value I would like to have my theorems published…I have not given the actual investigations nor the expressions I get but I have indicated the lines on which I proceed…”  He was trying to invoke both pity and admiration at the same time, but in effect was only seeking recognition of the international mathematical community.

  

Ramanujan’s appeal to Hardy and the eventual outcome of it was in many ways similar to Satyendranath Bose’s appeal to Albert Einstein much later, seeking the latter’s help in getting his own path breaking work in Quantum Physics published [See my earlier blog post: 30) Bose and Einstein – A Historic Collaboration (Jul 11)]

Hardy’s Response

This time Ramanujan’s effort met with the kind of response he was hoping for.  Though stunned and incredulous, even annoyed, with what appeared to him superficially as some outrageous claims of a lowly unschooled clerk from the backwaters of the British empire, Hardy had the patience and good sense to go through Ramanujan’s notes in detail and come to the conclusion that they must be either an elaborate hoax or the work of an incredibly gifted genius.  He sought a second opinion from his great friend and younger professional colleague, J E Littlewood, who also felt the same way.  With a mixture of perplexity and amazement, Hardy concluded that Ramanujan’s claims ‘must be true because, if they were not, no one could have the imagination to invent them’.

Replying to Ramnujan’s appeal, Hardy wrote very encouragingly and, at the same time pointed out a few major flaws and deficiencies in his findings that required further examination.  In particular, he complained that most of the findings, stated in the form of end results without the intermediate steps involved in arriving at them, required to be proved with the kind of rigour that was the hallmark of mathematics as a discipline.  He wanted Ramanujan to send such proofs to buttress his claims.  On his part, Ramanujan had not paid much attention to such finer points, having very often arrived at the final results through a gigantic leap of his intuition and imagination.  Many a time he had indeed worked out some of the more important intermediate steps, but only on a slate, and had not transferred them into the written form for he couldn’t even afford the paper required!  In most cases they had been spun out only in his mind, leaving the reader to figure out the details!  As it emerged later, few could indeed do this and it was left to other geniuses like Hardy and Littlewood to fill in the gaps!

Invitation to Cambridge

The recognition he secured from Hardy and his associates at Cambridge also meant that Ramanujan had gone up several notches in the eyes of officialdom in Madras who were now willing to vie with each other in conferring special privileges on him.  The University of Madras bestowed a special and generous scholarship on him with full freedom to pursue his mathematical researches.  With this, he was liberated from both his gnawing poverty and the dependence on others.

 

Unknown to Ramanujan, Hardy had initiated steps to get him invited to Cambridge to work with him and others at the famous Trinity College.   This was a great deal more than Ramanujan had hoped from Hardy and initially refused to consider the idea because of long held religious edicts that forbade orthodox Brahmins to ‘cross the seas’ to go to foreign lands.  Hardy was disappointed to hear this and assigned the delicate task of persuading Ramanujan to change his mind to his Cambridge colleague E H Neville who was bound for Madras to give a series of mathematical lectures there.  Before Neville could take up the issue with him Ramanujan had received ‘divine intervention’ through the medium of his mother lifting the objection to crossing the seas.  Neville was pleasantly surprised to discover that there was no need for any human intervention.  The coast was now clear for Ramanujan to go to Cambridge and show the world what he was worth.

Life and Work at Cambridge

Ramanujan’s departure from Madras to England on 17 March 1914 was an event of considerable excitement and most of his friends and supporters were on hand to see the prodigy off.  The ship’s captain promised to give him special treatment as long as he didn’t talk mathematics with him!

Neville himself was on hand in England to receive him four weeks later and see him settled in Cambridge after a brief acclimatization in London.

  

Despite his humble origin and background, Ramanujan’s fame had preceded his arrival in Cambridge and he was looked upon by most people in this historic town as someone special.  In India, Neville had assured him much to his great relief that he would be enrolled in the Trinity College as a research fellow and would not be required to pass any examinations.  He had little difficulty in adjusting himself to Cambridge’s academic atmosphere and in establishing a cordial working relationship with his great benefactor Hardy, Littlewood and others.  However, the same could not be said of his personal life.  As a strict vegetarian with distinctly south Indian food habits and a teetotaler, he faced considerable difficulties.  Soon he found it necessary to turn into a cook and prepare his own food in a kitchenette attached to his living room.  P C Mahalanobis who later founded the Indian Statistical Institute in Calcutta was of great help with some of his personal problems.

Hardy and Littlewood both got to examine all of Ramanujan’s notebooks in detail and what they discovered was even more amazing than what they had been led to believe from their previous correspondence with him.  While any professional touch was conspicuously absent in his writings, their authenticity and originality was never in question.  Here was someone without any formal education beyond high school coming up with some of the most astounding discoveries in the entire history of mathematics, a realization that could not have come to anyone but people of their own caliber.  Some of Ramanujan’s discoveries were already well known in the annals of mathematics though unknown to him because he had not learnt them anywhere; they had been just rediscovered by him.  A few of them were even wrong as they showed him.  But most of them were absolutely new and original and it was now left to these and other mathematical scholars to make them known in the world of mathematics.  There were many so utterly confounding to them that years and even decades were to pass before they could be deciphered, understood, and established.  In any case, there was a lot more to come during the following years of collaborative work with Hardy in Cambridge.

Hardy painstakingly edited many of Ramanujan’s manuscripts giving them the shape, rigour, and professional touch that he was so particular about, and got them published.  He added his name as a joint author only where he had also made some original contributions.

    

Here is a recent picture of the magnificent Trinity College in Cambridge where the great Newton himself had once worked and on whose hallowed grounds Ramanujan could now walk proudly as an insider.  The same man had no chance whatever of getting into the Presidency College in Madras a few years earlier.

Ramanujan and Hardy, a study in contrast

Rarely in the history of intellectual endeavour have two people of such contrasting backgrounds, characters, styles, and personalities as Hardy and Ramanujan teamed up to advance the cause of knowledge to the extent that they did over such a short period of collaboration.  Their life and work is a study in contrast that comes out conspicuously well in Robert Kanigel’s wonderful biography of Ramanujan and I urge interested readers to go through it.

Stunningly handsome yet a life-long bachelor, aristocratic in demeanor and cricket crazy, Godfrey Harold Hardy had a soft corner for the underprivileged (Lenin was one of the men he admired) as the Ramanujan episode so starkly brings out.  As an outstanding mathematician he was also an uncompromising purist in mathematics, with an obsessive emphasis on ‘proof’ as the defining characteristic of ‘real’ mathematics.  He looked upon applied mathematics and mathematical physics with some condescending tolerance.  He was a devoted atheist (if one can employ such a description), but never intolerant of conflicting viewpoints.  He wielded a considerable influence in the world of mathematicians, much of it due to the preeminent quality of his work.  Above all he could readily recognize and value any superior intellect.  Despite his own notable discoveries in mathematics, history remembers Hardy principally for his discovery of Ramanujan.

Srinivasa Ramanujan was the antithesis of Hardy in most respects.  He was devoutly religious, believed in and practiced all the precepts and rituals of his religion and caste, was highly superstitious, shed his traditional attire only because of the demands of radically altered circumstances and believed strongly that all his exceptional abilities were God-given. Being entirely self-taught, he had little idea of how mathematics was communicated among academicians, had little appreciation for the concepts of ‘rigour’ and ‘proof’ in mathematics till he interacted with Hardy, and had little interpersonal contact with other people in Cambridge.  He was well aware that, but for the benevolence of Hardy in particular, he might not have gained any professional recognition at all.

Soon after Ramanujan and Hardy met in Cambridge they started a very fruitful collaboration, each having had a great deal to learn from the other.  They also developed an enormous respect and admiration for each other even while realizing that they had little in common at the personal level.  In the matter of beliefs and practices, they carefully avoided treading on each other’s feet.

One of the best known incidents involving the two is the episode of the taxicab.  While visiting Ramanujan in his sanatorium years later, Hardy said he had travelled in dull weather in a taxicab bearing an equally dull number.  When asked, Hardy remembered the number to be 1729.  Ramanujan flew into a paroxysm of excitement saying it was anything but dull.  It was in fact the lowest integer known that could be expressed as the sum of two cubes in two different combinations!  Such was his hold over numbers every one of which was his personal friend!  (After I first read this story during my school days in Bangalore I realized that the house I was living in had the same door number as this; however, the house itself was far less exciting than the number!)

Recognition and Rewards

In March 1916 Ramanujan, who had been a college dropout at Madras earlier, received the highly valued B A degree of the Cambridge University based not on any coursework or examination from which he had been exempted, but on the basis of his published research work on ‘highly composite numbers’ that was one of his outstanding achievements in Cambridge.  Such a degree from Cambridge was easily the equivalent of a Ph D anywhere else.  A memorable group photograph taken on this occasion (see below) at Trinity College shows Ramanujan at the centre of a row of students standing stiffly and uncomfortably, with his mentor G H Hardy providing company at extreme right.  This is one of perhaps just a handful of photographs showing him that has survived.

Ramanujan was confidently expected to be conferred the even more highly valued Fellowship of the Trinity College the following year, but this did not materialize to his great disappointment and of his well-wishers because of some squabbles within the institution, with an undertone of racism thrown in.  Very sadly, this was also the beginning of a dark phase in his personal life that was to last the rest of his time in Cambridge.  A mysterious illness, later diagnosed as tuberculosis, had begun to engulf him.  This was to keep him confined to several sanatoria in England before forcing his departure back home and eventually to take his life as well.  The consequences of the ongoing First World War in Europe that had left some imprint on Cambridge too were to exacerbate his situation.

Ignoring the Trinity College fellowship fiasco, Hardy and eleven other reputed mathematicians decided that Ramanujan was worthy of an even greater honour – the fellowship of the Royal Society of England – and nominated him for this.  Earlier he had been elected a fellow of the London Mathematical Society in whose proceedings quite a number of his papers had been published.  Soon enough, Ramanujan received the enthralling news from Hardy that the Royal Society had indeed decided to confer him its fellowship in 1918.  Historically, the Fellowship of the Royal Society (FRS) is the highest academic honour a scientist can get in England and often regarded as next in importance only to the Nobel Prize. So the college dropout from South India now became an FRS, one of the youngest ever to get this honour and only the second Indian to do so.   Now that this had happened, it was easy for Trinity College to follow suit and confer him its own fellowship that he had been denied earlier – the lesser honour came Ramanujan’s way after the greater honour.  People back home in India were thrilled by the news and looked upon Ramanujan as their own and an international celebrity, waiting to bestow their own honours when Ramanujan returned to India.

Last days in England

Even before his health worsened, Ramanujan got into a state of depression caused by a variety of factors, including his increasing personal problems both in England and back home in Madras.  In one fatefully weak moment he tried to commit suicide by throwing himself on to a railway track in the London underground system. An alert railway guard spotted him and brought the train to a halt before any harm could be done.  The whole episode was hushed up through Hardy’s diplomatic intervention with the police authorities.  Ramanujan recovered from this incident for some time, buoyed up by the academic honours that came his way, but the reprieve was short lived.  His health deteriorated sharply and he had to spend time in several sanatoria for tuberculosis patients where the conditions, particularly the food, were intolerable for him.  Hardy advised that it was time for Ramanujan to return home, at least temporarily, a suggestion acceptable to all concerned since the world war had also come to an end.

Dying days in India

When he returned to India on 27 March 1919, Ramanujan’s health had deteriorated to such an extent that his old friends and admirers back home saw the writing on the wall clearly and tried to prop him up as much as possible, aided by numerous felicitations and honours showered upon him from all quarters in Madras, including a professorship at the university which he had not been able to dream of joining even as a student.

Ramanujan spent his last days in Madras in great agony, both physical and mental, but his mathematical productivity wasn’t affected much.  Unknown to most people he had been working on what was to be his magnum opus, a new and exciting class of functions called ‘Mock Theta’ functions.  When he met his end on 26 April 1920 in Madras, aged just 32, the papers containing these and other discoveries were found by his wife and passed on to Cambridge through various channels and got ‘lost’ somewhere there, fortunately to be rediscovered decades later.

 

Commemorating the genius

Ramanujan’s birth centenary in 1987 and the earlier decades leading up to this were heralded by numerous events and actions resurrecting and preserving the memory of this historic personality.  Here is a stamp issued by the Indian postal services to mark his 75th birthday:

Ramanujan’s Kumbakonam residence got a sort of face lift in 2003 and was dedicated to the nation as a memorial by the then president of India, A P J Abdul Kalam.  Its interior houses a bronze bust of Ramanujan and some memorabilia as seen in the following picture.  It is not clear if the preservation of the building, changed very little from its original condition, was an intentional act or one of indifference.

Several international prizes are being offered in memory of Ramanujan for distinguished original work in mathematics.  Among them is the annual ICTP (International Centre for Theoretical Physics founded in Trieste, Italy by physics Nobel Laureate Abdus Salaam) Ramanujan Prize for Young Mathematicians from Developing Countries.

 

A three-day international conference, ‘ Ramanujan 125,’  was held last month at the Florida State University in USA to mark the 125th birth anniversary celebration of Ramanujan.  Organized by Professors Krishnaswami Alladi and Frank Garvan of the University of Florida and Ae Ja Yee of Pennsylvania State University, it brought together about 70 researchers.  They delivered talks on current research work in several areas of mathematics influenced by Ramanujan’s work.  The three pre-eminent experts on the mathematical genius — Professors George Andrews, Bruce Berndt and Richard Askey all participated.

Ramanujan’s Biography

In 1991, Robert Kanigel, who was not a professional mathematician but a very competent and experienced science communicator and professor of science writing at the famed MIT, USA, wrote a voluminous, widely acclaimed and perhaps definitive biography of Ramanujan after extensive research on his life and work in both India and England, and titled it “The Man Who Knew Infinity – A Life of the Genius Ramanujan”.  He could more justifiably have changed the title to “The Man Who Played with Infinity.”  He has perhaps done more to bring out the authentic genius in Ramanujan than anyone else at the popular level.

The Ramanujan Scholars

Apart from Hardy, Littlewood and other contemporaries of Ramanujan, there have been several great names associated with the study of his monumental notebooks, filling the necessary gaps and publishing the voluminous results.  Three of them, all Americans, deserve particular mention.  They are: George Andrews of the Pennsylvania University, Bruce Berndt of the University of Illinois, and Richard Askey of the University of Wisconsin.  George Andrews was instrumental in rediscovering Ramanujan’s long-forgotten and now famous ‘Lost Notebook’ in Cambridge and making a special study of it, particularly his ‘mock theta’ functions.  He and Bruce Berndt together have brought out a fully edited and elaborated version of the ‘Lost Notebook’.  The other four notebooks have all been edited and brought out by Bruce Berndt.  As Kanigel puts it, these published notebooks today ‘sustain a veritable cottage industry of mathematicians’ all over the world devoted to their in-depth study.

Postscript

Some of the more prominent discoveries with which Ramanujan’s name is permanently engraved in the history of mathematics are: Landau-Ramanujan Constant, Mock Theta Functions, Ramanujan Conjecture, Ramanujan Prime, Ramanujan-Soldner Constant, Ramanujan theta Function, Ramanujan’s Sum, Rogers-Ramanujan Identities, and Ramanujan’s Master Theorem.  Here is a very small random sample of the expressions/equations/identities (stated without proof in the original Carr-inspired style of Ramanujan!) that a student or connoisseur of mathematics will find exceptionally beautiful and edifying.  To me they are just like listening to the last movement of Beethoven’s Ninth Symphony, the Ode to Joy.