My Search for Ramanujan

by Ken Ono

  Ramanujan was just a teenager, but he was already developing original research based on what he was reading. Of course, he was working in a relative mathematical desert, for despite its prestige in southeast India, Government College, with its mere dozen lecturers, was an intellectual backwater. Although unaware of what was known or unknown beyond the pages of Carr’s book, and although some of Ramanujan’s results, while original to him, were well known to the mathematical world, some of his results were new, often going beyond those obtained by some of the world’s great mathematicians, such as the brilliant eighteenth-century Swiss mathematician Leonhard Euler, arguably the most prolific mathematician of all time, whose collected mathematical papers fill about ninety volumes.

  So although Ramanujan dutifully attended the required lectures, his mind was elsewhere—engrossed in the riddles of Carr’s book, and “quite unmindful of what was going on around him,” as a classmate later described him to one of Ramanujan’s biographers. Although in high school he had excelled in many subjects, including English, he now failed his English composition paper, and as a result, he lost his scholarship. His mother went to the head of the college to plead, beg, cajole, and complain. But the scholarship was not reinstated.

  Ramanujan remained in school a few months longer, but without the scholarship, college was unaffordable. Torn between family loyalty, love of mathematics, the desire to please, and the pull of a greater force, Ramanujan was lost. He did the only thing he could, something with which I readily sympathize: he ran away from home. He took a train to the distant town of Vizagapatnam, seven hundred miles up the coast, leaving without a word at the beginning of August 1905. His family made frantic inquiries. They advertised and posted notices for their missing son, and they soon found him and brought him home.

  Ramanujan attempted to return to university to continue his education. In 1906, he traveled the two hundred miles by train to Madras, arriving dazed and confused, but eventually making his way to Pachaiyappa’s College, where he hoped to study and obtain his degree. Here, the experience from Kumbakonam repeated itself: Ramanujan excelled beyond expectations in mathematics—easily surpassing the knowledge of his professors, who held him in awe. But he just couldn’t concentrate on his other subjects. He failed his physiology exam miserably several times, and eventually had to withdraw from the college without a degree.

  © Springer International Publishing Switzerland 2016

  Ken Ono and Amir D. AczelMy Search for Ramanujan10.1007/978-3-319-25568-2_9

  9. The Goddess

  Ken Ono1 and Amir D. Aczel2

  (1)Department of Mathematics and Computer Science, Emory University, Atlanta, GA, USA

  (2)Center for Philosophy & History of Science, Boston University, Boston, MA, USA

  Mathematically talented children are frequently identified as such by their ability to perform very large calculations rapidly. Ramanujan was a prodigious calculator, but what set him apart was his creativity—his ability to conjure never before imagined mathematical formulas out of thin air. Where did they come from?

  Despite his status as an untrained amateur mathematician, and a college dropout to boot, the young Srinivasa Ramanujan was producing mathematical results that were so unexpected and so significant that they are striking to a mathematician who sees them for the first time even today, a century later. When I first saw them, they appeared to me as exquisite mathematical treasures, which had been revealed to Ramanujan as if by magic, as though spirited from the depths of some Ali Baba’s cave. And even at an early stage of my life as a professional mathematician, I found them utterly irresistible, and I would eventually tie my career to them. I just had to find out how Ramanujan had obtained them.

  How could he possibly know that the mathematical formulas he was conjuring were correct —formulas that would take some of the world’s leading mathematicians months and years to prove? There was something mystical, supernatural, perhaps even spiritual in the way Ramanujan obtained his results, and indeed, those are the sorts of adjectives that are frequently applied to him.

  When asked how he obtained his results, Ramanujan would reply that his family goddess, Namagiri, sent him visions in which mathematical formulas would unfold before his eyes.

  Brahmins were at the top of India’s caste system, and they prided themselves on their historical role in perpetuating and maintaining the Hindu religion. Ramanujan was a devout Hindu—likely not only because he was a Brahmin, but also through the influence of his mother, who was deeply involved with activities in the local temple to the goddess Namagiri. The Sarangapani temple was just up the street from Ramanujan’s family home in Kumbakonam, and he spent much of his time there doing his mathematics.

  He once explained to a friend that he saw in the mathematical expression “the primordial God and several divinities.” This was because when one plugs n = 0 into the expression, one gets 1−1 = 0, which represents nothingness. Plugging in 1 for n gives the result 2−1 = 1, which he said represented to him unity and the infinite God. When substituting 2 for n, one obtains 3, the Trinity, which to him meant the Hindu gods Shiva, Vishnu, and Brahma. Plugging in 3 for n yields the number 7, which represented the Saptha Rishis—the “Seven Sages” of Hinduism, described in the Vedas, sacred texts dating to as early as the second millennium b.c.e. Ramanujan also found religious symbolism for additional values of n.

  He once said to a friend, “An equation for me has no meaning unless it expresses a thought of God.” In this, he came close to an almost identical statement by Einstein, who in describing his work on general relativity and his attempts to find a theory that would capture the nature of gravitation and the physical laws of the universe, famously said, “I want to know God’s thoughts.”

  Beginning in 1907, having lost his college scholarship and at loose ends regarding everything in his life outside the pages of Carr’s book, Ramanujan began to keep a notebook of his mathematical results. He claimed that they came to him in dreams and visions: Namagiri would appear to him and draw mathematical statements over a red screen. In the morning, he would write them down, ponder them, analyze them, and study them further to see how they could be extended to new results. In fact, he would often wake up in the middle of the night to write down a formula that had appeared to him in a dream, and in the morning, he would write it out in greater detail.

  When I first heard this story about Ramanujan and his dreams, I rejected at once as fanciful poppycock the suggestion that his insights came to him as visions from a goddess. I, who had been raised with no religion and a strong belief that all phenomena have a rational explanation, viewed the idea of divine inspiration as a fable invented by others to add mystery to the legend of Ramanujan, or else cooked up by Ramanujan himself to elevate his accomplishments beyond the mundane world of hard work, the sort of thing my father did, scribbling all day on yellow pads of paper.

  But I have gone a very long distance out of my way since then, and today, I have quite different views.

  © Springer International Publishing Switzerland 2016

  Ken Ono and Amir D. AczelMy Search for Ramanujan10.1007/978-3-319-25568-2_10

  10. Purgatory

  Ken Ono1 and Amir D. Aczel2

  (1)Department of Mathematics and Computer Science, Emory University, Atlanta, GA, USA

  (2)Center for Philosophy & History of Science, Boston University, Boston, MA, USA

  Genius often finds itself rejected when it fails to fit into the mold designed for ordinary people. Einstein, for example, was at first denied admission to the prestigious Swiss Federal Institute of Technology, in Zurich, because he failed the nonscience portion of the entrance examination. Thomas Edison’s teachers told him that he was too stupid to learn anything, and he was fired from his first two jobs for not being productive enough. Ramanujan was in good company.

  When we last saw Ramanujan, he had just flunked out of Pachaiyappa’s College. He had hoped eventually to enter the University of Madras, since a degree fro
m that university would have qualified him as a working mathematician or given him the credentials to obtain some other respectable job with a decent salary. But having failed the physiology part of his exams repeatedly, he now found himself out of school and unemployed. No work meant no money, and Ramanujan did not have even enough to feed himself. In desperation, he returned home to Kumbakonam. But soon he was on the road again.

  Returning to Madras, Ramanujan continued to live in abject poverty. He had little to eat and was desperate to find some kind of employment. He began looking for any sort of job, and he finally obtained a temporary post as a clerk, but he left after a few weeks. He couldn’t keep his mind on his work; his addiction to mathematics was too strong. Then he decided to try to obtain a more suitable position—one that he hoped would leave him time for mathematics—by contacting men in the British administration of India, the famously efficient Indian Civil Service, who were also mathematicians or had some knowledge and appreciation of the subject. He began to build a network, contacting friends, friends of friends, family, family friends, and anyone who knew anyone who knew anyone who might appreciate his mathematical genius and decide that it would be a shame for someone so talented to be without a livelihood.

  Ramanujan had a pleasant personality. He was uncommonly likeable and engaging, which made even those who had met him only briefly happy to give him a reference. Given the name of someone who might be able to help him, he would go and knock on his door. He also approached professors at the colleges he had attended as well as at other colleges and universities in south India, showing them his mathematical work and hoping to interest them in providing him a post that would enable him to continue to do mathematics.

  His work was so astonishingly novel—but also seriously lacking in detail and anything resembling formal proof—that some of the professors at first doubted that it was his own. But once he showed them how he derived his equations, a few understood that he was not a fraud, and they wrote enthusiastic letters of reference on his behalf—but they came to nothing.

  Eventually, as his money dried up, his situation became more dire, and he had to think seriously about nonmathematical subjects, such as where his next meal was coming from. Desperate, he began to look for almost any form of employment. Eventually, he obtained a temporary post as clerk in the Office of the General Accountant, in Madras, a very minor civil service position with a very small salary. He could barely survive, and he continued looking for something better that would allow him time to pursue mathematics without starving for it. Finally, his networking began to pay off, and he went to meet a truly important man, Ramachandra Rao, who held a high position as revenue collector in the city of Nellore. And what was more, he was a mathematician, and indeed was currently serving as secretary of the Indian Mathematical Society. In a 1920 article in the Journal of the Indian Mathematical Society, Rao described his first meeting with Ramanujan:Several years ago, a nephew of mine perfectly innocent of mathematical knowledge said to me, “Uncle, I have a visitor who talks of mathematics; I do not understand him; can you see if there is anything in his talk?” And in the plenitude of my mathematical wisdom, I condescended to permit Ramanujan to walk into my presence. A short uncouth figure, stout, unshaved, not overclean, with one conspicuous feature—shining eyes—walked in with a frayed notebook under his arm. He was miserably poor. He had run away from Kumbakonam to get leisure in Madras to pursue his studies. He never craved for any distinction. He wanted leisure; in other words, that simple food should be provided for him without exertion on his part and that he should be allowed to dream on.

  He opened his book and began to explain some of his discoveries. I saw quite at once that there was something out of the way; but my knowledge did not permit me to judge whether he talked sense or nonsense. Suspending judgment, I asked him to come over again, and he did. And then he had gauged my ignorance and showed me some of his simpler results. These transcended existing books and I had no doubt that he was a remarkable man. Then, step by step, he led me to elliptic integrals and hypergeometric series and at last his theory of divergent series not yet announced to the world converted me. I asked him what he wanted. He said he wanted a pittance to live on so that he might pursue his researches.

  Rao was taken enough with the brilliance of Ramanujan that he determined that offering him a position in Nellore would not do—Ramanujan belonged in the larger Madras, where he could meet mathematicians and others—and instead offered him a stipend that would allow him to continue his mathematical work unhindered by worries of employment. This was a godsend. Ramanujan settled in Madras and continued to work on his identities.

  At some point, he returned home to his parents in Kumbakonam. There, completely oblivious to what was going on around him, he would sit on the porch, doing mathematics in total concentration, his eyes shining when he made a discovery. His parents supported him without complaint and showed no irritation or inclination to push him to look for a job. But his mother had certain ideas of her own about his future.

  © Springer International Publishing Switzerland 2016

  Ken Ono and Amir D. AczelMy Search for Ramanujan10.1007/978-3-319-25568-2_11

  11. Janaki

  Ken Ono1 and Amir D. Aczel2

  (1)Department of Mathematics and Computer Science, Emory University, Atlanta, GA, USA

  (2)Center for Philosophy & History of Science, Boston University, Boston, MA, USA

  What Ramanujan’s mother had in mind was a wife for her son, and like my parents’ marriage, Ramanujan’s was to be an arranged one.

  After consulting the goddess Namagiri, Komalatammal had decided that it was time for her son to get married. So she set out to find him a wife. Through an Indian tradition allowing child marriages, whose consummation was deferred, Komalatammal found her son a child bride, the ten-year-old S. Janaki Ammal.

  His mother had arranged everything. She began with a visit to distant relatives in Rajendram, seventy miles west of Kumbakonam, a village so small that it does not appear on most maps. There, she noticed a sprightly girl of nine whose family had five girls to marry off and was so poor that they could offer little by way of a dowry. She immediately consulted the girl’s horoscope in conjunction with that of her son—as was the custom in such matters—and decided that Janaki was perfect for Ramanujan. A few months later, on July 14, 1909, when Janaki was ten, the wedding took place in Rajendram, and at the same ceremony, one of Janaki’s sisters was also married. Unfortunately, this sister would die of a fever some months later.

  After the wedding, Ramanujan returned home with his mother to Kumbakonam, and over the next few years he would become a mathematical nomad—traveling by train throughout south India, to Madras, and elsewhere—while Janaki stayed home with her mother, who was teaching her how to be an obedient wife and perform household chores. The couple would not live together until 1912, in Madras, where Komalatammal would live with them.

  It was this child bride of Srinivasa Ramanujan who, seventy-five years later, as an eighty-five-year-old widow, sent my father that fateful thank-you letter.

  Janaki Ammal

  © Springer International Publishing Switzerland 2016

  Ken Ono and Amir D. AczelMy Search for Ramanujan10.1007/978-3-319-25568-2_12

  12. I Beg to Introduce Myself

  Ken Ono1 and Amir D. Aczel2

  (1)Department of Mathematics and Computer Science, Emory University, Atlanta, GA, USA

  (2)Center for Philosophy & History of Science, Boston University, Boston, MA, USA

  Though he was without a college degree and had essentially no formal training in mathematics, Ramanujan had accumulated a massive collection of formulas, all recorded in his notebooks without proof. Eager to share his work with others, he began to publish some of his findings, beginning in 1911 by submitting problems to the Journal of the Indian Mathematical Society. At first, nobody paid attention, or at least no one was able to solve his problems. For example, one of the problems that he challenged r
eaders to solve was to find the value of

  What number was represented by this infinite nested square root?

  Six months passed, and three new issues of the journal appeared, yet no one had proposed a solution. If any readers were paying attention, they were stumped.

  So Ramanujan simply revealed the answer: 3. But he did not reveal his methods, and in fact, finding the solution is difficult, requiring a long derivation. Ramanujan had developed a formula that took a sum of three numbers and expanded it into an infinite nested square root. The problem he posed was a special case of that formula for the sum 2 + 1 + 0.

  In 1912, Ramanujan finally obtained a more permanent position, as a clerk in the Madras Port Trust. He performed his job so efficiently that he had time to continue his research in mathematics and publish more papers. His brilliance was now more widely appreciated, and Ramanujan and his supporters began to realize that he would never receive the recognition he deserved in India, where little cutting-edge research was being done. Most Indian mathematicians were either pursuing mathematics as an avocation on top of another job or working in a college or university with heavy teaching duties and little time or reward for research. They would not have the background to understand and appreciate Ramanujan’s mathematical accomplishments.