My Search for Ramanujan

by Ken Ono

  A few months later, with my parents resigned to my decision, I moved out. They dropped me off at Penn Station in Baltimore, where I boarded a train bound for Montreal. I had my fancy Peugeot, and some clothes stuffed in an old blue canvas suitcase held together with duct tape.

  Who would have thought that some long-dead Indian dude named Ramanujan would be the talisman that would unlock the door of my cell? I had no way of knowing then that I would be hearing a lot more from that miraculous mathematician, that one day I would be drawn to search for the source of his mathematics, a search that would finally let me come to terms with my tough-loving Japanese tiger parents.

  Part II

  The Legend of Ramanujan

  © Springer International Publishing Switzerland 2016

  Ken Ono and Amir D. AczelMy Search for Ramanujan10.1007/978-3-319-25568-2_6

  6. Little Lord

  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

  Kumbakonam, India (1890s)

  Crowds ramble by on foot and rickshaw on this busy dirt road lined with small shops and peddlers seated on blankets selling vegetables, silks, tin plates, and almost anything else you might imagine. Most of the people are thin Indians clad in light, loose-fitting garments. Once in a while, a British officer or administrator passes by in a khaki uniform or formal suit. Cows and goats roam freely, eating whatever scraps and waste they can find in this hot and humid place. This is Sarangapani Street in late-nineteenth-century Kumbakonam, a town in the southeast Indian state of Tamil Nadu, Ramanujan’s childhood home.

  Map of South India (drawn by Aspen Ono)

  The year is 1887, and Komalatammal, Srinivasa Ramanujan’s mother, has left her husband behind in Kumbakonam to travel the 150 miles west to her mother’s home in Erode to give birth to her first child. She is a corpulent, authoritarian woman who, like my mother, runs her household with little assistance from her husband, who works as a clerk in a fabric store. And so it was in Erode, on December 22, 1887, that Srinivasa Ramanujan Iyengar was born. A year later, mother and son returned home to Kumbakonam.

  Ramanujan’s mother, Komalatammal

  Srinivasa was the baby’s father’s name, and Iyengar indicated his high priestly Brahmin caste. His father was K. Srinivasa Iyengar, and so the future mathematician would be known throughout his life simply as Ramanujan (the emphasis is on the middle syllable, Rah-MAN-ujan), the only part of his name that was uniquely his. The name derives from the Indian mythological hero Rama, on whose life and deeds the famous Indian epic the Ramayana is based. His doting mother, however, referred to him throughout his life by the endearing nickname Chinnaswami, which means “Little Lord.” And despite the family’s poverty, he grew up like one.

  Like another genius, Albert Einstein, born eight years earlier, Ramanujan began to talk unusually late—to the point that his parents worried whether he would ever be able to communicate. His mother feared that he was deaf, and following a suggestion made by a friend of her father’s, he was made to study the characters of the Tamil language—thus learning to read before he ever opened his mouth to speak. Finally, having memorized the characters of this ancient, linguistically pure Dravidian language, Ramanujan began to sound them out, and finally began to talk.

  Ramanujan’s father was rarely home—leaving early in the morning and returning home late in the evening—and like my father, he paid little attention to his son. Ramanujan was raised by his strong-willed mother, who helped him with his schoolwork and even advocated on his behalf by complaining volubly to the school principal whenever she was unhappy with how her son was being treated in school. Ramanujan grew to be a willful child who rarely agreed to do anything that didn’t suit his wishes, and he routinely ignored instructions from teachers and school administrators, indeed from anyone in authority. He was also exceptionally sensitive, taking offense where it may not have been intended, and possessing a perfectionistic attitude and a keen sense of shame. For him, any personal failure, however slight, was followed by severe mental anguish. If one of his classmates achieved a higher score on an exam in an area that he believed he understood better, Ramanujan would agonize over his “defeat.”

  Komalatammal would later give birth to three more children, but all of them died shortly after birth. That, unfortunately, was far from an unusual circumstance. Many children born in south India in that period died at birth or at a young age. Other children would be born to the family many years later, after Ramanujan had reached young adulthood, and so he was raised virtually as an only child.

  His mother, like mine, held a part-time job. She worked in a temple choir, singing hymns to the local goddess. This gave the family some much-needed extra income to supplement the father’s meager earnings. The family was poor, but richer than the average family in the peasant class of south India. As Brahmins, members of the highest Indian caste, they lived with pride despite their economic difficulties.

  Kumbakonam, which had about fifty thousand inhabitants at the time, is located within a vast agricultural region watered by the Cauvery River—a water source considered almost as holy as the famous Ganges hundreds of miles to the north. Mosquitoes proliferated in this area of ample standing water, and so malaria, a scourge that kills the young as well as the old—was rampant. So were many other diseases. A smallpox epidemic raged in this region in 1889, when Ramanujan was two years old, killing thousands of children. The boy fell seriously ill, but he beat the odds and survived.

  Just before reaching the age of five, Ramanujan enrolled in school in Kumbakonam. He disliked school. With the personality of a “Little Lord,” it was not easy for him to accept authority, and because he was an exceptionally bright child, his mind was logical and critical, and he was particularly unwilling to do anything that was asked of him for which he could see no good reason. Like me, he excelled in mathematics, and during this time in his schooling he was strong in all of his subjects, and he gained skills that would prove useful in later life. One of those skills was proficiency in English. Only about a tenth of the population of India at that time had good command of the English language—which was important for communication among the peoples in the subcontinent, who spoke well over one hundred different major native languages, most of them Dravidian or Indo-Aryan in origin. A proficiency in English opened up for a young person the possibility of obtaining a job in the civil service or some other well-paying profession.

  A month before his tenth birthday, Ramanujan took exams in Tamil, English, geography, and arithmetic. He scored first in the entire district and was praised for his outstanding performance. That achievement enabled him to enroll in the local high school, Town High, one of the best schools in the area. Classes were taught in English, which was a great advantage to its students.

  Already in the second year of high school, Ramanujan became known for his mastery of mathematics, and students from the school would often come to him for help with problems. In mathematics, he reigned supreme, and his knowledge quickly approached that of his teachers. Through books he had obtained, Ramanujan learned a great deal of trigonometry, geometry, and algebra on his own.

  Under the overbearing guidance of his mother, Ramanujan emerged as a headstrong and brilliant ten-year-old. He so excelled in his studies that nobody would have believed that the day would come when school would be a struggle for him.

  © Springer International Publishing Switzerland 2016

  Ken Ono and Amir D. AczelMy Search for Ramanujan10.1007/978-3-319-25568-2_7

  7. A Creative Genius

  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

  Just like the kids I knew from Julian Stanley’s John Hopkins study of talent
ed children, Ramanujan was identified at an early age for his gifts in mathematics. Like us, he was capable of flying through the usual mathematics curriculum. However, where we had a knack only for understanding formulas, he had a knack for creating them.

  On his own, Ramanujan was able to see past the formulas to the theory behind them. Indeed, he quickly exhausted the mathematical knowledge of his teachers at school and began to read books on mathematics independently, which he obtained from two university students who boarded with his family, who welcomed the additional income, which supplemented what Srinivasa was making as a clerk and what Komalatammal was paid for her temple singing.

  As Ramanujan was growing up, Komalatammal was spending less and less time with the choir, feeling that she needed to focus on nurturing her young lord. More than anyone else, she understood how brilliant her son was, and she would spend hours teaching him about the world, educating him at home in ways that were more suited to his personality than the rigid rote instruction that he was receiving at school. She played intellectual games with him, took him on long walks by the river, and even consulted with him about her own life. They were exceptionally close.

  Ramanujan quickly devoured the books lent to him by the two boarders, teaching himself so much that he had now advanced well beyond what the teachers at his school could offer. And he began to go beyond the mathematics presented in the books he had been given, developing his own ideas about trigonometry, geometry, number theory, and infinite series.

  In 1900, while only in the second year of secondary school (corresponding to our seventh grade), Ramanujan began to figure out on his own how to work with infinite series, that is, how to efficiently represent the sum of infinitely many terms, which may or may not represent a unique finite number. Infinite series and continued fractions would become an obsessive occupation for him throughout his life—as they would, decades later, in my own work.

  While still a child, Ramanujan was fast becoming an expert on those seemingly intractable mathematical entities. He spent more and more of his time trying to understand how infinite series work, which ones converge and which ones do not. He would soon find a method that allowed him to add up all of the positive integers, that is, the expression , and obtain a finite negative number! As crazy as this claim appears to be, one can make complete sense out of it, and it requires important theorems by some of the world’s great mathematicians, including Jacob Bernoulli, Leonhard Euler, and Bernhard Riemann, on a subject called the “analytic continuation of the Riemann zeta function.” The young, untrained Ramanujan had somehow obtained glimpses of the work of Bernoulli, Euler, and Riemann on his own.

  Ramanujan had such an amazing aptitude for mathematics that by age twelve, he had worked out new solutions to problems in number theory and analysis. Astonishingly, he was able to come up with mathematical facts and ideas in what was close to an intellectual vacuum.

  India had a long tradition, going back to the early Middle Ages, of producing important mathematical results without proof. And like some other Indian mathematicians, Ramanujan cared little about formal proof. He simply derived beautiful mathematics as if out of thin air—mostly identities and equations.

  In 1902, he learned about the method that Italian mathematicians had discovered in the sixteenth century for solving the cubic equation, and he derived on his own a method for the solution of quartic, or fourth-degree, equations—repeating a variation of a feat performed brilliantly by René Descartes three centuries earlier (though Descartes was much older at the time than Ramanujan).

  Ramanujan managed not to antagonize his teachers too much while still in school. But his mathematical prowess was growing without bound and consuming more and more of his time. Like Descartes, Galois, and Einstein, Ramanujan became known in his school as a math genius. He kept winning awards for excellence, and in 1904, he won the K. Ranganatha Rao Prize in mathematics. In announcing the honor to the gathered teachers and students, the principal, K. Iyer, described Ramanujan’s abilities in mathematics as being above an A+ level. At his graduation from Town High, Ramanujan won a scholarship to attend Government College, an excellent institute of higher education in Kumbakonam, considered by some the “Cambridge” of the region. At Government College, Ramanujan continued to perform amazingly well in mathematics, but he now began to do poorly in everything else. There was an unusual reason why this brilliant student, who was far better than everyone else around him in mathematics and whose knowledge exceeded even that of his teachers, stopped showing even minimal interest in any other subject.

  © Springer International Publishing Switzerland 2016

  Ken Ono and Amir D. AczelMy Search for Ramanujan10.1007/978-3-319-25568-2_8

  8. An Addiction

  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

  Imagine a college student who is so consumed by video games that he is failing all his classes. He even forgets to eat, shower, and sleep. If we replaced the video games by equations and formulas, then this would have been Ramanujan.

  During the last few months of his time at Town High School, Ramanujan became obsessed with mathematics. So much so, in fact, that had the obsession taken hold of him earlier, he might have jeopardized his graduation, because his addiction left room for nothing else in his life, no other subject whatsoever. What triggered his strange state of mind was a book published in 1886 titled A Synopsis of Elementary Results in Pure Mathematics, by George Shoobridge Carr. Carr was a professional mathematics tutor in London, and he had decided to put in book form all the formulas, theorems, and results known during his time that would be useful in tutoring English mathematics students.

  A friend had borrowed a library copy of this book for Ramanujan, and it is this book that appears to have unleashed Ramanujan’s true genius, and it has thus played an unlikely role in the history of mathematics. Carr was by no means a great mathematician, and his book—a compendium of results known in his time—was later described by the mathematician Godfrey Harold Hardy (who will play a major role in this story) in a 1937 article on Ramanujan as follows:The book is not in any sense a great one, but Ramanujan has made it famous, and there is no doubt that it influenced him profoundly and that his acquaintance with it marked the real starting point of his career.

  Carr’s book contained 6165 theorems and mathematical formulas, beginning with relatively simple ones, such as , and continuing to increasingly more complicated results. The characteristic trait of this book—unlike others that Ramanujan had read while in school—was its extreme terseness. The Synopsis contained few details of proofs. This had the distinct effect of encouraging Ramanujan to find “proofs” himself, and thus to develop his own mathematical chops. Within months—at which time he was a scholarship student at a prestigious local college—Ramanujan had become hooked on Carr and could focus on nothing else.

  While Carr’s highly condensed book allowed Ramanujan’s genius to emerge because he felt the need to supply the details in the theorems, and thus to explain to himself the reasoning behind the results Carr had presented, it had several negative effects. First, Carr had begun tutoring in the 1860s, and what he was preparing his pupils for was the Tripos examination, which required a great deal of rote learning of basic material that university students needed to know to obtain a bachelor’s degree. None of this was cutting-edge mathematics. Thus Ramanujan’s introduction to “modern mathematics” was not at all modern. It may have given him a wrong impression about the state of the art—like my father in Japan at the end of World War II, he was isolated from current trends—and when, some years later, he would greatly extend the formulas in Carr’s book, thinking perhaps that he was doing highly original work, it would appear to others that he was simply appropriating known facts and passing them off as his own.

  Second, the book’s extreme terseness—i
t was, after all, primarily a collection of mathematical facts that an undergraduate would need to know—may have given Ramanujan the false impression that this was how mathematics was done in the wider world, namely that there was no need of formal argumentation. Thus, when Ramanujan began to create his own mathematics, he, too, would provide no justifications for his results in the form of detailed derivations, as had become custom in mathematical writing since the mid-nineteenth century.

  The third problem with the book was perhaps the most unfortunate of all. It was too absorbing, too fascinating, and it deprived Ramanujan of the chance for a solid education. He discovered the book in late 1903, and he became totally obsessed with it. He spent all his time trying to supply the mathematical details left out by Carr.

  Ramanujan managed to coast during his last few months of high school without slipping too badly in any subject, even though he was spending all his time on his mathematical derivations and extensions of Carr’s material. He received his awards and a scholarship, which was badly needed, since his parents’ income was much too low to afford him a college education.

  Ramanujan entered Government College in 1904. From the outset, his situation was not auspicious. He continued to do nothing but mathematics. Nothing else interested him. He would sit in class, pretending to listen to the lecturer, with Carr’s Synopsis in his lap, his mind deep in thought about an infinite series or an infinite continued fraction. To what did these series or fractions converge? he would ask himself, completely unaware whether the professor was talking about the history of India or a Shakespearean tragedy. There was a curriculum that he was supposed to be following, but Carr’s book, and the flood of mathematical ideas that it inspired, had eclipsed everything else.