The Man Who Knew Infinity

by Robert Kanigel

  Through arrangements made by the university, Ramanujan was staying in a house located on East Agraharam Street, near the town registrar’s office; agraharam meant the Brahmin quarter. Crossing the T of Agraharam Street was a street that dropped down to the Cauvery at a spot where men and women came to wash clothes and bathe. And there, on August 11, 1919, at the time of the Sravanam ceremony that marked the annual changing of the sacred thread, Ramanujan openly rebelled against his mother.

  Trouble had been brewing for weeks between the two of them, from even before they’d arrived in town; Ramanujan had wanted to travel first class, but his mother had insisted on second or third class. Now, Ramanujan was heading down to the river to bathe, as part of the Sravanam rites. Janaki wanted to go with him. Ramanujan said yes. Komalatammal said no.

  And Ramanujan insisted, yes.

  It was a turning point. All the hostility pent up over the past few years, the diverted letters, the endless quarrels—and now this. He spoke respectfully, but firmly: Janaki was going with him.

  After that, Janaki began to occupy a larger place in her husband’s life. He grew freer in her company. He told her, more than once, “If only you had come with me to England perhaps I would not have fallen ill.” On Friday afternoons, he would sometimes watch her oil and wash her hair, shaking it out and letting it dry in the sun. The sight gave him great pleasure, sometimes enough to draw him away from his work. In coming months, it would be Janaki who, increasingly, would care for him during the day, give him his medication, nurse him at night. When, at one point, his mother urged him to send Janaki packing, back to her parents, Ramanujan refused.

  For two months Ramanujan stayed in Kodumudi. Every Sunday the doctor visited—a C. F. Fearnside, Divisional Medical Officer. And it may only have been now, in questioning him, that Komalatammal grasped the full extent of her son’s condition. At one point Fearnside suggested that Ramanujan go to Coimbatore. At another, Thanjavur, near Kumbakonam, was the recommendation. (Ramanujan would not hear of it. “They want me to go to Tan-savu-ur,” he said, punning on the word’s Tamil components, “the place-of-my-death.”) But with the worst of summer over, the pressure was strong to go somewhere less out-of-the-way.

  Komalatammal decreed that it be Kumbakonam.

  In late August, perhaps grateful to get away from a domestic situation that had turned against her, she went on ahead to look for a house, the family home on Sarangapani Sannidhi Street apparently being deemed inappropriate for her son’s care. On September 3, Ramanujan and the rest of the family left Kodumudi and, around dusk of the following day, arrived in Kumbakonam.

  The surroundings of his childhood and adolescence may have mellowed Ramanujan a little because stories from this period bear a softer quality, even a nostalgic tinge. One of Ramanujan’s former classmates told how “he insisted on seeing my aunt, who is an old widow on the wrong side of 70, whom he knew while at school some 20 years ago.” Old friends dropped by. Balakrishna Iyer, one of those Ramanujan had approached for a job around 1910, sent him dried brinjal slices. Radhakrishna Iyer, the Pachaiyappa’s College classmate who had nursed him while he was sick in 1909, recalled Ramanujan as just “a bundle of bones,” lying on his cot. “Ramanujan had powerful and penetrating eyes. For a moment his eyes flashed a look of recognition,” and he mumbled Radhakrishna’s name.

  Ramanujan had a new doctor now, P. S. Chandrasekar, brought down from Madras especially to care for him. Hardy, through the Royal Society, had approached the authorities in Madras. The surgeon general had turned to Chandrasekar, fifty, a professor of hygiene and physiology at Madras Medical College and a prominent tuberculosis specialist. At one point, Chandrasekar accompanied one of Ramanujan’s oldest friends, Sarangapani (the boy who had once bested him in an arithmetic exam) to the house on Bhaktapuri Street where Ramanujan lay sick. For more than an hour Chandrasekar examined him. It was tuberculosis, he was sure, notions of blood poisoning notwithstanding. “I have a friend who loves me more than all of you, who does not want to leave me at all,” Ramanujan told Sarangapani later. “It’s this tuberculosis fever.”

  When Ramanujan boarded the Nagoya to return to India early that spring, hopes for recovery, at least in Hardy’s estimation, had been high. But the kind of sanatoriums treating him for the past two years had one essential failing: they didn’t cure tuberculosis. Of those discharged from English sanatoriums, one study found, between one-and two-thirds were dead within five years, most of them within the first two. In Ramanujan, Chandrasekar now concluded, the disease had reached an advanced stage. His fate lay more with God than in anything medicine could do.

  Since coming down sick two and a half years before, Ramanujan had been through the wringer. In England, he’d seen many doctors, heard countless conflicting diagnoses, been shuffled around to numerous hospitals and nursing homes. Now in India, it was the same thing. Two places in Madras. Kodumudi. Perhaps briefly in Coimbatore. Kumbakonam. All in nine months. More doctors. More examinations. More diagnoses. Through it all, his condition grew gradually worse. Now, in Kumbakonam, the downward slide accelerated.

  Just before Christmas of 1919, while writing Hardy with details of Ramanujan’s accounts, Dewsbury added: “He is still in very bad health and difficult to deal with, living up country with his family. Mr. Ramachandra Rao is doing what he can for him, but Mr. Ramanujan himself will not consent to live in a suitable environment under proper treatment. It is a great pity.” By one account, he refused to be further treated, by another he had simply given up the will to live. It was like pulling teeth even to get him to go to Madras, where Chandrasekar wanted him. It was the coolest, most pleasant time of the year; the heat that had originally driven him to Kodumudi had given way to daytime temperatures in the eighties. Finally, Ramanujan relented.

  Sometime probably a little after the first of the year, Ramanujan, his mother, wife, and brother-in-law, Srinivasa, set off for Madras—but, for the moment, got only as far as the railway station. They had missed the train, the next one not due for another six or seven hours. Rather than subject Ramanujan to another bumpy, two-mile ride to Bhaktapuri Street, only to turn around a few hours later, they camped out at the station on the edge of town. Periodically, Janaki and Komalatammal returned to the house for food or other provisions.

  The train ultimately came, of course, and they reached Madras. On January fifteenth, when Dewsbury wrote Hardy again, he gave Ramanujan’s address as “Kudsia,” Harrington Road, Chetput, Madras.

  3. THE FINAL PROBLEM

  Ramanujan was living in one of a group of substantial stucco houses built for the British around the turn of the century in a western suburb of the city. Given names like Sydenham, Ravenscroft, and Lismoyle, and sometimes supplied with tennis courts, they were popular with middle-rank Indian Civil Service officers and company officials. Several dozen of them had gone up along a lane perpendicular to Harrington Road, now called Fifteenth Avenue, but during Ramanujan’s time unnamed. “Kudsia” was one of them, though apparently Ramanujan didn’t stay there long. For a time, he was installed in a considerably larger one, “Crynant,” in a culde-sac at the end of the dusty lane. Ultimately, he would be moved to another, known as “Gometra.”

  It was from one of these houses that on January 12, 1920, Ramanujan wrote Hardy for the first time in almost a year.

  I am extremely sorry for not writing you a single letter up to now… . I discovered very interesting functions recently which I call “Mock” theta functions. Unlike the “False” theta functions (studied partially by Prof. Rogers in his interesting paper) they enter into mathematics as beautifully as the ordinary theta functions. I am sending you with this letter some examples.

  In how it would excite the active interest of mathematicians up until the present, it was a letter much like the one Ramanujan had written Hardy seven years earlier almost to the day. By one estimate, it represented “one of the most original pieces of mathematics, and in some ways the very best, which Ramanujan did.”


  As with so many other subjects that interested Ramanujan, theta functions could be represented as infinite series. Strict rules governed the formation of these series, and so long as you observed them, they always had particularly intriguing properties. For example, they were “quasi doubly periodic” and they were “entire functions.” Over the years since Jacobi had first studied them, theta functions had come to exert a profound impact on fields ranging from mathematical physics to number theory. Books devoted to them typically wound up bridging vast terrain, yielding connections to fields to which at first they might seem wholly unrelated.

  But theta functions were fragile. The properties that made them so intriguing to mathematicians vanished if, in trying to construct them, you departed even slightly from their canonical form—for example, by merely changing the signs of some of the terms; at a glance, they might seem indistinguishable from theta functions themselves, yet they lost most of their interesting properties. True, certain “false theta functions” explored by Ramanujan’s intellectual cousin, L. J. Rogers, in a paper a few years before, held some interest; later study would apply them, for example, to partitions. But compared to the real, or “classical,” theta functions that mathematicians found so rich with meaning, they were pale substitutes.

  Now, in his letter to Hardy, Ramanujan was telling how a new class of functions could be constructed that were not theta functions but bore crucial similarities to them and were just as interesting. They were not “false” theta functions, like those Rogers had investigated, but what Ramanujan termed “mock” theta functions.

  Classical theta functions could be put into a variant “Eulerian” form, in which terms like (1 − q) or (1 − q3) appeared in the denominator. That meant that when q = 1, say, or q3 = 1, the functions were mathematically undefined and had to be explored in alternative ways (in a way loosely analogous to how Ramanujan and Hardy had explored the partition function through their “circle method”). Offering two theta functions as examples, Ramanujan noted that at such “singularities”—at the points demanding special study—“we know how beautifully the asymptotic form of the function can be expressed in a very neat and closed exponential form.”

  All this was spadework, with which Hardy was familiar. But now Ramanujan ranged beyond it:

  Now a very interesting question arises. Is the converse of the statements concerning the [two theta functions] true? That is to say: Suppose there is a function in the Eulerian form and suppose that all or an infinity of points are exponential singularities, and also suppose …

  and on he went like this, fashioning his function before Hardy’s eyes. When it had satisfied a host of such special conditions it was, he said, a “mock theta function.” Then he went on to offer a whole slew of them—four third-order mock theta functions, ten fifth-order functions, and so on.

  “The first three pages in which Ramanujan explained what he meant by a ‘mock theta function’ are very obscure,” G. N. Watson, one of the first of those to study them, would allow. But hazy and poorly expressed as they were, they held within them profound mathematical truths that became the subject of Watson’s presidential address to the London Mathematical Society sixteen years later and would keep other mathematicians busy for years. Wrote Watson:

  Ramanujan’s discovery of the mock-theta functions makes it obvious that his skill and ingenuity did not desert him at the oncoming of his untimely end. As much as any of his earlier work, the mock-theta functions are an achievement sufficient to cause his name to be held in lasting remembrance. To his students such discoveries will be a source of delight and wonder until the time shall come when we too shall make our journey to that Garden of Proserpine where

  Pale, beyond porch and portal

  Crowned with calm leaves, she stands

  Who gathers all things mortal

  With cold, immortal hands.

  In 1893, Arthur Conan Doyle, creator of Sherlock Holmes, wrote a story in which the famous detective, grappling with his archfoe Professor Moriarty at Reichenbach Falls in Austria, plunges to his death on the rocks below—or so Holmes’s friend, Dr. Watson, is led to believe. Now, four decades later, discussing Ramanujan’s surge of mathematical creativity in the months before his death, this other Watson, the mathematician, resurrected the title Conan Doyle had used for Sherlock Holmes’s last case. Mock theta functions, he said, were Ramanujan’s “Final Problem.”

  All that year in India Ramanujan worked on mock theta functions, “q-series,” and related areas, filling page after page with theorems and computational fragments and infinite series going off in every direction, by one reckoning about 650 formulas in all. When the American mathematician George Andrews began poring through them half a century later, he was stunned by their richness, the surprises they offered, was left mystified over how anyone could think them up. “It is very difficult even for somebody trained in mathematics, but not an expert, to tell them apart,” he would say. Among one group of five superficially similar formulas, “the first one took me fifteen minutes to prove, the second an hour. The fourth one followed from the second. The third and fifth took me three months.”

  Ramanujan’s work during the year before he died could be seen to support an old nostrum of the tuberculosis literature—that the tuberculous patient, as he succumbed, was driven to an ever-higher creative pitch; that approaching death inspired a final flurry of creativity impossible during normal times. The idea has been soundly trounced by modern scholars quick to cite artists and other creative workers whose greatest work long preceded their illness and whose deathbed productivity was nil. Still, during Ramanujan’s time, the idea had a following. And at least one account of Ramanujan’s last days, by P. V. Seshu Iyer, bore its flavor: “There are no papers and researches of his more valued nor more intuitive,” he wrote, “than those which he thought out during these fateful days. His physical body was failing no doubt, but his intellectual vision grew proportionately keener and brighter.”

  But George Andrews, for one, could explain Ramanujan’s remarkable productivity in more prosaic, if doubtless sounder, terms. “He had the full power of his collaboration with Hardy behind him and the sort of wild eccentric genius of his youth behind him,” too. Ramanujan, in other words, should have been at the height of his powers. “And this notebook,” said he of the product of that final year, “bears it out.”

  • • •

  Sometime during late winter or early spring of 1920, while staying in “Crynant,” Ramanujan complained to his mother that the “cry” of that name seemed inauspicious. Komalatammal went to Namberumal Chetty, owner of the bungalows (as well as of a small railroad) and a friend of Sir Francis Spring. Omitting Ramanujan’s real reasons, Komalatammal told him her son needed a quieter place. Namberumal obliged, and Ramanujan was moved a little down the road to “Gometra,” which means something like “Friend of Cows,” a reference to Krishna.

  It was a house full of doors. There were banks of doors everywhere, doors leading back outside to an arched portico, doors to an interior court where the cooking was done in a separate outbuilding, literally dozens of doors ranging over the house. When they were all open, the inside of the house, with its floor of large granite slabs, was like a cool, shaded glen open to any stray passing breeze. You’d enter through what from the street seemed a side entrance but which, leading you under one of several arches, actually brought you into the center of the house. There stood a stairway, its banisters fashioned from Burma teak, that led up to the second floor. To your right, toward the front of the house, was a large living room. Ramanujan may briefly have stayed there, but in time he was shifted to a small corner room on the other side of the stairs at the back of the house.

  Enhancing its sense of openness was the bungalow’s scant furniture. Ramanujan’s bed was a mattress and a pillow laid out upon the bare granite floor. He scarcely budged from it. Lying there, he could look up and see a massive wood beam, from which, like a spine, the second floor was supported, spanning th
e length of the room. It was quiet back here away from the street. Visitors moved silently so as not to disturb him. Tirunarayanan, who fondly recalled his older brother horsing around with him, telling him stories and carrying him on his shoulders, now found access to him jealously guarded, with no one allowed to bother him.

  Still, Ramanujan could scarcely have gone unaware of the tensions that gripped the household. The whole family was in a state of jangled nerves. And that he was sick, miserable, angry, and prone to sudden outbursts did nothing to ease matters. Again and again, Janaki and Komalatammal clashed, absorbing energies that might better have gone toward Ramanujan’s care. Now, moreover, the contest was not so unequal as before. Janaki was twenty, enjoyed a new and stronger position in the household, and had her brother Srinivasa by her side, too.

  Around March, Komalatammal appeared at the door of G. V. Narayanaswamy Iyer, a former student of mathematician P. V. Seshu Iyer, Ramanujan’s long-time friend and champion, whose note of introduction she now bore. Narayanaswamy Iyer was a high school teacher in Triplicane, but it was not for his pedagogic abilities that Komalatammal wanted to see him. He was also a noted astrologer.

  Narayanaswamy Iyer asked for a copy of the horoscope she wanted read. Komalatammal dictated it from memory. This, he said after studying it for a while, was the chart either of a man of worldwide reputation apt to die at the height of his fame, or one who, if he did live long, would remain obscure. “Who is this gentleman? What is his name?” he asked.

  It was, of course, Ramanujan, and Komalatammal, in tears, replied that she had drawn the same inference from his horoscope.

  Narayanaswamy backpedaled. Ramanujan! “I am sorry I was so hasty,” he said. “Please do not carry what I said to any of his relatives.”

  “I am the mother of Ramanujan,” she replied.

  Groping for a way out of the uncomfortable scene, Narayanaswamy suggested that maybe his wife’s horoscope might mitigate the bad news. “Can you bring her horoscope next time?” he asked. A separate visit was hardly necessary: Komalatammal rattled it off for him.