The Man Who Knew Infinity

by Robert Kanigel

  It was into this nascent new world that Ramanujan “came out,” as it were, as a mathematician in 1911. He had met Ramaswami Iyer, the society’s founder, the previous year when, in search of a job, he had traveled to Tirukoilur. Now Ramanujan’s work was appearing in volume 3 of Ramaswami Iyer’s new Journal—which, like most mathematics publications, opened its pages to provocative or entertaining problems from its readers.

  One of two problems Ramanujan posed, as question 289, simply asked the reader to evaluate

  Seemingly straightforward arithmetic, with not so much as an x or y to complicate it? Well, three issues of the Journal came and went—six months—with no solution offered; in the end, Ramanujan supplied it himself. The problem was those three little dots, indicating that the nesting of square roots, and the sequence of numbers begun, was to continue ad infinitum.

  Ramanujan had generated the problem years before in the form of an example illustrating a more general theorem. The fourth equation down in chapter 12, on page 105 of his first notebook, read:

  Break any number into three components, x, n, and a, the equation said, and you could represent it in the form of those endlessly nesting square roots. For example, 3 could be imagined as (x + n + a) in which x = 2, n = 1, and a = 0. Plug those values into the equation and you wind up with just what Ramanujan had, in question 289, asked to have evaluated. The answer, in other words, was just plain 3. Of course, how you’d figure it out without Ramanujan’s equation was scarcely obvious.

  In its small way, Ramanujan’s deceptively difficult problem rose up from mathematical terrain that had fascinated him for as long as he had worked in mathematics. A superficially similar problem might have asked the reader to take the nested square roots only so far, out to, say, 10, or 100, or 1000. That way, you could plug in the numbers, run through the computation, and be finished. But this wasn’t the problem Ramanujan posed. He asked, what happened if you were never finished? What if the number of nested square roots was infinite?

  This was, in a sense, a contradiction in terms: how could any number “equal” infinity? Infinity was no place you could reach, no quantity you could plug into an equation; there was no “last number.” So to understand how a mathematical expression behaved “at” infinity was to explore an elusive and mysterious terrain out beyond all seeing.

  And no one explored this terrain more ardently, or knew it more intimately, than Ramanujan.

  Many kinds of mathematical processes can be ordered to proceed ad infinitum. Ramanujan’s problem in the Mathematical Society Journal was built around “nested radicals”—square roots of square roots of square roots of … , an area little studied then or now. But Ramanujan also studied continued fractions—fractions of fractions of fractions of … Most of all, he explored infinite series, which appeared on virtually every page of his notebooks and would fairly litter his first substantial paper in the Journal later in 1911. “Infinite series,” one mathematician has written, “were Ramanujan’s first love.”

  The telltale three dots heralding their presence showed up early in his notebooks, though more often he simply wrote “&c,” which meant the same thing: numbers or algebraic terms, following some particular pattern, were to be added to one another forever. Thus,

  1 + 2 + 3 + …

  clearly suggests that the next term is to be 4, then 5, and so on.

  Of course, this is not very interesting because, here, an infinite number of terms just adds up to infinity. What makes infinite series so intriguing, so valuable, and the object of so much study, is when they don’t build up without bound, when they add up to something finite. As mathematicians put it, the series “converges” to a particular value. For example,

  1 + 1/2 + 1/4 + 1/8 + …

  Here, the next term is 1/16, the next 1/32, and so on. And the curious thing, known even to the Greeks, is that even though you add terms forever, each term diminishes so rapidly from the preceding one that the sum, even after an infinite number of terms, is a quite manageable 2—the value to which the series is said to converge. The more terms you add, the closer you get to 2.

  But just because each successive term in a series is smaller than the one before doesn’t mean it converges. For example,

  1 + 1/2 + 1/3 + 1/4 + …

  is superficially similar to the earlier convergent series, but doesn’t converge; go as far out into the series as you like, but so soon as you think it’s adding up to something, more terms always take you beyond it. For example, is the sum of this series perhaps 2, as it was for the previous one? No—four terms are enough to exceed it. Maybe 3? Here it takes a little longer, but already by the eleventh term, you’ve passed it. Perhaps 10? Twelve thousand, three hundred ninety terms add up to more than 10. It turns out—and can be proven—that whatever number you pick, the same holds: The sum of the series is infinite; it doesn’t converge.

  The series of interest to mathematicians, then, are those that do converge, or that converge under certain circumstances. And it’s what they converge to that is often the wonderful thing about them.

  Take, for example, the trigonometric functions that many remember from high school, which teachers practically always first introduce as the ratios of the legs of a right triangle. These functions—named sine, cosine, tangent, and so on—are all computed by taking the length of one or another side of a right triangle and dividing it by the length of another. The backs of trigonometry books are stuffed with long lists of angles with the corresponding values of their trig functions. Give me an angle, the tables say, and I’ll give you the value, for example, of the sine of that angle. The angle is 30 degrees? Its sine is .5000. And so on. Such tables in hand, navigators cross oceans, engineers design machines.

  And yet these same trigonometric functions, historically rooted in right triangles and ratios, can be evaluated in a way seemingly unrelated—as the sums of infinite series. If, say, the angle θ (theta, the Greek letter) is expressed not in degrees but in another measure of angularity that mathematicians find more convenient (radians), then:

  (Here, 5!, read “factorial five,” just means 5 × 4 × 3 × 2 × 1 = 120.)

  Want the sine of 30 degrees? Just plug into the equation its radian equivalent (π/6, or about .5236), and add up as many terms as you want to get a value as accurate as you want. Here, even three terms are enough to get you to .500002—quite close to the correct .500000; the series converges rapidly.

  Thus, this alternating infinite series—adding a bit here, subtracting something a little smaller there, and so on through an infinite number of terms—inexplicably equals just what you get from dividing one leg of a triangle by another.

  Just this sort of seemingly unexpected connection shows up all the time with infinite series, which is what has made them so attractive to mathematicians, Ramanujan most particularly. Bernoulli numbers, the subject of his first published paper, were defined in terms of infinite series. And every page of his paper was riddled with more of them.

  Jacob Bernoulli was among the first in a line of eminent seventeenth-and eighteenth-century mathematicians derived from a merchant family that had fled anti-Protestant massacres in Antwerp and settled in Switzerland. He helped extend calculus, the powerful set of mathematical tools for dealing with continuously varying quantities, beyond the point that Germany’s Gottfried von Leibniz, along with England’s Sir Isaac Newton, had taken it two decades before. Along the way, he derived the numbers that have since borne his name.

  Bernoulli numbers are intimately tied to the quantity e which, like π, is a number whose special properties make it ubiquitous in mathematics. It is defined as

  Now, when a particular algebraic expression involving e is expressed as an infinite series, the coefficients of each term turn out to have special significance. (Coefficients are just the ordinary numbers by which the algebraic parts are multiplied; in the equation 3x + 1/2x2 = 12, 3 and 1/2 are the coefficients.) These coefficients were the Bernoulli numbers, which first appeared in his book Ars Con
jectandi, published after his death in 1713. Notational inconsistencies confuse matters some, but in one system the first few Bernoulli numbers, generically Bn, are B1 = − 1/2, B2 = 1/6, B4 = − 1/30, B6 = 1/42. (The odd-numbered ones, except for the first, are all zero.)

  Ramanujan had stumbled on Bernoulli numbers for the first time about eight years before, though probably without having ever heard of them as such. The second volume of Carr’s Synopsis contained references to them in various guises, but Ramanujan may not have seen it until 1904, when he was at Government College—a year after he apparently began working with them. In any case, he’d worked with them ever since, using them repeatedly, through the Euler-Maclaurin summation formula, to approximate the values of mathematical entities known as “definite integrals” (unrelated to “integers”) in calculus. Pages 30 and 31 of the first notebook cited them. So did much of chapter 5 of his second.

  Now Ramanujan was making them the subject of his first formal contribution to the mathematical literature. Children take the Salk or the Sabin vaccine to protect them against polio. Supersonic aircraft exceed Mach 1, the speed of sound in the measurement system named for Austrian physicist Ernst Mach. In science and medicine, immortality is having something—a treatment, a unit of measurement, a theory—named after you. So, too, in mathematics. Bernoulli numbers bear his name because they appear again and again in a wide variety of mathematical applications. They weren’t just flukes of mathematical nature, meaningless chains of digits; there were relationships among them, and Ramanujan had discovered—or, in some cases, rediscovered—what some of them were.

  “Some Properties of Bernoulli’s Numbers,” he called his paper, and it was an apt title. The physical properties of a metal, like its melting point or specific gravity, appear in any chemical handbook; Ramanujan was discovering mathematical properties of these numbers. Bernoulli numbers were expressed as fractions; for example, . Well, Ramanujan found that the denominators (the bottom parts) of those fractions were always divisible by 6. He found alternative ways of calculating Bn based on earlier Bernoulli numbers. The sixth of eighteen numbered sections began:

  6. It will be observed that if n is even but not equal to zero,

  (i) Bn is a fraction and the numerator of in its lowest terms is a prime number,

  (ii) the denominator of Bn contains each of the factors 2 and 3 once and only once,

  (iii) is an integer and consequently 2(2n − 1)Bn is an odd integer.

  On and on Ramanujan’s paper went like that, filling seventeen pages of the Journal. By one reckoning, it stated eight theorems, offering proofs, of a sort, for three of them; two were stated as corollaries of two other theorems, three more as mere conjectures.

  Ramanujan’s manuscript had problems when it first reached the editor’s desk. “Mr. Ramanujan’s methods were so terse and novel and his presentation so lacking in clearness and precision,” it would later be observed, “that the ordinary [mathematical] reader, unaccustomed to such intellectual gymnastics, could hardly follow him.” In other words, his paper was a mess. And this was written by a champion of Ramanujan’s work. M. T. Narayana Iyengar, a math professor at Central College, Bangalore and the Journal’s editor during the early years, confessed later “that the editor’s work in connection with Ramanujan’s contributions was by no means light,” and that the manuscript went back and forth between him and its author three times.

  In this first paper, as all through his work, Ramanujan found connections between things that seemed unconnected. Other mathematicians would later prove most of them true; Ramanujan, though, either couldn’t be bothered, or didn’t see the need to. What proofs he did offer were sketchy or incomplete.

  A testament to the influence on him of George Shoobridge Carr? That’s what most scholars later concluded. Carr, writing a synopsis of results rather than making original contributions, had given proofs, where he did so at all, only in bare outline. Now Ramanujan, who was making original contributions, clung to the pattern. He had asserted, for example, that the numerator of the nth Bernoulli number divided by n was always a prime number. Proof? Not a shred. Another mathematician later observed, “He takes the numerical evidence as sufficient, and there is no trace of any suggestion that there is need of other proof.” Whether Ramanujan cared about proof is debatable; that normally he didn’t furnish it, sometimes offering the most provocative results without a scintilla of evidence to support them, is not.

  In this particular case, as it happens, Ramanujan was wrong. For example, the numerator of B20 / 20 = 174611, which is not a prime number at all, as he claimed, but equal to 283 × 617.

  This, though, was the exception to prove the rule; much more often, Ramanujan’s trust in himself was wholly justified. In his paper on Bernoulli numbers, in the notebooks, in his mathematical correspondence, in his other published papers—he was, with astounding consistency, right.

  5. THE PORT TRUST

  Appearing in the Journal of the Indian Mathematical Society, Ramanujan was on the world’s mathematical map at last, if tucked into an obscure corner of it. He was starting to be noticed.

  Early the following year, K. S. Srinivasan, a student at Madras Christian College who’d known Ramanujan back in Kumbakonam, dropped by to see him at Summer House.

  “Ramanju,” he said, “they call you a genius.”

  Hardly a genius, replied Ramanujan, “Look at my elbow. That will tell you the story.” It was rough, dirty, and black. Working from his large slate, he found the quick flip between writing hand and erasing elbow a lot faster, when he was caught up in the throes of his work, than reaching for a rag. “My elbow is making a genius of me,” he said.

  Why, Srinivasan asked, didn’t he just use paper? Can’t afford it, replied Ramanujan. He was getting money from Ramachandra Rao. But that only went so far. Paper? He’d need four reams of it a month.

  Another friend from the Summer House days, N. Ramaswami Iyer [no relationship to the “Professor”] also recalled Ramanujan’s “huge appetite” for paper. Ramaswami pictured him lying on a mat, his shirt torn, “his long hair carelessly bound up with a piece of thin string,” working feverishly, notebooks and loose sheets of plain white paper piled up around him. A friend from Pachaiyappa’s who met him in Madras a little later, T. Srinivasacharya, recalled that, for want of paper, Ramanujan would sometimes write in red ink on paper already written upon.

  It was during this period that, apparently worried something might happen to Ramanujan’s notebook, Ramachandra Rao prevailed on him to copy it over. Ramanujan did so, though not without revising and expanding it as he did, incorporating the notes appended to it into appropriate sections of the new one—which comes down to us today as the “second” Notebook.

  Half a century after Ramanujan was dead, in one of the many memorial books honoring him, one sponsor would be a manufacturer of writing and printing papers in Erode, Ramanujan’s place of birth. “Paper, The Great Immortalizer,” its one-page ad was headed. “Good Paper,” it went on, “has helped preserve and propagate the great thoughts of Man.” It would be a fitting tribute.

  • • •

  For about a year, Ramanujan lived on Ramachandra Rao’s generosity. He was mathematically productive, peppering the Mathematical Society Journal with one interesting new problem after another, and completing a second paper. But he was, after all, unemployed, and this grew to bother him. Not long before, through one of his patrons, Ramanujan got a temporary job in the Madras Accountant General’s Office, making twenty rupees per month, but held it only a few weeks. Now, early in 1912, Ramachandra Rao had turned to others among his influential friends, and Ramanujan was applying for a new job:

  Sir,

  I understand there is a clerkship vacant in your office, and I beg to apply for the same. I have passed the Matriculation Examination and studied up to the F.A. but was prevented from pursuing my studies further owing to several untoward circumstances. I have, however, been devoting all my time to Mathematics and developing
the subject. I can say I am quite confident I can do justice to my work if I am appointed to the post. I therefore beg to request that you will be good enough to confer the appointment on me.

  I beg to remain,

  Sir,

  Your most obedient servant,

  S. Ramanujan

  Ramanujan’s letter was written in a neat, schoolboy hand unremarkable in every way save for t’s whose horizontal arms rarely intersected the vertical stems they were meant to cross, but floated off instead to their right. Appended to it was a hand-copied version of a recommendation by a mathematics professor at Presidency College, E. W. Middlemast, who described him as “a young man of quite exceptional capacity in Mathematics.” In fact, recommendation and letter were probably both a matter of form, the job for which he applied doubtless his all along, thanks to Ramachandra Rao.

  The letter was dated 9th February 1912. It listed Ramanujan’s return address as 7, Summer House, Triplicane. It was addressed to The Chief Accountant, Madras Port Trust.

  • • •

  From its beginnings, Madras had been a trading settlement and port, though nature had equipped it poorly for the job. An uncommonly rough surf crashed relentlessly. There were insidiously tricky ocean currents and peculiar sand buildups. It had no natural harbor at which cargo vessels might unload. Instead, ships would anchor a quarter mile offshore, and masula boats—flat-bottomed craft about twenty-five feet long made from thin planks stitched together with coconut fiber—piloted by daring men expert in reading the surf, would row out to the ships, load cargo a few tons at a time, and return with it to the beach. Ninety percent of all losses suffered by ships calling at Madras came in that treacherous last quarter mile.

  Yet somehow they’d made a port out of it. In 1796, a lighthouse had been built; it burned coconut oil in lamps visible from seventeen miles away. In 1861, a pier extending eleven hundred feet into the Bay of Bengal was completed, and 1876 saw construction begin on a rectangular artificial harbor, twelve hundred yards on a side, built up from twenty-seven-ton concrete blocks. The new harbor improved matters little. Unloading losses remained high, and the entrance rapidly began silting up, the high-water line on the south side of the harbor advancing seventy feet a year into the bay.