Ramanujan, it need hardly be said, flunked physiology. Except for math he did poorly in all his subjects, but in physiology he reached particularly impressive lows, often scoring less than 10 percent on exams. He’d take the three-hour math exam and finish it in thirty minutes. But that got him exactly nowhere. In December 1906, he appeared again for the F.A. examination and failed. The following year, he took it again. And failed again.
Government College, Kumbakonam, 1904 and 1905 … Pachaiyappa’s College, Madras, 1906 and 1907 … In the first decade of the twentieth century, there was no room for Srinivasa Ramanujan in the higher education system of South India. He was gifted, and everyone knew it. But that hardly sufficed to keep him in school or get him a degree.
The System wouldn’t budge.
• • •
Describing the obsession with college degrees among ambitious young Indians around this time, an English writer, Herbert Compton, noted how “the loaves and fishes fall far short of the multitude, and the result is the creation of armies of hungry ‘hopefuls’—the name is a literal translation of the vernacular generic term omedwar used in describing them—who pass their lives in absolute idleness, waiting on the skirts of chance, or gravitate to courses entirely opposed to those which education intended.” Ramanujan, it might have seemed in 1908, was just such an omedwar. Out of school, without a job, he hung around the house in Kumbakonam.
Times were hard. One day back at Pachaiyappa’s, the wind had blown off Ramanujan’s cap as he boarded the electric train for school, and Ramanujan’s Sanskrit teacher, who insisted that boys wear their traditional tufts covered, asked him to step back out to the market and buy one. Ramanujan apologized that he lacked even the few annas it cost. (His classmates, who’d observed his often-threadbare dress, chipped in to buy it for him.)
Ramanujan’s father never made more than about twenty rupees a month; a rupee bought about twenty-five pounds of rice. Agricultural workers in surrounding villages earned four or five annas, or about a quarter rupee, per day; so many families were far worse off than Ramanujan’s. But by the standards of the Brahmin professional community in which Ramanujan moved, it was close to penury.
The family took in boarders; that brought in another ten rupees per month. And Komalatammal sang at the temple, bringing in a few more. Still, Ramanujan occasionally went hungry. Sometimes, an old woman in the neighborhood would invite him in for a midday meal. Another family, that of Ramanujan’s friend S. M. Subramanian, would also take him in, feeding him dosai, the lentil pancakes that are a staple of South Indian cooking. One time in 1908, Ramanujan’s mother stopped by the Subramanian house lamenting that she had no rice. The boy’s mother fed her and sent her younger son, Anantharaman, to find Ramanujan. Anantharaman led him to the house of his aunt, who filled him up on rice and butter.
To bring in money, Ramanujan approached friends of the family; perhaps they had accounts to post, or books to reconcile? Or a son to tutor? One student, for seven rupees a month, was Viswanatha Sastri, son of a Government College philosophy professor. Early each morning, Ramanujan would walk to the boy’s house on Solaiappa Mudali Street, at the other end of town, to coach him in algebra, geometry, and trigonometry. The only trouble was, he couldn’t stick to the course material. He’d teach the standard method today and then, if Viswanatha forgot it, would improvise a wholly new one tomorrow. Soon he’d be lost in areas the boy’s regular teacher never touched.
Sometimes he would fly off onto philosophical tangents. They’d be discussing the height of a wall, perhaps for a trigonometry problem, and Ramanujan would insist that its height was, of course, only relative: who could say how high it seemed to an ant or a buffalo? One time he asked how the world would look when first created, before there was anyone to view it. He took delight, too, in posing sly little problems: If you take a belt, he asked Viswanatha and his father, and cinch it tight around the earth’s twenty-five-thousand-mile-long equator, then let it out just 2π feet—about two yards—how far off the earth’s surface would it stand? Some tiny fraction of an inch? Nope, one foot.
Viswanatha Sastri found Ramanujan inspiring; other students, however, did not. One classmate from high school, N. Govindaraja Iyengar, asked Ramanujan to help him with differential calculus for his B.A. exam. The arrangement lasted all of two weeks. You can think of calculus as a set of powerful mathematical tools; that’s how most students learn it and what most exams require. Or else you can appreciate it for the subtle questions it poses about the nature of the infinitesimally small and the infinitely large. Ramanujan, either unmindful of his students’ practical needs or unwilling to cater to them, stressed the latter. “He would talk only of infinity and infinitesimals,” wrote Govindaraja, who was no slouch intellectually and wound up as chairman of India’s public service commission. “I felt that his tuition [teaching] might not be of real use to me in the examination, and so I gave it up.”
Ramanujan had lost all his scholarships. He had failed in school. Even as a tutor of the subject he loved most, he’d been found wanting.
He had nothing.
And yet, viewed a little differently, he had everything. For now there was nothing to distract him from his notebooks—notebooks, crammed with theorems, that each day, each week, bulged wider.
5. THE NOTEBOOKS
“In proving one formula, he discovered many others, and he began to compile a notebook” to record his results. That’s how Ramanujan’s friend Neville put it many years later, and it remains as concise a distillation as any of how his notebooks came to be. Certainly, it was in working through Carr’s Synopsis, as he tottered through college during the years from 1904 to 1907, that he began keeping them in earliest form.
After Ramanujan’s death, his brother prepared a succession of handwritten accounts of the raw facts, data, and dates of his life. And preserved in their original form as they are, they remind us of a world before computers and word processors made revision easy and routine: we see rude scrawls growing neater, more digested and refined, as they are copied and recopied through successive versions.
Such was the likely genesis of Ramanujan’s notebooks.
The first of the published Notebooks that come down to us today, which Ramanujan may have prepared around the time he left Pachaiyappa’s College in 1907, was written in what someone later called “a peculiar green ink,” its more than two hundred large pages stuffed with formulas on hypergeometric series, continued fractions, singular moduli …
But this “first” notebook, which was later expanded and revised into a second, is much more than mere odd notes. Broken into discrete chapters devoted to particular topics, its theorems numbered consecutively, it suggests Ramanujan looking back on what he has done and prettying it up for formal presentation, perhaps to help him find a job. It is, in other words, edited. It contains few outright errors; mostly, Ramanujan caught them earlier. And most of its contents, arrayed across fifteen or twenty lines per page, are entirely legible; one needn’t squint to make out what they say. No, this is no impromptu record, no pile of sketches or snapshots; rather, it is like a museum retrospective, the viewer being guided through well-marked galleries lined with the artist’s work.
Or so they were intended. At first, Ramanujan proceeded methodically, in neatly organized chapters, writing only on the right-hand side of the page. But ultimately, it seems, his resolve broke down. He began to use the reverse sides of some pages for scratch work, or for results he’d not yet categorized. Mathematical jottings piled up, now in a more impetuous hand, with some of it struck out, and sometimes with script marching up and down the page rather than across it. One can imagine Ramanujan vowing that, yes, this time he is going to keep his notebook pristine … when, working on an idea and finding neither scratch paper nor slate at hand, he abruptly reaches for the notebook with its beckoning blank sheets—the result coming down to us today as flurries of thought transmuted into paper and ink.
In those flurries, we can imagine the very ear
liest notebooks, those predating the published ones, coming into being. Ramanujan had set out to prove the theorems in Carr’s book but soon left his remote mentor behind. Experimenting, he saw new theorems, went where Carr had never—or, in many cases, no one had ever—gone before. At some point, as his mind daily spun off new theorems, he thought to record them. Only over the course of years, and subsequent editions, did those early, haphazard scribblings evolve into the published Notebooks that today sustain a veritable cottage industry of mathematicians devoted to their study.
• • •
“Two monkeys having robbed an orchard of 3 times as many plantains as guavas, are about to begin their feast when they espy the injured owner of the fruits stealthily approaching with a stick. They calculate that it will take him 21/4 minutes to reach them. One monkey who can eat 10 guavas per minute finishes them in 2/3 of the time, and then helps the other to eat the plantains. They just finish in time. If the first monkey eats plantains twice as fast as guavas, how fast can the second monkey eat plantains?”
This charming little problem had appeared some years before Ramanujan’s time in an Indian mathematical textbook. Exotic as it might seem at first, one has but to change the monkeys to foxes, and the guavas to grapes, to recognize one of those exercises, beloved of some educators, supposed to inject life and color into mathematics’ presumably airless tracts. Needless to say, this sort of trifle, however tricky to solve, bears no kinship to the brand of mathematics that filled Ramanujan’s notebooks.
Ramanujan needed no vision of monkeys chomping on guavas to spur his interest. For him, it wasn’t what his equation stood for that mattered, but the equation itself, as pattern and form. And his pleasure lay not in finding in it a numerical answer, but from turning it upside down and inside out, seeing in it new possibilities, playing with it as the poet does words and images, the artist color and line, the philosopher ideas.
Ramanujan’s world was one in which numbers had properties built into them. Chemistry students learn the properties of the various elements, the positions in the periodic table they occupy, the classes to which they belong, and just how their chemical properties arise from their atomic structure. Numbers, too, have properties which place them in distinct classes and categories.
For starters, there are even numbers, like 2, 4, and 6; and odd numbers, like 1, 3, and 5.
There are the integers—whole numbers, like 2, 3, and 17; and nonintegers, like 17 1/4 and 3.778.
Numbers like 4, 9, 16, and 25 are the product of multiplying the integers 2, 3, 4, and 5 by themselves; they are “squares,” whereas 3, 10, and 24, for example, are not.
A 6 differs fundamentally from a 5, in that you can get it by multiplying two other numbers, 2 and 3; whereas a 5 is the product only of itself and 1. Mathematicians call 5 and numbers like it (2, 3, 7, and 11, but not 9) “prime.” Meanwhile, 6 and other numbers built up from primes are termed “composite.”
Then, there are “irrational” numbers, which can’t be expressed as integers or the ratio of integers, like , which is approximately 1.414 … , but which, however many decimal places you take to express it, remains approximate. Numbers like 3, 1/2, and 911/16, on the other hand, are “rational.”
And what about numbers, like the square root of − 1, which seem impossible or absurd? A negative number times a negative number, after all, by mathematical convention is positive; so how can any number multiplied by itself give you a negative number? No ordinary number, of course, can; those so defined are called “imaginary,” and assigned the label i; . Such numbers, it turns out, can be manipulated like any other and find wide use in such fields as aerodynamics and electronics.
That happens often in mathematics; a notion at first glance arbitrary, or trivial, or paradoxical turns out to be mathematically profound, or even of practical value. After an innocent childhood of ordinary numbers like 1, 2, and 7, one’s initial exposure to negative numbers, like − 1 or − 11, can be unsettling. Here, it doesn’t require much arm-twisting to accept the idea: If t represents a temperature rise, but the temperature drops 6 degrees, you certainly couldn’t assign the same t = 6 that you would for an equivalent temperature rise; some other number, − 6, seems demanded. Somewhat analogously, imaginary numbers—as well as many other seemingly arbitrary or downright bizarre mathematical concepts—turn out to make solid sense.
Ramanujan’s notebooks ranged over vast terrain. But this terrain was virtually all “pure” mathematics. Whatever use to which it might one day be put, Ramanujan gave no thought to its practical applications. He might have laughed out loud over the monkey and the guava problem, but he thought not at all, it is safe to say, about raising the yield of South Indian rice. Or improving the water system. Or even making an impact on theoretical physics; that, too, was “applied.”
Rather, he did it just to do it. Ramanujan was an artist. And numbers—and the mathematical language expressing their relationships—were his medium.
• • •
Ramanujan’s notebooks formed a distinctly idiosyncratic record. In them even widely standardized terms sometimes acquired new meaning. Thus, an “example”—normally, as in everyday usage, an illustration of a general principle—was for Ramanujan often a wholly new theorem. A “corollary”—a theorem flowing naturally from another theorem and so requiring no separate proof—was for him sometimes a generalization, which did require its own proof. As for his mathematical notation, it sometimes bore scant resemblance to anyone else’s.
In mathematics, the assignment of x’s and y’s need conform to no particular rule; while an equation may reveal profound mathematical truths, just how it is couched—the letters and symbols assigned to its various entities, for example—is quite arbitrary. Still, in a mature field, one or very few notational systems normally take hold. A mathematician laying open a new field picks the Greek letter π, say, to stand for a certain variable; soon, through historical accident or force of habit, it’s become enshrined in the mathematical literature.
To pick an example familiar from high school algebra, the two roots of a quadratic equation (which describes the geometric figure known as a parabola) are given by
where a, b, and c are constants, x a variable. So entrenched is this form of the equation that it’s hard to imagine anything else. And yet, there’s no reason why the constants couldn’t be p, q, and r. Or m1, m2, and m3. And the quantity within the square root sign could be seen as the difference of two squares and broken up into two terms. And the square root itself could be expressed as a fractional power. And each of the two roots could get its own equation. The result would be:
The mathematical gymnastics don’t matter here, only that this is identical to the more canonical version—and yet, on its face, unrecognizable. Someone coming up with the result on his own, and expressing it in this alien notation because he did not know the established one, would face extra roadblocks to being understood and might be written off as unorthodox or strange.
Which is just how Ramanujan’s notebooks would tend to be regarded by mathematicians, both of his own day and of our own. In the area of elliptic functions, where everybody used k for the modulus, an important constant, Ramanujan used the Greek letter or . Sometimes n was, in his notebooks, a continuous variable, which for professional mathematicians it never was. As for the quantity π(x), by which everyone else meant the number of prime numbers among the first x integers, it never appeared at all.
There was nothing “wrong” in what Ramanujan did; it was just weird. Ramanujan was not in contact with other mathematicians. He hadn’t read last month’s Proceedings of the London Mathematical Society. He was not a member of the mathematical community. So that today, scholars citing his work must invariably say, “In Ramanujan’s notation,” or “Expressing Ramanujan’s idea in standard notation,” or use similar such language.
He was like a species that had branched off from the main evolutionary line and, like an Australian echidna or Galápagos tortoise, had come to occ
upy a biological niche all his own.
• • •
If offbeat to other mathematicians, the parade of symbols in Ramanujan’s notebooks amounted to a foreign language to most lay people. And yet, as arcane a language as it was, the concepts it expressed often turned out to be surprisingly straightforward.
Take, for example, the f(x)’s and other examples of “functional notation” that litter Ramanujan’s notebooks. Here, f(x), read “ef of ex,” doesn’t mean f times x, but rather some unspecified function of x; something, in other words, depends on x. Without defining the function we don’t know how it depends. Later, we may specify, for example, that f(x) = 3x + 1. Then we do know how it depends on x; the algebraic formula tells us, describing its mathematical behavior: In this case, when x = 1, f(x) = 4; when x = 2, f(x) = 7; and so on. But often, the mathematician doesn’t want to get down to specifics. Functional notation lets him work in the more abstract realms he prefers, free from slavery to particular cases.
In functional notation, φ (a,b), read “phi of ay and bee,” just means some unspecified function, φ, that depends on the variables a and b. And f(3) just means f(x) evaluated when x = 3. And g(− x) just means g(x) with − 1 plugged into the equation whenever x = 1, − 2 when x = 2, and so on. With such broad brush strokes, sometimes never stooping to particular functions at all, the mathematician fashions his world.
Or sometimes he does make f(x) a specific function, then goes on to discover its odd or revealing properties. On page 75 of the first notebook, for example, Ramanujan writes
φ(x) + φ(−x) = 1/2 φ (−x2)
for a particular function defined previously. Evaluate φ(x) at, say, x = 1/2, then at x = − 1/2. Add up the two results. And that will equal half of what you get if you evaluate the function at x = − 1/4. But Ramanujan’s equation says it more generally, reveals the function’s mathematical idiosyncrasies. And says it without so many words.
The Man Who Knew Infinity Page 8
