“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 earliest
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 2 1/4 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.
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.”
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.”
