The time was set for rewriting
history; in 1916, Einstein’s General Theory of Relativity, and in 1917,
Ramanujan and Hardy’s Analytical Number Theory came to light. There is a tingle
in the blood even now when we think of those days. Einstein has taken the world
by storm and has initiated a novel way of looking at life from the smallest
particles to that of the largest cosmic bodies. To match with that, Ramanujan’s
Number Theory and magical equations could define a fresh perspective to understand
zero and infinity. Posterity would treasure his memory as that of not just a
mathematical wizard, but as a mystic visionary. This may sound a bit farfetched
as mathematics is supposed to be a scientific subject, measuring values of each
thing exactly and precisely. But Ramanujan is a contemporary of Einstein and a
predecessor of the Quantum Physicists in the sub-atomic world, where everything
is abstract and invisible to the naked eye. In order to predict the definite position
of a particle at a particular place and time, one has to depend on numerous
possibilities and intricate calculations in that multi-dimensional world. This
paper is a modest attempt to understand Ramanujan in the light of that torch
that he holds for the times, far ahead of him. In a world where conjectures and
complications are the rule of the day in order to minimize errors and approach
a point, rather than reach it, Ramanujan’s congruencies and identities are of
immense value.
Ramanujan said, “an equation for me has no meaning unless it expresses a
thought of God”. He belonged to a deeply rooted religious family, his mother
and grandmother being ardent devotees of Namakkal, their family deity. It had
so happened that, only after getting approval from this Goddess, did he set out
to London to pursue his research in Mathematics. The story goes that Namakkal‘s
consort. Narasimha Swamy came in Ramanujan’s dreams and revealed to him many
mystical equations. Till now, Ramanujan’s equations bear evidence to the fact--
they hold a mystic background. The purpose of his equations had been clear to
him, but the proof, provided by him, wasn’t felt to be adequate, even by his
well-informed contemporaries like Hardy. Hardy has always had a challenge to
guard his co-partner’s originality. The beaten track of formalities, expected
from the publishers of the journals used to be a sort of straight jacket for
Ramanujan. Whatever Ramanujan did was clear to him. Even his basic equation, 2n-1
is not just a mathematical formula, but holds an inner meaning for him.
2n-1;
20 -1 =0 (Sunya), n=0;
21-1=1
(Shiv, a unifying principle in the Infinite God), n=1,
22-1=3 (Trinity, the three
Godheads - Shiv, Vishnu and Brahma), n=2,
23-1=7 (The Saptarishis – i.e.,
the seven famous saints), n=3,
and so on.
Ramanujan would build a
theory of reality around zero and Infinity; for him, in zero the Absolute
Reality is represented. The mathematical product of ∞ and 0 is not one number,
but all numbers, each of which corresponds to individual acts of creation. Ramanujan could visualize the Infinity as the
basic background behind all numbers. His formulae are an attempt at finding a
link between the finite and the infinite, weaving a web of connectivity between
the seemingly opposite entities, the sunya, the emptiness and the Infinity,
Shiv, the Divinity. All numbers, with their varied values, do emerge from that
abstraction, that subtle concept of zero, the one with an infinitesimally small
value like the atom. But like the atom, zero also yields to an unexplored
world, another cosmos. In the same way, the journey from zero to Infinity is an
abstract, subtle one through various ways and sequences; Ramanujan’s formulae reveal
the wonder of kaleidoscopic joining together of numbers in magical patterns.
According to Ramanujan, integers
are not separate wholes, but are a part and parcel of an active and dynamic relationship,
vibrating with values of varying associations in a huge network of partitions,
exponential complications, functional intricacies and so on, extending over
geometric shapes. He believed in the fascinating principle of fastening behind
the divisive forms of numbers and their nature to tend to rigidities; his
equations evince an adhesive compulsion which makes a disintegrating
multiplicity only a subordinate term. Is each integer a partition by any mode (
+, -, ×, or ÷) of infinity? Does zero also have a place in this magnetic field
of magical relationships, linking vast, abstract regions into invincible, but
meaningful formulae? Zero and Infinity are not contradictory terms, but are
co-existent correlatives and intelligent complements.
Ramanujan’s Infinity
consists in its body- a number of connections – several series of numbers – in
aggregates and divisions, partitions, equations and so on. The visionary has
conceived of an infinite body of sequences of numbers, summable in various ways through magical formulae. The
secret of his success is – he has seen a unity behind the apparent drama of
diversity. While Einstein was connecting cosmic bodies through space-time,
Ramanujan was establishing a unifying process of multiplexes. If Infinity is
the Absolute, numbers are variable and relative, serving for “n” in his
equations; if the former is ineffable, the
latter is expression; they are not contradictory, but complementary to each
other. There are innumerable ways of finding this medium between the two – a
seemingly endless journey, but executed through a number of equations. The
countless numbers are the rays of the sun; to understand that Infinity, the
sun, the numbers, the rays, are to be marshalled, and one has to arrive at or
approach a point of zero. One can’t stop there, but can bring together an
infinite wealth – how? It can only happen through varied sequences of numbers,
directed at the unaccountable infinity, grasped through countable entities, an
absolute summability of infinite series.
Ramanujan’s number theory
is a bottomless reservoir of raw data; it provides huge scope for finding
sustainable links in number systems themselves. The wizard can witness numerous
possibilities of a mysterious process, going on in two opposite directions –
joining up and splitting down of numbers. This visionary challenges himself to
find a connection, a sequence, a joining thread of expression, say, of 4 as
2+2, 1+3,1+1+2, 1+1+1+1, and to be scrupulously complete about it, just plain
4+0, expressed as p(4) = 5. Partition Theory of Ramanujan has started like
that, leading to further complexities or unravelling nature’s mysteries. In
Arithmetical functions, Ramanujan tries to find formulae for certain properties
of numbers, pi(n), the number of prime numbers, the number of partitions and
the tau functions, T(n). All these are not simple formulae, to be easily
understood, but are huge mathematical labyrinths. During his time, Hardy called
T (n) a hypothesis, and later, it was called tau conjecture; in fact, it is not
an explicit formula. Rather, it is a hint at something, a way of stating that
it is an approximation, a probability,
one order, chosen out of a few, available. This, like the prime number theorem,
is a kind of approximation; even in Mathematics, the values like that of pie , can only be calculated to the nearest point, not beyond
that. The same is the case with Partition Theory; later, scientists have come
to know that the Victorian module of thinking of atoms as basic building blocks
and measurements as pertaining to rigid numbers has yielded to flexible bodies
of constantly changing sizes. In 1974, a Belgian Mathematician, Pierre Deligne
proved this conjecture, using powerful new tools, supplied by the field, known
as algebraic geometry. Similarly, just before leaving London for India, in his
last days, Ramanujan worked on the congruence properties of Partition Function.
Based on this Function, two identities, called the first and second Rogers-Ramanujan
identities, became popular in the field of Mathematics. Later, Ramanujan
himself found a link between the two identities.
Even the latest nuclear scientists make use of the mathematical tools, found by Ramanujan, long ago. String Theory imagines the universe as populated by infinitesimally short string-like packets whose movement produces particles. The mathematics, required to describe these strings demand 26 dimensions,23 more than our ordinary 3 dimensions. Partition Theory and Ramanujan’s work in the area, known as modular forms, have proved essential in this analysis. Again, Ramanujan’s mathematics has helped in untangling an important problem in statistical mechanics. For example, there is a theoretical model that explains how liquid helium disperses through a crystal lattice of carbon. As it happens, the sites, helium molecules may occupy in a sheet of graphite, can never be adjacent to one another. Since each potential site is surrounded by six neighbors in a hexagonal array, once it is filled, the six around it define an unbreachable hexagonal wall. R. J. Baxter, the Australian scientist, acknowledges that the thinking behind his “hard hexagon model” is mathematically dependent on a particular set of infinite series that occur in the famous Rogers-Ramanujan identities. Based on them, the scientist is able to find a way to determine the probability that any particular site a helium molecule may harbor. This mathematical prediction, based on the model, takes the scientist close to the experiment in the lab.
Even the latest nuclear scientists make use of the mathematical tools, found by Ramanujan, long ago. String Theory imagines the universe as populated by infinitesimally short string-like packets whose movement produces particles. The mathematics, required to describe these strings demand 26 dimensions,23 more than our ordinary 3 dimensions. Partition Theory and Ramanujan’s work in the area, known as modular forms, have proved essential in this analysis. Again, Ramanujan’s mathematics has helped in untangling an important problem in statistical mechanics. For example, there is a theoretical model that explains how liquid helium disperses through a crystal lattice of carbon. As it happens, the sites, helium molecules may occupy in a sheet of graphite, can never be adjacent to one another. Since each potential site is surrounded by six neighbors in a hexagonal array, once it is filled, the six around it define an unbreachable hexagonal wall. R. J. Baxter, the Australian scientist, acknowledges that the thinking behind his “hard hexagon model” is mathematically dependent on a particular set of infinite series that occur in the famous Rogers-Ramanujan identities. Based on them, the scientist is able to find a way to determine the probability that any particular site a helium molecule may harbor. This mathematical prediction, based on the model, takes the scientist close to the experiment in the lab.
Ramanujan has an
aesthetic sense of creating bonds amidst numerous seemingly unsolvable mysteries.
His is a creative process of tracking the course of illusive numbers to capture
sequence and consequence .The course, this mathematician’s mental agility has
taken in defining the relative values of numbers in their connection with that
of the Absolute Infinity, brings in the importance of deep-rooted culture in
shaping Intuition and instinct. In the Gita, it is said that God is “Indivisible,
but as if divided in beings”; He “is there in beings, indivisible and as if
divided”. The same may be said of Ramanujan’s Infinity, theoretically
indivisible, but practically split into varied values of several numbers. Just as
the same Divinity imparts His essence to all beings, Infinity spreads value
system into each and every existent thing. To put it the other way, the Absolute,
the Infinity, is basically indeterminable and undefinable, but is the source of
all determinates and can be defined only through them. He is an innovator, not only
in mathematics and other related fields, but in setting a trend of thinking. Like
a rare artist, Ramanujan enjoys a niche at that seldom conjoint of exact
science and mystical musings. Ramanujan’s perception of infinity and numbers is
like that of a painter, bringing in a perfect piece of art with each stroke of
a brush. In his case, each integer, fraction, or even zero becomes plastic
enough to be a tool to see through it not only multitudes and boundless
totalities, but a unifying essentiality.
