Ramanujan, Greatest Indian Mathematician

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COMPUTING THE MATHEMATICAL FACE OF GOD: S. RAMANUJAN

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Subject: COMPUTING THE MATHEMATICAL FACE OF GOD: S. RAMANUJAN
Posted by: jai@aloha.com (Dr. Jai Maharaj)

HINDUISM TODAY February 1990
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Computing the Mathematical Face of God: S. Ramanujan

He died on his bed after scribbling down
revolutionary mathematical formulas that bloomed
in his mind like ethereal flowers -- gifts, he
said, from a Hindu Goddess.

He was 32 the same age that the advaitan advocate
Adi Shankara died. Shankara, born in 788, left
earth in 820. Srinivasa Ramanujan was born in
1887. He died in 1920 -- an anonymous Vaishnavite
brahmin who became the first Indian mathematics
Fellow at Cambridge University. Both Shankara and
Ramanujan possessed supernatural intelligence, a
well of genius that leaves even brilliant men
dumb-founded. Ramanujan was a meteor in the
mathematics world of the World War I era. Quiet,
with dharmic sensibilities, yet his mind blazed
with such intuitive improvisation that British
colleagues at Cambridge -- the best math brains in
England -- could not even guess where his ideas
originated. It irked them a bit that Ramanujan
told friends the Hindu Goddess Namagiri whispered
equations into his ear. Today's mathematicians --
armed with supercomputers -- are still
star-struck, and unable to solve many theorems the
young man from India proved quickly by pencil and
paper.

Ramanujan spawned a zoo of mathematical creatures
that delight, confound and humble his peers. They
call them "beautiful," "humble," "transcendent,"
and marvel how he reduced very complex terrain to
simple shapes.

In his day these equations were mainly pure
mathematics, abstract computations that math sages
often feel describe God's precise design for the
cosmos. While much of Ramanujan's work remains
abstract, many of his theorems are now the
mathematical power behind several 1990's
disciplines in astrophysics, artificial
intelligence and gas physics. According to his
wife -- Janaki, who still lives outside Madras --
her husband predicted "his mathematics would be
useful to mathematicians for more than a
century." Yet, before sailing to England,
Ramanujan was largely ignorant of the prevailing
highest-level math. He flunked out of college in
India. Like Albert Einstein, who toiled as a
clerk in a Swiss patent office while evolving his
Special Theory of Relativity at odd hours,
Ramanujan worked as a clerk at a port authority in
Madras, spending every spare moment contemplating
the mathematical face of God. It was here in
these sea-smelling, paper-pushing offices that he
was gently pushed into destiny -- a plan that has
all the earmarks of divine design.

Ramanujan was born in Erode, a small, rustic town
in Tamil Nadu, India. His father worked as a
clerk in a cloth merchant's shop. his namesake is
that of another medieval philosophical giant --
Ramanuja -- a Vaishnavite who postulated the
Vedanta system known as "qualified monism." the
math prodigy grew up in the overlapping
atmospheres of religious observances and ambitious
academics. He wasn't spiritually preoccupied, but
he was steeped in the reality and beneficence of
the Deities, especially the Goddess Namagiri.
Math, of course, was his intellectual and
spiritual touchstone. No one really knows how
early in life ramanujan awakened to the psychic
visitations of Namagiri, much less how the
interpenetration of his mind and the Goddess'
worked. By age twelve he had mastered
trigonometry so completely that he was inventing
sophisticated theorems that astonished teachers.
In fact his first theorems unwittingly duplicated
those of a great mathematician of a hundred years
earlier. This feat came after sifting once
through a trigonometry book. he was disappointed
that his "discovery" has already been found. then
for four years there was numerical silence. At
sixteen a copy of an out-of-date math book from
Cambridge University came into his hands. It
listed 5,000 theorems with sparse, short-cut
proofs. Even initiates in the arcane language of
mathematics could get lost in this work.
Ramanujan entered it with the giddy ambition and
verve of an astronaut leaping onto the moon. It
subconsciously triggered a love of numbers that
completely saturated his mind. He could envision
strange mathematical concepts like ordinary people
see the waves of an ocean.

Ironically, his focus on math became his academic
undoing. he outpaced his teachers in numbers
theory, but neglected all other subjects. He
could speak adequate English, but failed in it and
history and other science courses. He lost a
scholarship, dropped out, attempted a return but
fell ill and quit a second time. By this time he
was married to Janaki, a young teenager, and was
supporting his mother. Often all night he
continued his personal excursions into the math
universe - being fed rice balls by his wife as he
wrote lying belly-down on a cot. During the day
he factored relatively mundane accounts at the
post office for 20 pounds a year. He managed to
publish one math paper.

As mathematicians would say, one branch of
potential reality could have gone with Ramanujan
squandering his life at the port. But with one
nudge from the invisible universe, Namagiri sent
him Westward. A manager at the office admire the
young man's work and sensed significance. He
talked him into writing to British mathematicians
who might sponsor him. Ramanujan wrote a simple
letter to the renowned G. W. Hardy at Cambridge,
hinting humbly at his breakthroughs and describing
his vegetarian diet and spartan needs if he should
come to the university. He enclosed one hundred
of his theorem equations.

Hardy was the brightest mathematician in England.
Yet, as he knew and would write later at the
conclusion of his life, he had done no original,
mind-bending work. At Cambridge he collaborated
with an odd man named Littlewood, who was so
publicly retiring that people joked Hardy made him
up. The two, though living within a hundred yards
of each other, communicated by exchange of terse,
math-laden letters. Ramanujan's letter and
equations fell to them like a broadcast from alien
worlds. AT first they dismissed it as a
curiosity. Then, they suddenly became intrigued
by the Indian's musings. Hardy later wrote: "A
single look at them is enough to show that they
could only be written down by a mathematician of
the highest class. They must be true, for if they
were not true, no one would have the imagination
to invent them."

Hardy sensed an extremely rare opportunity, a
"discovery," and quickly arranged a scholarship
for the then 26-year-old Ramanujan. The
invitation came to India and landed like a bomb in
Ramanujan's family and community circle. His
mother was horrified that he would lose caste by
traveling to foreign shores. She refused to let
him go unless it was sanctioned by the Goddess.
According to one version of the story, the aged
mother then dreamt of the blessing from Namagiri.
But Janaki says her husband himself went to the
namagiri temple for guidance and was told to make
the voyage. Ramanujan consulted the astrological
data for his journey. He sent is mother and wife
to another town so they wouldn't see him with his
long brahmin's hair and bun trimmed to British
short style and his Indian shirt and wrapcloth
swapped for European fashion. He left India as a
slightly plump man with apple-round cheeks and
eyes like bright zeroes.

Arriving in 1914 on the eve of World War I,
Ramanujan experienced severe culture shock at
Cambridge. he had to cook for himself and
insisted on going bare foot Hindu style on the
cold floors. But Hardy, a man without airs or
inflated ego, made him feel comfortable amidst the
stuffy Cambridge tradition. Hardy and Littlewood
both served as his mentors for it took two
teachers to keep pace with his advances. Soon, as
Hardy recounts, it was Ramanujan who was teaching
them, in fact leaving them in the wake of
incandescent genius.

Within a few months war broke out. Cambridge
became a military college. vegetable and fruit
shortages plagued Ramanujan's already slim diet.
The war took away Littlewood to artillery
research, and Ramanujan and Hardy were left to
retreat into some of the most recondite math
possible. One of the stunning examples of this
endeavor is a process called partitioning,
figuring out how many different ways a whole
number can be expressed as the sum of other whole
numbers. Example: 4 is partitioned 5 ways (4
itself, 3+1, 2+2, 2+1+1, 1+1+1+1), expressed as
p(4)=5. The higher the number, the more the
partitions. Thus p(7)=15. Deceptively though,
even a marginally larger number creates
astronomical partitions. p(200)=397,999,029,388.
Ramanujan -- with Hardy offering technical checks
-- invented a tight, twisting formula that
computes the partitions exactly. To check the
theorem a fellow Cambridge mathematician tallied
by hand the partitions for 200. It took one
month. Ramanujan's equation was precisely
correct. U.S. mathematician George Andrews, who
in the late 1960's rediscovered a "lost notebook"
of Ramanujan's and became a lifetime devotee,
describes his accuracy as unthinkable to even
attempt. Ramanujan's partition equation helped
later physicists determine the number of electron
orbit jumps in the "shell" model of atoms.

ANother anecdote demonstrates his mental
landscape. By 1917, Ramanujan had fallen
seriously ill and was convalescing in a country
house. Hardy took a taxi to visit him. As math
masters like to do he noted the taxi's number --
1729 -- to see if it yielded any interesting
permutations. To him it didn't and he thought to
himself as he went up the steps to the door that
it was a rather dull number and hoped it was not
an inauspicious sign. He mentioned 1729 to
Ramanujan who immediately countered, "Actually, it
is a very interesting number. It is the smallest
number expressible as the sum of two cubes in two
different ways."

Ramanujan deteriorated so quickly that he was
forced to return to India -- emaciated -- leaving
his math notebooks at Cambridge. He spent his
final year face down on a cot furiously writing
out pages and pages of theorems as if a storm of
number concepts swept through his brain. Many
remain beyond today's best math minds.

Debate still lingers as to the origins of
Ramanujan's edifice of unique ideas.
Mathematicians eagerly acknowledge surprise states
of intuition as the real breakthroughs, not
logical deduction. There is reticence to accept
mystical overtones, though, like Andrews, many can
appreciate intuition *in the guise* of a Goddess.
But we have Ramanujan's own testimony of feminine
whisperings from a Devi and there is the sheer
power of his achievements. Hindus cognize this
reality. As an epilogue to this story, a seance
held in 1934 claimed to have contacted Ramanujan
in the astral planes. Asked if he was continuing
his work, he replied, "No, all interest in
mathematics dropped out after crossing over."

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The teacher was asking some simple questions in arithmetic. The class was learning the simple operation of division. When the teacher asked how many bananas would each boy get if three bananas were divided equally among three boys, someone had an answer. One each. Thousand bananas divided equally among thousand boys? The answer was still the same. One. The class was progressing thus, questions being asked by the teacher and answers being provided by the student. But there was a boy who had a question. If none of the bananas was divided among no boys, how much would each boy get?
The whole class burst into laughter at what the students thought was a fast one or a silly question. But the teacher seemed to have been impressed. He took it upon himself to explain to the boys that what the student had asked was not a silly question but rather a profound one. He was questioning the teacher about the concept of infinity. A concept that had baffled mathematicians for centuries, until the Indian scientist Bhaskara had provided some light. He had proved that zero divided by zero was neither zero nor one, but infinity.

The student was Srinivas Ramanujam, the genius who introduced the concept of zero to the world. Ramanujam was born Erode in Tamil Nadu on December 22, 1887. His mathematical genius began to show at a very early age and soon senior students began to haunt his house for clarifying doubts. When he was merely thirteen years of age, he mastered a book on Trigonometry. So taken by the subject was he that he launched his own research work. He put forward theorems and formulae that had been discovered earlier by great mathematicians but were not covered in the book.
Srinivas Ramanujan


The real turning point that triggered off his own creations came two years later, when a friend introduced the book Synopsis of Elementary Results in Pure and Applied Mathematics by George Shoobridge Carr to Ramanujam. Where any other person at the age of fifteen may have recoiled from the book, Ramanujam became delighted at the introduction. He began solving problems given in the book. With the floodgates now open, ideas began to pour forth. Such was the gush of ideas that Ramanujam found it difficult to write them all down. Can you hazard a guess on the number of papers that Ramanujam required per month for jotting his ideas? Two thousand! He scribbled his results in loose sheets and notebooks. In fact, before he went abroad for pursuing his studies at the Cambridge University, he had filled three notebooks with his jottings, which later came to be known as Ramanujam s Frayed Notebooks.

Ramanujam s father, a clerk, however, could never fathom the boy s obsession for numbers. Although the boy had secured a first class in his matriculation examination and had also been awarded the Subramanyan scholarship, he had failed in his first year college examinations. This was because, being obsessed with mathematics, he had neglected all other subjects. Desiring to bring his mad son back on the course of normalcy , the worried father got him married to a young girl of eight!

This put Ramanujam in real dilemma. He needed to find money to support self, wife and buy paper for his jottings. Oh yes, marriage did not distract him from his magnificent obsession. Driven to desperation, he began reusing papers, now writing on them in blue and rewriting over it in red so as to be able distinguish between two trains of thoughts. Ramanujam approached several offices and applied for a clerical job, displaying his now famous frayed notebooks and papers and claiming that he was good in mathematics. However, nobody could follow his work and he was turned away. Luckily for him, he came across one Francis Spring, who did seem to understand what was in the notebooks and who appointed him at the Madras Port Trust where he (Spring) was the Director. Soon after, some educationists took up the cause of Ramanujam and in May 1913, the University of Madras awarded him a fellowship although he had no formal degree.

In the meantime, Ramanujam had approached the great mathematician G. H. Hardy and presented to him a set of one hundred and twenty theorems and formulae. A part of it was the Reimann series, a topic in definite integral in calculus. Ignorant of Reimann s original work, Ramanujam had reproduced the work all over again.

Yet another intriguing portion of the collection sent to Hardy was Ramanujam s interpretation about the equations called modular . It was later proved that Ramanujam s conjectures were indeed correct. The collection also included a formula in hypergeometric series, which later came to be named after him.

Hardy and his colleague, J. E. Littlewood, recognised the genius in Ramanujam and made arrangements for him to travel to Cambridge University to study. Hardy was amused to find that Ramanujam was an unsystematic mathematician, who played with maths much as a child played with toys. His mathematical truths were not explained and it was left to other mathematicians to prove them.

Ramanujam was elected Fellow of the Royal society in February 1918. He was the second Indian to be honoured with this fellowship and the first Indian to be elected Fellow of the Trinity College, Cambridge. His contributions to the field of mathematics included the Hardy-Ramanujam Littlewood circle method in number theory, Roger-Ramanujam s identities in partition of integers, list of highest composite numbers and some work on the algebra of inequalities and the number theory.

Unfortunately, Ramanujam fell victim to tuberculosis and had to be sent home to India. Fighting pain and death, Ramanujam kept himself pre-occupied by playing with numbers. He succumbed to the illness at the tender age of thirty-two. Within the short life span, Ramanujam had earned repute as an astrologer and an orator too.

I have recently read in an English newspaper that the 'lost' note book of Ramanujan is being researched by eminent mathematicians of our times in a certain western University.

Both the people who have expressed their opinion here before me are not doing any justice to the memory of a great Indian mathematician due to the simple reason that long posts are seldom read. :D

I would have expressed my admiration for someone like Ramanujan in a better way,so that people feel like reading it. :?

I have read through Hardy's own account of Ramanujan. While Hardy does not give any credit to Namagiri's assistance, he does not deny that Ramanujan's originality was unsurpassed. Mathematics is a very broad field; there is a tendency to specialize in one branch or another. But according to Hardy, Ramanujan's potential was not at all limited like this. He explored numerous branches, and made many original contributions. Also according to Hardy, Ramanujan came up with famous theorems, on his own, that were already well known. Now this is amazing, if you consider that one theorem was maybe discovered a 100 yrs after another. But Ramanujan discovered these theorems within what, a 15 yr period! Also these were famous theorems; in other words, the product of the long efforts put forth by the best Western minds. So how does one with little mathematical training reach such conclusions? As Ramanujan said, he was inspired by Namagiri. I have done some higher maths myself; there is a very large difference between mastering calculations and proving a theorem. These days, there are entire classes devoted to learning to prove theorems. Anyone can master the calculations, with the right effort - this is part of a physicists or engineers normal training. But proving the difficult theorems requires far more abstract thought. That one with no proper training was doing this at a very young age is incredible.