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    Ramanujan, Greatest Indian Mathematician

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    COMPUTING THE MATHEMATICAL FACE OF GOD: S. RAMANUJAN

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    To: Multiple recipients of list <talk@SARASWATI.MIT.EDU>
    Subject: COMPUTING THE MATHEMATICAL FACE OF GOD: S. RAMANUJAN
    From: jit <abdutta@icaen.uiowa.edu>
    Date: Fri, 17 Mar 1995 15:44:41 -0500
    From rwja.UMDNJ.EDU!cbando@SARASWATI.MIT.EDU Fri Mar 17 15: 39:35 1995
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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.

    The realization of happiness happens only after experience of pain.

    If we desire to blossom like a flower in the garden of life,
    Then we must learn the art of adjusting our life with the thorns!

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    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.

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

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    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.

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