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niranjan · 03-09-2007, 06:02 AM

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."

03-09-2007, 06:02 AM	#19 (permalink)
niranjan Senior Member Join Date: Feb 2007 Posts: 589	RAmanujan-2 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."