Kevin Report This Comment Date: January 28, 2005 02:40AM
It is one of the most romantic stories in the history of mathematics: in 1913,
the English mathematician G. H. Hardy received a strange letter from an unknown
clerk in Madras, India. The ten-page letter contained about 120 statements of
theorems on infinite series, improper integrals, continued fractions, and number
theory (Here is a .dvi file with a sample of these results). Every prominent
mathematician gets letters from cranks, and at first glance Hardy no doubt put
this letter in that class. But something about the formulas made him take a
second look, and show it to his collaborator J. E. Littlewood. After a few
hours, they concluded that the results "must be true because, if they were
not true, no one would have had the imagination to invent them".
Thus was Srinivasa Ramanujan (1887-1920) introduced to the mathematical world.
Born in South India, Ramanujan was a promising student, winning academic prizes
in high school. But at age 16 his life took a decisive turn after he obtained a
book titled A Synopsis of Elementary Results in Pure and Applied Mathematics.
The book was simply a compilation of thousands of mathematical results, most set
down with little or no indication of proof. It was in no sense a mathematical
classic; rather, it was written as an aid to coaching English mathematics
students facing the notoriously difficult Tripos examination, which involved a
great deal of wholesale memorization. But in Ramanujan it inspired a burst of
feverish mathematical activity, as he worked through the book's results and
beyond. Unfortunately, his total immersion in mathematics was disastrous for
Ramanujan's academic career: ignoring all his other subjects, he repeatedly
failed his college exams.
As a college dropout from a poor family, Ramanujan's position was precarious. He
lived off the charity of friends, filling notebooks with mathematical
discoveries and seeking patrons to support his work. Finally he met with modest
success when the Indian mathematician Ramachandra Rao provided him with first a
modest subsidy, and later a clerkship at the Madras Port Trust. During this
period Ramanujan had his first paper published, a 17-page work on Bernoulli
numbers that appeared in 1911 in the Journal of the Indian Mathematical Society.
Still no one was quite sure if Ramanujan was a real genius or a crank. With the
encouragement of friends, he wrote to mathematicians in Cambridge seeking
validation of his work. Twice he wrote with no response; on the third try, he
found Hardy.
Hardy wrote enthusiastically back to Ramanujan, and Hardy's stamp of approval
improved Ramanujan's status almost immediately. Ramanujan was named a research
scholar at the University of Madras, receiving double his clerk's salary and
required only to submit quarterly reports on his work. But Hardy was determined
that Ramanujan be brought to England. Ramanujan's mother resisted at
first--high-caste Indians shunned travel to foreign lands--but finally gave in,
ostensibly after a vision. In March 1914, Ramanujan boarded a steamer for
England.
Ramanujan's arrival at Cambridge was the beginning of a very successful
five-year collaboration with Hardy. In some ways the two made an odd pair: Hardy
was a great exponent of rigor in analysis, while Ramanujan's results were (as
Hardy put it) "arrived at by a process of mingled argument, intuition, and
induction, of which he was entirely unable to give any coherent account".
Hardy did his best to fill in the gaps in Ramanujan's education without
discouraging him. He was amazed by Ramanujan's uncanny formal intuition in
manipulating infinite series, continued fractions, and the like: "I have
never met his equal, and can compare him only with Euler or Jacobi."
One remarkable result of the Hardy-Ramanujan collaboration was a formula for the
number p(n) of partitions of a number n. A partition of a positive integer n is
just an expression for n as a sum of positive integers, regardless of order.
Thus p(4) = 5 because 4 can be written as 1+1+1+1, 1+1+2, 2+2, 1+3, or 4. The
problem of finding p(n) was studied by Euler, who found a formula for the
generating function of p(n) (that is, for the infinite series whose nth term is
p(n)xn). While this allows one to calculate p(n) recursively, it doesn't lead to
an explicit formula. Hardy and Ramanujan came up with such a formula (though
they only proved it works asymptotically; Rademacher proved it gives the exact
value of p(n)).
Ramanujan's years in England were mathematically productive, and he gained the
recognition he hoped for. Cambridge granted him a Bachelor of Science degree
"by research" in 1916, and he was elected a Fellow of the Royal
Society (the first Indian to be so honored) in 1918. But the alien climate and
culture took a toll on his health. Ramanujan had always lived in a tropical
climate and had his mother (later his wife) to cook for him: now he faced the
English winter, and he had to do all his own cooking to adhere to his caste's
strict dietary rules. Wartime shortages only made things worse. In 1917 he was
hospitalized, his doctors fearing for his life. By late 1918 his health had
improved; he returned to India in 1919. But his health failed again, and he died
the next year.
Besides his published work, Ramanujan left behind several notebooks, which have
been the object of much study. The English mathematician G. N. Watson wrote a
long series of papers about them. More recently the American mathematician Bruce
C. Berndt has written a multi-volume study of the notebooks. In 1997 The
Ramanujan Journal was launched to publish work "in areas of mathematics
influenced by Ramanujan".
geezer Report This Comment Date: January 28, 2005 03:17AM
So, where's the romantic part?
Kevin Report This Comment Date: January 28, 2005 03:30AM
In a nutshell: This man was a largely self taught mathematician. Much of his
work did cover previously established ground, but he rediscovered and studied
hundreds of years of progressive Mathematics without direction. Sort of a David
and Goliath story really. Genius often comes from the places you least expect
it.
It is also rumored that Ramanujan credited his mathematical abilities to the
deity he worshipped (sorry I can't recall which one at the current time). He
said that the deity actually visited him in his dreams.
geezer Report This Comment Date: January 28, 2005 04:01AM
Oh, I was thinking women's romance, lmao.
Kevin Report This Comment Date: January 29, 2005 05:50PM
Anonymous@1007:
I have a life. Perhaps you should expand your horizons by learning how to
formulate a complete sentence.