Transcript Slide 1

Born 22 December
Erode, British India Died
26 April 1920 (aged 32)
Chetput, (Madras), British
Residence Tamil Nadu,
India Fields Mathematics
Alma mater Government
Arts College,
Pachaiyappa's College,
Trinity College,
Cambridge Academic
advisors G. H. Hardy and
J. E. Littlewood
Known for Landau–
Ramanujan constant
Mock theta functions
Ramanujan conjecture
Ramanujan prime
Ramanujan theta function
Ramanujan's sum
Srīnivāsa Aiyangār Rāmānujan FRS, better known as
Srinivasa Iyengar Ramanujan (22 December 1887 – 26
April 1920) was an Indian mathematician and
autodidact who, with almost no formal training in
pure mathematics, made substantial contributions to
mathematical analysis, number theory, infinite series
and continued fractions. Rāmānujan's talent was said,
by the prominent English mathematician G.H. Hardy,
to be in the same league as legendary mathematicians
such as Euler, Gauss, Newton and Archimedes [1].
Born and raised in Erode, Tamil Nadu, India,
Ramanujan first encountered formal mathematics at
age 10. He demonstrated a natural ability, and was
given books on advanced trigonometry written by S.
L. Loney.[2] He had mastered them by age 12, and even
discovered theorems of his own. He demonstrated
unusual mathematical skills at school, winning
accolades and awards. By 17, Ramanujan conducted
his own mathematical research on Bernoulli numbers
and the Euler–Mascheroni constant.
He received a scholarship to study at Government College in
Kumbakonam, but lost it when he failed his non-mathematical
coursework. He joined another college to pursue independent
mathematical research, working as a clerk in the AccountantGeneral's office at the Madras Port Trust Office to support
himself.[3] In 1912–1913, he sent samples of his theorems to three
academics at the University of Cambridge. Only G. H. Hardy
recognized the brilliance of his work, subsequently inviting
Ramanujan to visit and work with him at Cambridge. He became
a Fellow of the Royal Society and a Fellow of Trinity College,
Cambridge, dying of illness, malnutrition and possibly liver
infection in 1920 at the age of 32.
During his short lifetime, Ramanujan independently compiled
nearly 3900 results (mostly identities and equations). Although a
small number of these results were actually false and some were
already known, most of his claims have now been proven correct.
He stated results that were both original and highly
unconventional, such as the Ramanujan prime and the
Ramanujan theta function, and these have inspired a vast
amount of further research. However, some of his major
discoveries have been rather slow to enter the mathematical
mainstream. Recently, Ramanujan's formulae have found
applications in crystallography and string theory. The
Ramanujan Journal, an international publication, was launched
to publish work in all areas of mathematics influenced by his
Ramanujan was born on 22 December 1887 in the city Erode, Tamil
Nadu, India, at the residence of his maternal grandparents .His
father, K. Srinivasa Iyengar worked as a clerk in a sari shop and
hailed from the district of Thanjavur. His mother, Komalatammal or
Komal Ammal was a housewife and also sang at a local temple.They
lived in Sarangapani Street in a traditional home in the town of
Kumbakonam. The family home is now a museum. When Ramanujan
was a year and a half old, his mother gave birth to a son named
Sadagopan, who died less than three months later. In December 1889,
Ramanujan had smallpox and recovered, unlike thousands in the
Thanjavur district who succumbed to the disease that year.He moved
with his mother to her parents' house in Kanchipuram, near Madras
(now Chennai). In November 1891, and again in 1894, his mother gave
birth, but both children died in infancy.
On 1 October 1892, Ramanujan was enrolled at the local school.[ In
March 1894, he was moved to a Telugu medium school. After his
maternal grandfather lost his job as a court official in Kanchipuram,
Ramanujan and his mother moved back to Kumbakonam and he was
enrolled in the Kangayan Primary School.[ After his paternal
grandfather died, he was sent back to his maternal grandparents, who
were now living in Madras. He did not like school in Madras, and he
tried to avoid going to school. His family enlisted a local constable to
make sure he attended school. Within six months, Ramanujan was
back in Kumbakonam.
Since Ramanujan's father was at work most of the day, his
mother took care of him as a child. He had a close
relationship with her. From her, he learned about tradition
and puranas. He learned to sing religious songs, to attend
pujas at the temple and particular eating habits – all of
which are part of Brahmin culture. At the Kangayan
Primary School, Ramanujan performed well. Just before
the age of 10, in November 1897, he passed his primary
examinations in English, Tamil, geography and arithmetic.
With his scores, he finished first in the district.[That year,
Ramanujan entered Town Higher Secondary School where
he encountered formal mathematics for the first time.
By age 11, he had exhausted the mathematical knowledge
of two college students who were lodgers at his home. He
was later lent a book on advanced trigonometry written by
S. L. Loney. He completely mastered this book by the age of
13 and discovered sophisticated theorems on his own. By
14, he was receiving merit certificates and academic
awards which continued throughout his school career and
also assisted the school in the logistics of assigning its 1200
students (each with their own needs) to its 35-odd teachers.
He completed mathematical exams in half the allotted time, and
showed a familiarity with infinite series. When he was 16, Ramanujan
came across the book A Synopsis of Elementary Results in Pure and
Applied Mathematics by George S. Carr. This book was a collection of
5000 theorems, and it introduced Ramanujan to the world of
mathematics. The next year, he had independently developed and
investigated the Bernoulli numbers and had calculated Euler's
constant up to 15 decimal places.[ His peers of the time commented
that they "rarely understood him" and "stood in respectful awe" of him.
When he graduated from Town Higher Secondary School in 1904,
Ramanujan was awarded the K. Ranganatha Rao prize for
mathematics by the school's headmaster, Krishnaswami Iyer
introduced Ramanujan as an outstanding student who deserved
scores higher than the maximum possible marks.He received a
scholarship to study at Government College in Kumbakonam,
However, Ramanujan was so intent on studying mathematics that he
could not focus on any other subjects and failed most of them, losing
his scholarship in the process.In August 1905, he ran away from home,
heading towards Visakhapatnam. He later enrolled at Pachaiyappa's
College in Madras. He again excelled in mathematics but performed
poorly in other subjects such as physiology. Ramanujan failed his Fine
Arts degree exam in December 1906 and again a year later. Without a
degree, he left college and continued to pursue independent research
in mathematics. At this point in his life, he lived in extreme poverty
and was often on the brink of starvation
On 14 July 1909, Ramanujan was married to a nine-year old bride,
Janaki Ammal. – in the branch of Hinduism to which Ramanujan
belonged, marriage was a formal engagement that was consummated
only after the bride turned 17 or 18, as per the traditional calendar.
After the marriage, Ramanujan developed a hydrocele testis, an
abnormal swelling of the tunica vaginalis, an internal membrane in the
testicle.The condition could be treated with a routine surgical
operation that would release the blocked fluid in the scrotal sac. His
family did not have the money for the operation, but in January 1910, a
doctor volunteered to do the surgery for free. After his successful
surgery, Ramanujan searched for a job. He stayed at friends' houses
while he went door to door around the city of Madras (now Chennai)
looking for a clerical position. To make some money, he tutored some
students at Presidency College who were preparing for their F.A. exam.
In late 1910, Ramanujan was sick again, possibly as a result of the
surgery earlier in the year. He feared for his health, and even told his
friend, R. Radakrishna Iyer, to "hand these [my mathematical
notebooks] over to Professor Singaravelu Mudaliar [mathematics
professor at Pachaiyappa's College] or to the British professor Edward
B. Ross, of the Madras Christian College." After Ramanujan recovered
and got back his notebooks from Iyer, he took a northbound train from
Kumbakonam to Villupuram, a coastal city under French control.
He met deputy collector V. Ramaswamy Aiyer, who had recently
founded the Indian Mathematical Society. Ramanujan, wishing
for a job at the revenue department where Iyer worked, showed
him his mathematics notebooks. As Iyer later recalled:
I was struck by the extraordinary mathematical results contained
in it [the notebooks]. I had no mind to smother his genius by an
appointment in the lowest rungs of the revenue department.Iyer
sent Ramanujan, with letters of introduction, to his
mathematician friends in Madras. Some of these friends looked at
his work and gave him letters of introduction to R. Ramachandra
Rao, the district collector for Nellore and the secretary of the
Indian Mathematical Society. Ramachandra Rao was impressed
by Ramanujan's research but doubted that it was actually his own
work. Ramanujan mentioned a correspondence he had with
Professor Saldhana, a notable Bombay mathematician, in which
Saldhana expressed a lack of understanding for his work but
concluded that he was not a phony. Ramanujan's friend, C. V.
Rajagopalachari, persisted with Ramachandra Rao and tried to quell any
doubts over Ramanujan's academic integrity.
Rao agreed to give him another chance, and he listened as
Ramanujan discussed elliptic integrals, hypergeometric series, and
his theory of divergent series, which Rao said ultimately
"converted" him to a belief in Ramanujan's mathematical
brilliance. When Rao asked him what he wanted, Ramanujan
replied that he needed some work and financial support. Rao
consented and sent him to Madras. He continued his mathematical
research with Rao's financial aid taking care of his daily needs.
Ramanujan, with the help of V. Ramaswamy Aiyer, had his work
published in the Journal of Indian Mathematical Society. . One of
the first problems he posed in the journal was:
He waited for a solution to be offered in three issues, over six
months, but failed to receive any. At the end, Ramanujan supplied
the solution to the problem himself. On page 105 of his first
notebook, he formulated an equation that could be used to solve
the infinitely nested radicals problem.
 Using
this equation, the answer to the question
posed in the Journal was simply 3. Ramanujan
wrote his first formal paper for the Journal on the
properties of Bernoulli numbers. One property he
discovered was that the denominators (sequence
A027642 in OEIS) of the fractions of Bernoulli
numbers were always divisible by six. He also
devised a method of calculating Bn based on
previous Bernoulli numbers. One of these methods
went as follows:
It will be observed that if n is even but not equal to
(i) Bn is a fraction and the numerator of
in its
lowest terms is a prime number,
(ii) the denominator of Bn contains each of the
factors 2 and 3 once and only once,
is an integer and
consequently is an odd integer.
 In
his 17–page paper, "Some Properties of
Bernoulli's Numbers", Ramanujan gave three
proofs, two corollaries and three
conjectures.[Ramanujan's writing initially had
many flaws. As Journal editor M. T. Narayana
Iyengar noted:
 Mr. Ramanujan's methods were so terse and novel
and his presentation so lacking in clearness and
precision, that the ordinary [mathematical
reader], unaccustomed to such intellectual
gymnastics, could hardly follow him.Ramanujan
later wrote another paper and also continued to
provide problems in the Journal.In early 1912, he
got a temporary job in the Madras Accountant
General's office, with a 20-rupee-a-month salary.
He lasted for only a few weeks. Toward the end of
that assignment he applied for a position under
the Chief Accountant of the Madras Port Trust. In
a letter dated 9 February 1912, Ramanujan wrote:
I understand there is a clerkship vacant in your office,
and I beg to apply for the same. I have passed the
Matriculation Examination and studied up to the F.A.
but was prevented from pursuing my studies further
owing to several untoward circumstances. I have,
however, been devoting all my time to Mathematics
and developing the subject. I can say I am quite
confident I can do justice to my work if I am
appointed to the post. I therefore beg to request that
you will be good enough to confer the appointment on
 Attached to his application was a recommendation
from E. W. Middlemast, a mathematics professor at the
Presidency College, who wrote that Ramanujan was "a
young man of quite exceptional capacity in
Mathematics".Three weeks after he had applied, on 1
March, Ramanujan learned that he had been accepted
as a Class III, Grade IV accounting clerk, making 30
rupees per month.At his office, Ramanujan easily and
quickly completed the work he was given, so he spent
his spare time doing mathematical research.
Ramanujan's boss, Sir Francis Spring, and S.
Narayana Iyer, a colleague who was also treasurer of
the Indian Mathematical Society, encouraged
Ramanujan in his mathematical pursuits.
Spring, Narayana Iyer, Ramachandra Rao and E. W. Middlemast
tried to present Ramanujan's work to British mathematicians.
One mathematician, M. J. M. Hill of University College London,
commented that Ramanujan's papers were riddled with holes. He
said that although Ramanujan had "a taste for mathematics, and
some ability", he lacked the educational background and
foundation needed to be accepted by mathematicians. Although
Hill did not offer to take Ramanujan on as a student, he did give
thorough and serious professional advice on his work. With the
help of friends, Ramanujan drafted letters to leading
mathematicians at Cambridge University.
The first two professors, H. F. Baker and E. W. Hobson, returned
Ramanujan's papers without comment. On 16 January 1913,
Ramanujan wrote a letter to G. H. Hardy who received it in late
January along with a large envelope, which began as:
Dear Sir,
I beg to introduce myself as a clerk in Accounts Department of
Port Trust Office at Madras...I am 23 years of age...I have had no
university education but undergone the ordinary school course.
After leaving school I have been employing the spare time at my
disposal to work at mathematics. I have not trodden through
conventional regular course of a university but I am striking out a
new path for myself. I have made a special investigation of
divergent series in general and the results I get are termed by
local mathematicians as ‘startling’.
The manuscript, all of 10 pages including the letter, was
written out in large and legible school boy handwriting.
Hardy, who was used to get bizarre letters from strangersas his friend C. P. Snow later put it, “pretended to prove
the prophetic wisdom of the Great Pyramids, revelations of
the Elders of the Zion or the cryptograms that Bacon had
inserted in the plays of Shakespeare”- gave a perfunctory
glance and put the manuscript aside and went about his
work as usual. Yet in his mind throughout the day ‘wild
theorems’ he had seen in the Indian’s manuscript scrapped
and tugged his composure-theorems he had never seen
before nor imagined. Was it all a practical joke or a hoax
wondered Hardy. Many Englishmen holding high positions
in Indian Civil Service well versed in mathematics, were
perhaps pulling off a stunt to dupe an old Cambridge friend
Hardy? With such vagrant thoughts bubbling, Hardy went
back to his room in the evening and again sat with the
letter from India.
Coming from an unknown mathematician, the nine pages
of mathematical wonder made Hardy originally view
Ramanujan's manuscripts as a possible "fraud". Hardy
recognized some of Ramanujan's formulae but others
"seemed scarcely possible to believe.“ One of the theorems
Hardy found so incredible was found on the bottom of page
three (valid for 0 < a < b + 1/2):
Hardy was also impressed by some of Ramanujan's
other work relating to infinite series:
The first result had already been determined by a mathematician named
Bauer. The second one was new to Hardy. It was derived from a class of
functions called a hypergeometric series which had first been researched
by Leonhard Euler and Carl Friedrich Gauss. Compared to Ramanujan's
work on integrals, Hardy found these results "much more intriguing".
After he saw Ramanujan's theorems on continued fractions on the last
page of the manuscripts, Hardy commented that the "[theorems] defeated
me completely; I had never seen anything in the least like them before.“
He figured that Ramanujan's theorems "must be true, because, if they
were not true, no one would have the imagination to invent them."Hardy
asked a colleague, J. E. Littlewood, to take a look at the papers. Little
wood was amazed by the mathematical genius of Ramanujan. After
discussing the papers with Little wood, Hardy concluded that the letters
were "certainly the most remarkable I have received" and commented
that Ramanujan was "a mathematician of the highest quality, a man of
altogether exceptional originality and power.“ One colleague, E. H.
Neville, later commented that "not one [theorem] could have been set in
the most advanced mathematical examination in the world."
On 8 February 1913, Hardy wrote a letter to Ramanujan, expressing his
interest for his work. Hardy also added that it was "essential that I
should see proofs of some of your assertions.“ Before his letter arrived in
Madras during the third week of February, Hardy contacted the Indian
Office to plan for Ramanujan's trip to Cambridge. Secretary Arthur
Davies of the Advisory Committee for Indian Students met with
Ramanujan to discuss the overseas trip. In accordance with his Brahmin
upbringing, Ramanujan refused to leave his country to "go to a foreign
land." Meanwhile, Ramanujan sent a letter packed with theorems to
Hardy, writing, "I have found a friend in you who views my labour
To supplement Hardy's endorsement, a former mathematical lecturer at
Trinity College, Cambridge, Gilbert Walker, looked at Ramanujan's work
and expressed amazement, urging him to spend time at Cambridge. As a
result of Walker's endorsement, B. Hanumantha Rao, a mathematics
professor at an engineering college, invited Ramanujan's colleague
Narayana Iyer to a meeting of the Board of Studies in Mathematics to
discuss "what we can do for S. Ramanujan."[The board agreed to grant
Ramanujan a research scholarship of 75 rupees per month for the next
two years at the University of Madras. While he was engaged as a
research student, Ramanujan continued to submit papers to the Journal
of the Indian Mathematical Society. In one instance, Narayana Iyer
submitted some theorems of Ramanujan on summation of series to the
above mathematical journal adding “The following theorem is due to S.
Ramanujan, the mathematics student of Madras University” Later in
November, British Professor Edward.B.Ross of Madras Christian
College, whom Ramanujan had met few years ago, stormed into his class
one day with his eyes glowing, asking his students, “Does Ramanujan
know Polish?” The reason was that in one paper, Ramanujan had
anticipated the work of a Polish mathematician whose paper had just
arrived by the day’s mail. In his quarterly papers, Ramanujan drew up
theorems to make definite integrals more easily solvable. Working off
Giuliano Frullani's 1821 integral theorem, Ramanujan formulated
generalizations that could be made to evaluate formerly unyielding
integrals.[Hardy's correspondence with Ramanujan soured after
Ramanujan refused to come to England. Hardy enlisted a colleague
lecturing in Madras, E. H. Neville, to mentor and bring Ramanujan to
England. Neville asked Ramanujan why he would not go to Cambridge.
Ramanujan apparently had now accepted the proposal; as Neville put it,
"Ramanujan needed no converting and that his parents' opposition had
been withdrawn." Apparently, Ramanujan's mother had a vivid dream in
which the family Goddess Namagiri commanded her "to stand no longer
between her son and the fulfillment of his life's purpose.
Ramanujan boarded the S.S. Nevasa on 17 March 1914,
and at 10 o'clock in the morning, the ship departed from
Madras. He arrived in London on 14 April, with E. H.
Neville waiting for him with a car. Four days later,
Neville took him to his house on Chesterton Road in
Cambridge. Ramanujan immediately began his work
with Littlewood and Hardy. After six weeks, Ramanujan
moved out of Neville's house and took up residence on
Whewell's Court, just a five-minute walk from Hardy's
room. Hardy and Ramanujan began to take a look at
Ramanujan's notebooks. Hardy had already received
120 theorems from Ramanujan in the first two letters,
but there were many more results and theorems to be
found in the notebooks. Hardy saw that some were
wrong, some had already been discovered, while the
rest were new breakthroughs. Ramanujan left a deep
impression on Hardy and Littlewood. Littlewood
commented, "I can believe that he's at least a Jacobi",
while Hardy said he "can compare him only with
[Leonhard] Euler or Jacobi."
Ramanujan spent nearly five years in Cambridge
collaborating with Hardy and Littlewood and published a
part of his findings there. Hardy and Ramanujan had highly
contrasting personalities. Their collaboration was a clash of
different cultures, beliefs and working styles. Hardy was an
atheist and an apostle of proof and mathematical rigour,
whereas Ramanujan was a deeply religious man and relied
very strongly on his intuition. While in England, Hardy tried
his best to fill the gaps in Ramanujan's education without
interrupting his spell of inspiration.
Ramanujan was awarded a B.A. degree by research (this
degree was later renamed PhD) in March 1916 for his work on
highly composite numbers, which was published as a paper in
the Journal of the London Mathematical Society. The paper
was over 50 pages with different properties of such numbers
proven. Hardy remarked that this was one of the most
unusual papers seen in mathematical research at that time
and that Ramanujan showed extraordinary ingenuity in
handling it. On 6 December 1917, he was elected to the
London Mathematical Society. He became a Fellow of the
Royal Society in 1918, becoming the second Indian to do so,
following Ardaseer Cursetjee in 1841, and he was the
youngest Fellow in the entire history of the Royal Society. He
was elected "for his investigation in Elliptic functions and the
Theory of Numbers." On 13 October 1918, he became the first
Indian to be elected a Fellow of Trinity College, Cambridge.
Plagued by health problems all throughout his life, living
in a country far away from home, and obsessively involved
with his mathematics, Ramanujan's health worsened in
England, perhaps exacerbated by stress and by the scarcity
of vegetarian food during the First World War. He was
diagnosed with tuberculosis and a severe vitamin
deficiency and was confined to a sanatorium.
Ramanujan returned to Kumbakonam, India in 1919 and
died soon thereafter at the age of 32. His widow, S. Janaki
Ammal, lived in Chennai (formerly Madras) until her death
in 1994.
A 1994 analysis of Ramanujan's medical records and
symptoms by Dr. D.A.B. Young concluded that it was much
more likely he had hepatic amoebiasis, a parasitic infection
of the liver. This is supported by the fact that Ramanujan
had spent time in Madras, where the disease was
widespread. He had two episodes of dysentery before he left
India. When not properly treated, dysentery can lie
dormant for years and lead to hepatic amoebiasis. It was a
difficult disease to diagnose, but once diagnosed, could have
been readily cured.
Ramanujan has been described as a person with a
somewhat shy and quiet disposition, a dignified
man with pleasant manners. He lived a rather
Spartan life while at Cambridge. Ramanujan's first
Indian biographers describe him as rigorously
orthodox. Ramanujan credited his acumen to his
family Goddess, Namagiri of Namakkal, and looked
to her for inspiration in his work. He often said,
"An equation for me has no meaning, unless it
represents a thought of God.“ Hardy cites
Ramanujan as remarking that all religions seemed
equally true to him. Hardy further argued that
Ramanujan's religiousness had been romanticized
by Westerners and overstated—in reference to his
belief, not practice—by Indian biographers. At the
same time, he remarked on Ramanujan's strict
observance of vegetarianism.
In mathematics, there is a distinction between
having an insight and having a proof.
Ramanujan's talent suggested a plethora of
formulae that could then be investigated in depth
later. It is said that Ramanujan's discoveries are
unusually rich and that there is often more in it
than what initially meets the eye. As a by-product,
new directions of research were opened up.
Examples of the most interesting of these
formulae include the intriguing infinite series for
π, one of which is given below
This result is based on the negative fundamental
discriminant d = −4×58 with class number h(d) = 2
(note that 5×7×13×58 = 26390 and that 9801=99×99;
396=4×99) and is related to the fact that
Compare to Heegner numbers, which have class
number 1 and yield similar formulae.
Ramanujan's series for π converges
extraordinarily rapidly (exponentially) and forms
the basis of some of the fastest algorithms
currently used to calculate π. Truncating the sum
to the first term also gives the approximation
for π, which is correct to six decimal places
 One
of his remarkable capabilities was the
rapid solution for problems. He was sharing a
room with P. C. Mahalanobis who had a
problem, "Imagine that you are on a street with
houses marked 1 through n. There is a house in
between (x) such that the sum of the house
numbers to left of it equals the sum of the house
numbers to its right. If n is between 50 and 500,
what are n and x." This is a bivariate problem
with multiple solutions. Ramanujan thought
about it and gave the answer with a twist: He
gave a continued fraction. The unusual part was
that it was the solution to the whole class of
problems. Mahalanobis was astounded and
asked how he did it. "It is simple. The minute I
heard the problem, I knew that the answer was
a continued fraction. Which continued fraction,
I asked myself. Then the answer came to my
mind", Ramanujan replied
His intuition also led him to derive some previously
unknown identities, such as
for all θ, where Γ(z) is the gamma function. Equating
coefficients of θ0, θ4, and θ8 gives some deep identities for
the hyperbolic secant.
 In 1918, Hardy and Ramanujan studied the partition
function P(n) extensively and gave a non-convergent
asymptotic series that permits exact computation of the
number of partitions of an integer. Hans Rademacher, in
1937, was able to refine their formula to find an exact
convergent series solution to this problem. Ramanujan and
Hardy's work in this area gave rise to a powerful new
method for finding asymptotic formulae, called the circle
method.He discovered mock theta functions in the last year
of his life. For many years these functions were a mystery,
but they are now known to be the holomorphic parts of
harmonic weak Maass forms.
Although there are numerous statements that
could bear the name Ramanujan conjecture,
there is one statement that was very influential
on later work. In particular, the connection of
this conjecture with conjectures of André Weil
in algebraic geometry opened up new areas of
research. That Ramanujan conjecture is an
assertion on the size of the tau function, which
has as generating function the discriminant
modular form Δ(q), a typical cusp form in the
theory of modular forms. It was finally proven
in 1973, as a consequence of Pierre Deligne's
proof of the Weil conjectures. The reduction
step involved is complicated. Deligne won a
Fields Medal in 1978 for his work on Weil
While still in India, Ramanujan recorded the bulk of his results in four
notebooks of loose leaf paper. These results were mostly written up
without any derivations. This is probably the origin of the
misperception that Ramanujan was unable to prove his results and
simply thought up the final result directly. Mathematician Bruce C.
Berndt, in his review of these notebooks and Ramanujan's work, says
that Ramanujan most certainly was able to make the proofs of most of
his results, but chose not to.
This style of working may have been for several reasons. Since paper
was very expensive, Ramanujan would do most of his work and
perhaps his proofs on slate, and then transfer just the results to paper.
Using a slate was common for mathematics students in India at the
time. He was also quite likely to have been influenced by the style of
G. S. Carr's book, which stated results without proofs. Finally, it is
possible that Ramanujan considered his workings to be for his
personal interest alone; and therefore only recorded the results.The
first notebook has 351 pages with 16 somewhat organized chapters
and some unorganized material. The second notebook has 256 pages in
21 chapters and 100 unorganized pages, with the third notebook
containing 33 unorganized pages. The results in his notebooks
inspired numerous papers by later mathematicians trying to prove
what he had found. Hardy himself created papers exploring material
from Ramanujan's work as did G. N. Watson, B. M. Wilson, and Bruce
Berndt. A fourth notebook, the so-called "lost notebook", was
rediscovered in 1976 by George Andrews.
common anecdote about Ramanujan relates
to the number 1729. Hardy arrived at
Ramanujan's residence in a cab numbered
1729. Hardy commented that the number
1729 seemed to be uninteresting. Ramanujan
is said to have stated on the spot that it was
actually a very interesting number
mathematically, being the smallest natural
number representable in two different ways as
a sum of two cubes:
 Submitted