Srinivasa Ramanujan
Srinivasa Ramanujan FRS (
pronunciation (help·info)) (22 December 1887 – 26 April 1920) was an Indian
mathematician and
autodidact who, with almost no formal training in
pure mathematics, made extraordinary contributions to
mathematical analysis,
number theory,
infinite series, and
continued fractions.
Living in India with no access to the larger mathematical community,
which was centered in Europe at the time, Ramanujan developed his own
mathematical research in isolation. As a result, he sometimes
rediscovered known theorems in addition to producing new work. Ramanujan
was said to be a natural genius by the English mathematician
G.H. Hardy, in the same league as mathematicians like
Euler and
Gauss.
Born in a poor
Brahmin family, Ramanujan's introduction to formal
mathematics began at age 10. He demonstrated a natural ability, and was given books on advanced
trigonometry written by
S. L. Loney that he mastered by the age of 12; he even discovered
theorems of his own, and re-discovered
Euler's identity independently.
He demonstrated unusual mathematical skills at school, winning
accolades and awards. By 17, Ramanujan had conducted his own
mathematical research on
Bernoulli numbers and the
Euler–Mascheroni constant.
Ramanujan 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 Accountant-General's office at the
Madras Port Trust Office to support himself.
In 1912–1913, he sent samples of his theorems to three academics at the
University of Cambridge.
G. H. Hardy, recognizing the brilliance of his work, invited Ramanujan to visit and work with him at
Cambridge. He became a
Fellow of the Royal Society and a Fellow of
Trinity College, Cambridge. Ramanujan died 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).
Nearly all his claims have now been proven correct, although a small
number of these results were actually false and some were already known.
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, the mathematical mainstream has been rather slow in absorbing some of his major discoveries. The
Ramanujan Journal, an international publication, was launched to publish work in all areas of mathematics influenced by his work.
In December 2011, in recognition of his contribution to mathematics,
the Government of India declared that Ramanujan's birthday (22 December)
should be celebrated every year as
National Mathematics Day, and also declared 2012 the
National Mathematical Year.
Early life
Ramanujan's home on Sarangapani Street, Kumbakonam.
Ramanujan was born on 22 December 1887 in
Erode,
Madras Presidency, 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, 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 died from 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 to two children, 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.
When 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 attending. 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 stood 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
geometry and
infinite series.
Ramanujan was shown how to solve cubic equations in 1902 and he went on
to find his own method to solve the quartic. The following year, not
knowing that the quintic could not be solved by radicals, he tried (and
of course failed) to solve the quintic. In 1903 when he was 16,
Ramanujan obtained from a friend a library-loaned copy of a book by
G. S. Carr.
The book was titled
A Synopsis of Elementary Results in Pure and Applied Mathematics and was a collection of 5000 theorems. Ramanujan reportedly studied the contents of the book in detail.
The book is generally acknowledged as a key element in awakening the genius of Ramanujan.
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 at 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. Iyer
introduced Ramanujan as an outstanding student who deserved scores
higher than the maximum possible marks.
He received a scholarship to study at
Government Arts College, 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 and stayed in
Rajahmundry for about a month.
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.
Adulthood in India
On 14 July 1909, Ramanujan was married to a ten-year old bride,
Janaki Ammal (21 March 1899 - 13 April 1994).
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 [Ramanujan's mathematical
notebooks] over to Professor Singaravelu Mudaliar [the 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.
Attention from mathematicians
Ramanujan met deputy collector
V. Ramaswamy Aiyer, who had recently founded the Indian Mathematical Society.
Ramanujan, wishing for a job at the revenue department where Ramaswamy
Aiyer worked, showed him his mathematics notebooks. As Ramaswamy Aiyer
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.
Ramaswamy Aiyer 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 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.
[44] 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 zero,
(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,
(iii)
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 salary of 20 rupees per month. 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:
Sir,
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 me.
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.
Contacting English mathematicians
On the spring of 1913, 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 to
G. H. Hardy.
Coming from an unknown mathematician, the nine pages of mathematics
made Hardy initially view Ramanujan's manuscripts as a possible "fraud".
Hardy recognised 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, and 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. Littlewood was amazed by the mathematical
genius of Ramanujan. After discussing the papers with Littlewood, 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
sympathetically."
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 a few years before, 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 generalisations 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 fulfilment of his life's purpose".
Life in England
Ramanujan (centre) with other scientists at Trinity College
Whewell's Court, Trinity College, Cambridge
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, others 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, the first part of which was published as a paper in the
Proceedings 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.
[citation needed] 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 one of the youngest Fellows in the 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.
Illness and return to India
Plagued by health problems 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,
Madras Presidency
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 widespread in Madras, where Ramanujan had spent time. 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,
a difficult disease to diagnose, but once diagnosed readily cured.
Personality and spiritual life
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. He looked to her for inspiration in his work,
and claimed to dream of blood drops that symbolised her male consort,
Narasimha, after which he would receive visions of scrolls of complex mathematical content unfolding before his eyes.
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
romanticised 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.
Mathematical achievements
In mathematics, there is a distinction between having an insight and
having a proof. Ramanujan's talent suggested a plethora of formula that
could then be investigated in depth later. It is said that Ramanujan's
discoveries are unusually rich and that there is often more to them than
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.
[85][86]
His intuition also led him to derive some previously unknown
identities, such as
![\left [ 1+2\sum_{n=1}^\infty \frac{\cos(n\theta)}{\cosh(n\pi)} \right ]^{-2} + \left [1+2\sum_{n=1}^\infty \frac{\cosh(n\theta)}{\cosh(n\pi)} \right ]^{-2} = \frac {2 \Gamma^4 \left ( \frac{3}{4} \right )}{\pi}](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_vcDCYX-BeUrKVSGrrrzdNMDMZVC51ynQ8-j9y_XpZF3QxSnN13pTJGG5ivUBFTEkezmB79UAFHTZHTTctQ8VSBAGImLyaRd_GrWLYkCfKwe5hIp1jZuS5iHhXlQ1hUxsunW3-YzQ-UC9x6tOVLXg=s0-d)
for all
, where
is the
gamma function. Expanding into series of powers and equating coefficients of
,
, and
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.
[87]
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.
The Ramanujan conjecture
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 conjectures.
Ramanujan's notebooks
While still in Madras, 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 the
Madras Presidency at the time. He was also quite likely to have been influenced by the style of
G. S. Carr's
book studied in his youth, which stated results without proofs.
Finally, it is possible that Ramanujan considered his workings to be for
his personal interest alone; and therefore recorded only 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 unorganised pages, with the third notebook containing
33 unorganised 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 with 87 unorganised pages, the so-called
"lost notebook", was rediscovered in 1976 by
George Andrews.
Notebooks 1, 2 and 3 were published as a two volume set in 1957 by the
Tata Institute of Fundamental Research (TIFR), Mumbai, India. This was a photocopy edition of the original manuscripts, in his own handwriting.
In December 2011, as part of Ramanujan's 125th birth centenary
celebrations, TIFR republished the notebooks in a colored two volume
collector's edition. These were produced from scanned and microfilmed images of the original manuscripts by expert archivists of
Roja Muthiah Research Library, Chennai.
Ramanujan–Hardy number 1729
Main article:
1729 (number)
The number 1729 is known as the Hardy–Ramanujan number after a famous anecdote of the British mathematician
G. H. Hardy regarding a visit to the hospital to see Ramanujan. In Hardy's words:
| “ |
I remember once going to see him when he was ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen.
"No," he replied, "it is a very interesting number; it is the smallest
number expressible as the sum of two cubes in two different ways." |
” |
The two different ways are
- 1729 = 13 + 123 = 93 + 103.
Generalizations of this idea have created the notion of "
taxicab numbers". Coincidentally, 1729 is also a
Carmichael number.
Other mathematicians' views of Ramanujan
Hardy said : "The limitations of his knowledge were as startling as its profundity. Here was a man who could work out
modular equations
and theorems... to orders unheard of, whose mastery of continued
fractions was... beyond that of any mathematician in the world, who had
found for himself the functional equation of the
zeta function and the dominant terms of many of the most famous problems in the analytic theory of numbers; and yet he had never heard of a
doubly periodic function or of
Cauchy's theorem, and had indeed but the vaguest idea of what a function of a
complex variable was...".
When asked about the methods employed by Ramanujan to arrive at his
solutions, Hardy said that they were "arrived at by a process of mingled
argument, intuition, and induction, of which he was entirely unable to
give any coherent account."
He also stated that he had "never met his equal, and can compare him only with Euler or Jacobi."
Quoting K. Srinivasa Rao,
"As for his place in the world of Mathematics, we quote Bruce C. Berndt: '
Paul Erdős
has passed on to us Hardy's personal ratings of mathematicians. Suppose
that we rate mathematicians on the basis of pure talent on a scale from
0 to 100, Hardy gave himself a score of 25, J.E. Littlewood 30,
David Hilbert 80 and Ramanujan 100.'"
Professor Bruce C. Berndt of the
University of Illinois, during a lecture at
IIT Madras
in May 2011, stated that over the last 40 years, as nearly all of
Ramanujan's theorems have been proven right, there had been a greater
appreciation of Ramanujan's work and brilliance. Further, he stated
Ramanujan's work was now pervading many areas of modern mathematics and
physics.
In his book
Scientific Edge, noted physicist
Jayant Narlikar
spoke of "Srinivasa Ramanujan, discovered by the Cambridge
mathematician Hardy, whose great mathematical findings were beginning to
be appreciated from 1915 to 1919. His achievements were to be fully
understood much later, well after his untimely death in 1920. For
example, his work on the
highly composite numbers (numbers with a large number of factors) started a whole new line of investigations in the theory of such numbers."
During his lifelong mission in educating and propagating mathematics among the school children in India, Nigeria and elsewhere,
P.K. Srinivasan has continually introduced Ramanujan's mathematical works.
Recognition
Ramanujan's home state of
Tamil Nadu
celebrates 22 December (Ramanujan's birthday) as 'State IT Day',
memorializing both the man and his achievements, as a native of Tamil
Nadu. A stamp picturing Ramanujan was released by the
Government of India in 1962 – the 75th anniversary of Ramanujan's birth – commemorating his achievements in the field of number theory,
and a new design was issued on December 26, 2011, by the
India Post.
Since the Centennial year of Ramanujan, every year 22 Dec, is celebrated as Ramanujan Day by the
Government Arts College, Kumbakonam
where he had studied and later dropped out. It is celebrated by the
Department Of Mathematics by organising one-, two-, or three-day seminar
by inviting eminent scholars from universities/colleges, and
participants are mainly students of Mathematics, research scholars, and
professors from local colleges. It has been planned to celebrate the
125-th birthday in a grand manner by inviting the foreign Eminent
Mathematical scholars of this century viz., G E Andrews. and Bruce C
Berndt, who are very familiar with the contributions and works of
Ramanujan.
Ramanujan's work and life are celebrated on 22 December at The
Indian Institute of Technology (IIT), Madras in
Chennai.
The Department of Mathematics celebrates this day by organising a
National Symposium On Mathematical Methods and Applications (NSMMA) for
one day by inviting Eminent Indian and foreign scholars.
A prize for young mathematicians from developing countries has been created in the name of Ramanujan by the
International Centre for Theoretical Physics (ICTP), in cooperation with the
International Mathematical Union, who nominate members of the prize committee. The
Shanmugha Arts, Science, Technology & Research Academy (SASTRA), based in the state of Tamil Nadu in South India, has instituted the
SASTRA Ramanujan Prize
of $10,000 to be given annually to a mathematician not exceeding the
age of 32 for outstanding contributions in an area of mathematics
influenced by Ramanujan. The age limit refers to the years Ramanujan
lived, having nevertheless still achieved many accomplishments. This
prize has been awarded annually since 2005, at an international
conference conducted by SASTRA in
Kumbakonam, Ramanujan's hometown, around Ramanujan's birthday, 22 December.
On the 125th anniversary of his birth, India declared the birthday of
Ramanujan, December 22, as 'National Mathematics Day.' The declaration
was made by
Dr. Manmohan Singh in
Chennai on December 26, 2011.
Dr Manmohan Singh also declared that the year 2012 would be celebrated as the
National Mathematics Year.
