Srinivas Ramanujan: Namagiri Gifted Math Genius for whom “Every positive integer was one of his personal friend”.
India celebrates 22nd, December, the birthday of the
legendary mathematician - Srinivasa Ramanujan, as the National Mathematics day.
The legendary Ramanujan was born on 22nd December in Erode, Tamil Nadu
in the year 1887 and in the brief period that he lived (1887-1920) he has left
behind a legacy, which will perpetuate for generations to come not just in
India but globally. The genius of Ramanujan and his goddess – Namagiri - gifted
mathematics has remained an enigma, which is evidenced from the foreword that
was written by CP Snow, friend of GH Hardy who was the mentor of Ramanujan.
C.P. Snow, in his preface to Hardy’s remarkable memoir, ‘A Mathematician’s
Apology’, writes; ‘Hardy did not forget that he was in the presence of a genius
- Ramanujan’. GH Hardy, considered as one of the leading mathematicians of the
world, had profound respect for Ramanujan’s genius in maths which is evidenced
in another incident that Hardy narrates to his friend CP Snow. Hardy says ‘Ramanujan
really had the natural genius. in the sense that the greatest mathematicians
had it’. Hardy on another occasion modestly says ‘I have done one thing that
most others could never have done, and that is to have collaborated with the
greats like Ramanujan on something like equal terms’. The legend and the myth
that surrounds Ramanujan and his Namagiri goddess gifted genius in mathematics,
therefore, comes from such examples, which are seen from the statements and
experiences of GH Hardy and other great mathematicians of his times.
Our admiration for Ramanujan grew exponentially while
researching for curation and development of an exhibition on the life and works
of Ramanujan, during his 125th birth anniversary - 2012. The government of
India had announced celebration of the 125th birth anniversary of the great
Srinivasa Ramanujan and while announcing the celebrations, the then Prime
Minister of India, Shri Manmohan Singh, declared that the birth day of
Ramanujan will be celebrated and commemorated as the National Mathematics Day, and
ever since 22nd December - the Birthdate of Ramanujan - is celebrated as the
National Mathematics Day.
The first thing that came to our mind when developing an
exhibition on Ramanujan was the well-researched biography book on Ramanujan,
which was published by Robert Kanigel entitled “Srinivas Ramanujan: The Man Who
Knew Infinity”. During the early days of discussion on the curatorial concept and
the approach that we should adopt for developing this exhibition, was to ensure
that the subject of maths, which is considered abstract and hard to comprehend,
must be made simple, experiential and interactive. We also felt that apt title
for the exhibition would be the title of Robert Kanigel’s Book - Srinivas
Ramanujan: The Man Who Knew Infinity. We therefore sought permission from Dr
Kanigel for using the title of his book for our exhibition. Kanigel was very
kind to permit us to use this title for our exhibition. We worked on a
different presentation style for the exhibition and for the first time
attempted some new digital interactive techniques to present the complex math
that Ramanujan carried out in a manner which could be appreciated by our visitors,
particularly, school students. This exhibition was successfully opened in December
2012 - the 125th birth anniversary of Ramanujan – by Padma Bhushan, Prof M S
Narasimhan, FRS and a Member, National Committee, for National Mathematics Year,
at Visvesvaraya Industrial and Technological Museum, Bangalore.
The exhibition received overwhelming response and
appreciation, particularly the digital immersive experience and interactive
presentations to present the life and works of the legend. The exhibition also
had a lesson for our youngsters that failure is something which has not eluded
even the great Ramanujan and therefore students must not be afraid of failures.
In his authoritative biography of Ramanujan, Kanigel states that
‘Ramanujan appeared for the Intermediate examinations four times and failed in
all of them’. “Except for maths he did poorly in all his subjects. … He would take
the three-hour maths exam and finish it in thirty minutes.” T.V. Rangaswami’s
Tamil biography (‘Ragami’) on which Kanigel’s account of Ramanujan’s early life
is largely based, states that ‘Ramanujan appeared for the F.A. examination
three times and failed’. Ragami, however, adds that in his last attempt, in
1907, Ramanujan got a centum in mathematics. David Leavitt in his book ‘The
Indian Clerk (2007)’, underlines the repeated failures of Ramanujan in his
intermediate examinations, a point reiterated by the Ramanujan Museum’s
website: “Appeared privately for F.A. examination, secured centum in
mathematics, but failed to secure pass marks in other subjects.” Our exhibition
effectively communicated a lesson from Ramanujan that failures are integral to one’s
life and that includes some of the greatest of scientists as well.
Notwithstanding his multiple failures in intermediate
exam one thing remained certain for Ramanujan. His love and passion for maths
never attenuated, rather he pursued his passion for maths with that much more
focus and managed to publish quite a number of papers in leading mathematical
journals in India before managing to earn a job as a Clark in the Madras Port
Trust. It was here that Ramanujan wrote that famous letter to his mentor, G H
Hardy in the year 1913 and the rest what they say is now history. The letter of
Ramanujan, though evoked mixed response from Hardy initially, it befittingly
earned Ramanujan an invitation from Hardy to the Trinity College, Cambridge. In just five years of his stay in Cambridge,
Ramanujan, with mentoring from Hardy for being more structured in his approach
to solving mathematical problems, made profound contributions in mathematics
and that too during the most testing times of the World War 1. Befittingly
his outstanding contributions at the Cambridge University, won Ramanujan, the
B.A. degree ‘by research’ in 1916. This was a momentous occasion for Ramanujan
who had no formal college degree until this time. This was the first of many
great recognitions and honours, which were destined to follow Ramanujan in the
years ahead at England and in India. The Trinity College, London and so also
the prestigious Royal Society conferred their prestigious fellowship on
Ramanujan. Incidentally Ramanujan became the first Indian to be so honoured
with the fellowship of the Trinity College when he was only thirty.
The legend of Ramanujan is now almost a household name in
India and most students in India will certainly have heard his name, though not
quite familiar with his works. Notwithstanding the inspiration that the great
Ramanujan has provided for our youngsters, most unfortunately, maths has been
overly segmented as a subject meant only for the so called intellectuals,
distancing it from common folks. Hopefully declaration of the National
Mathematics Day will help change this scenario in India. Mathematics is one of
the most important subjects, which acts as a tool to solve problems of every
other science subjects. It provides students an opportunity to think in her own
way and seek solutions to the problems. It makes a student systematic and
methodical and encourages them to make their lives orderly. It is perhaps for
this reason that Mathematics is often called as the mother of all sciences and
it is befitting that the birthday of Ramanujan, the greatest of mathematicians
of India, is celebrated as the National Mathematics Day in India.
Ramanujan was a non-traditional
mathematician who has been befittingly hailed as an all-time great
mathematician of India in modern times and is famously clubbed with the other
international greats like Euler, by G H Hardy. It is very well known that Hardy
was a diehard cricket fan and used cricketing parlance in every field,
including in rating scientists and mathematicians in a scale he termed ‘Bradman
Scale’. He included the likes of the great mathematician Euler and Newton in
the highest scale - the Bradman scale, measured in memory of Hardy’s all-time
favourite cricket player - Don Bradman from Australia. What his rating for
Ramanujan would have been can best be seen in the quote of another
mathematician, Bruce C Berndt, who says ‘Paul Erdos 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, Littlewood 30, Hilbert 80 and Ramanujan 100’. From this yardstick
of marks assigned by the great Hardy to different mathematicians including
himself and Ramanujan it is a given that Ramanujan was a Bradman scale
mathematician for Hardy and that Ramanujan fell in the league of the greats
like Euler and Newton. It is so heartening to know that Hardy, who was one of
the great all time mathematicians of his time and also the man who is credited
to be the discoverer of Ramanujan, has rated his prodigy -Ramanujan, far higher
than what he rates himself and his close associate Littlewood, another great
mathematician of their time.
Although Ramanujan lived only for 32 years
(22nd December 1887 – 26th April 1920) out of which he
spent just five years in the company of Hardy and Littlewood in Cambridge, he
has left behind a very large volume (4000 original theorems) of his works
(including the famous works rediscovered in his ‘lost note books’) which
continue to fascinate greatest of mathematicians of the world, even today.
Legend has it that Srinivas Ramanujan’s mathematical genius came from his
goddess Namagiri - his family deity in Kumbakonam - in whose reverence
Ramanujan had undying faith. It is often said that Ramanujan credited his ideas
and solutions to Namagiri, his family deity, who helped him decipher
mathematical theorems in his dreams. An evidence to Ramanujan’s goddess gifted
ability to provide solutions to problems can be seen in another anecdotal story
associated with the number 1729. G H Hardy, his mentor, in his memoir, says
that he once went to see Ramanujan who was ill and lying in the hospital bed in
Putney. Hardy rode to the hospital in a taxi with ‘dull and unimpressive number
1729’, and he hoped that it was no bad omen number for him. This so called dull
and unimpressive taxi number has now been immortalised by the genius of
Ramanujan’s goddess gifted ability to seek problems and find solutions. On
meeting Ramanujan in hospital, Hardy informs Ramanujan of this ‘unimpressive
number’, but then, lo and behold, Ramanujan turns around to say that it is not
a dull number, rather it is a very interesting and quite a unique number.
Ramanujan, with some mental calculations told Hardy that 1729 is the smallest
number expressible as sum of two cubes in two different ways. He instantly gave
solution to his thoughts; 1729 = 12cube + 1cube = 10cube + 9
cube. (1³ + 12³ = 1 + 1,728 = 1,729 and 9³ + 10³ = 729 + 1,000 =
1,729. Because of this incident, 1729 is now known as the Ramanujan - Hardy
number. Such was his goddess gifted talent for numbers. It is therefore
no wonder that, Littlewood, an associate of Hardy, who also collaborated and
worked with Ramanujan, once said ‘Ramanujan could remember the idiosyncrasies
of numbers in an uncanny way ‘courtesy Namagiri Goddess’ and for Ramanujan
‘every positive integer was one of his personal friend’.
Srinivas
Ramanujan was born to a poor orthodox Tamil Brahmin family on the 22nd of
December, 1887 in his grandmother's house in Erode, Tamil Nadu. His father, Kuppuswamy Srinivasa Iyengar, worked as a clerk
in a sari shop in Kumbakonam and his mother, Komalatammal, was
a housewife who sang devotional songs at a local temple. They
lived in a small traditional home on Sarangapani Sannidhi Street in the town
of Kumbakonam. When Ramanujan
was just 2 years old, he contracted smallpox, whose marks were conspicuous
during his childhood days. Ramanujan confronted a life of extreme poverty
during his younger days. His early studies were in different schools in
Kumbakonam from the age of five until entering the Town High School at the age
of 11 years. From his childhood, Ramanujan had a huge passion for mathematics.
So much so that at the age of 12 he had mastered trigonometry (SL Loney:
Plane Trigonometry) and developed many theorems on his own with no assistance.
Ramanujan was a precocious child and did very well in school and hardly evinced
any interest in any other activities or games other than his studies in maths.
While his friends played, Ramanujan engaged himself in nothing but academics.
The first sign of his extraordinary talent in maths was noticed when he was 13
years. It was when he began to work on his own on summing geometric and
arithmetic series, far beyond his class. He engaged himself in solving cubic
and other fairly complicated problems some of which he failed. He would engage
his teachers in some unorthodox questions. He once questioned a teacher, who
was teaching the class that any number divided by that number equals one. He
asked him whether zero divided by zero would be one.
A major turning point in the life of
Ramanujan came when he came across a mathematics book by G S Carr called
‘Synopsis of elementary results in pure mathematics’. This book was simply a
compilation of thousands of mathematical results, with most of the results not
explained properly with adequate proof. This book was written as an aid to
coaching English mathematics students facing the notoriously difficult Tripos
examination, which involved a great deal of rote memorization. However, this
book greatly influenced Ramanujan and it inspired him to pursue his passion in
maths with vigour and at a feverish pace. He worked through the book's results
and beyond at the cost of other subjects. The style and approach of Carr to
write the equations and solutions without giving mandated proof for the
problems and equations became a trade mark of Ramanujan, who had fallen in love
with this book. He worked extensively on his slate trying to find solutions to
the problems. It must be remembered that working on slate was a necessity for
him since he could not have afforded pen and paper, which is one reason why we
don’t know what went into the mind of Ramanujan while solving problems.
In 1904, Ramanujan joined Government Arts
College in Kumbakonam. Unfortunately, by now he was completely engrossed in
maths with Carr Synopsis, which left him no time for other subjects. The
outcome was on expected lines. He failed in all the subjects in college except
excelling in maths. The failure did not help his cause and he had to lose the
much needed scholarship that he had managed to get while joining the college.
Failure played on his mind and he ran away from home to Andhra Pradesh. On his
return he enrolled at Pachaiyappa’s college in Chennai. Here, too, he engaged
himself mostly in maths and couldn’t comprehend subjects like physiology and
once again failed in the BA Fine Arts exam. He had no way but to leave his
college, without attaining a degree. Failure in BA did not however deter him
from continuing his independent research in maths.
Failure in BA Fine Arts exams, resulted in
Ramanujan concentrating completely on his maths research and it was during this
period, 1903-1914, that Ramanujan meticulously kept a record of the final
results of his original research work in the form of entries in two large-sized
Note Books. He worked extensively on his slate and recorded his final results
in note books. With time his profound works in maths were beginning to be
noticed and he was lucky that he could show some of his recorded notebooks to
eminent citizens in the city; Diwan, Bahadur Ramachandra Rao, V. Ramaswamy Iyer
(Founder of Indian Mathematical Society) R. Narayana Iyer (Madras Port Trust),
and to many others to convince them of his abilities as a Mathematician. Fortunately,
it helped him get an employment at the Madras Port Trust as a clerk, at a
salary equivalent to about 25 Pounds a year. By this time, Ramanujan had
already established himself as a fairly well recognised scholar in maths and
reports of his unusual abilities had begun to spread far and wide. Fortunately,
his talent in maths came to the notice of Dr. G. T. Walker at the Madras
University and courtesy his influence, Ramanujan obtained a small scholarship,
which relieved him and set him free for research.
During this period Ramanujan published
several papers in influential Indian Mathematical journals. He also sent his
long list of complex theorems to three academics of Cambridge. One of them was
G H Hardy. Ramanujan had come across a book by Hardy titled ‘Orders of
Infinity’ which motivated him to write his now famous letter to Hardy.
Ramanujan’s life took a decisive turn in 1913, when his 10-page letter
containing 100 statements of theorems on infinite series, improper integrals,
continued fractions, and number theory, reached Professor GH Hardy. The letter
was a collection of Ramanujan’s self-derived equations and unproven theorems.
Hardy was perplexed to see this letter from an ordinary clerk from India who
had professed to have discovered some infinite series and had posted some 100
odd findings in his letter. Hardy knew that the letter warranted some merit but
was also sceptical. After his dinner he met his compatriot mathematician
Littlewood. Hardy mentioned to Littlewood some of the claims he had received in
the mail from an unknown Indian clerk who was working at the Madras Port Trust.
Hardy opined to Littlewood that some of the assertions made in the letter of
Ramanujan were well known, others could be proven, while some others they could
disprove. Hardy and Littlewood decided to have a relook of the letter, which
they did and agreed that many of the statements made in Ramanujan’s letter were
not only fascinating and unusual but also impossible to resolve.
Hardy and Littlewood continued their
discussion on this letter over the next couple of days and soon they were
convinced that the clerk who wrote this innocent letter must be a genius. Hardy
therefore replied to Ramanujan, encouraging him on his works. This was the
beginning of a series of letter exchanges between the two of them. Although
Hardy by now was sure that Ramanujan was an exceptional mathematician, however,
in spite of his amazing feats in mathematics, Hardy realised that Ramanujan
lacked the basic tools of the trade of a professional mathematician. Hardy knew
that if Ramanujan was to fulfil his potential, he had to have a solid
foundation in mathematics, which are normally possessed by the best of
Cambridge graduates. For this Hardy extended an invitation to Ramanujan to come
to Cambridge. Hardy was completely taken aback when Ramanujan could not make up
his mind to accept his invitation to come to Cambridge. He later realised that
as a Brahmin, Ramanujan was not expected to cross the ‘seven seas’. His mother
was totally opposed to the idea of Ramanujan’s sea voyage. But fortunately
Ramanujan could manage to convince his mother by invoking Goddess Namagiri to
get his mother’s consent. Hardy soon swung into action. He asked E.H Neville,
another fellow of Trinity College, who was on a trip to Madras, to secure
Ramanujan a scholarship from the University of Madras. Neville’s wrote in a
letter to the university that “the discovery of the genius of S. Ramanujan of
Madras promises to be the most interesting event of our time in the
mathematical world ..."
What followed next is now a legend that
has entered into the annals of history. Courtesy Hardy and Neville, Madras
University offered Ramanujan the first research scholarship of the University.
Hardy also ensured that Ramanujan was offered a scholarship of 250 pounds a
year for five years with 100 pounds for passage by ship and for initial outfit
to go to England in 1914. Ramanujan, at the age of 26 years, set sail to
England and reached Cambridge, just before the outbreak of World War I. In the
very first year of his stay in Cambridge- 1914 - Ramanujan made some path
breaking contributions. However, his journey to success faced major hurdles in
the initial part of his stay in Cambridge. Ramanujan who was majorly influenced
by Carr’s style of summing up his end findings and stating the formula for an
infinite series or such other mathematical problem, without assigning any
deductive solution, was something which did not please his mentor Hardy nor any
other mathematicians. This incident has been so beautifully depicted in a scene
in the film “Ramanujan The Man Who Knew Infinity”. The scene shows the
excitement of Ramanujan to publish his new findings and in his discussion with
Hardy, Ramanujan presents to him two of his note books adding to the 100
formulas, which he had already sent to him and other problems which he had been
corresponding with Hardy. One of this was an interesting series which was
mind boggling for Hardy to comprehend how Ramanujan could even visualise it.
The very equation looks quite deceptive and it goes as follows; 1+2+3+4+5......
= -1/12. Interestingly Hardy immediately had recognised this to be the theory
of analytic continuation (Riemann Zeta Function) from complex analysis. Hardy
was convinced that Ramanujan must learn to produce mathematical solutions for
the answers that so naturally come to Ramanujan so that other mathematicians
take him seriously. The scene shows a brief argument that ensues between the
two in which Ramanujan- role played by Dev Patel, is seen arguing with Hardy, a
role played in the film by Jeremy Irons. Ramanujan expresses his desire to
publish while Hardy insists that he must first learn to communicate his mind in
the exacting standards that befit the standards of Cambridge. Hardy advises
Ramanujan to take some basic lectures to learn a more formal language in maths
which he could use to communicate his findings to his fellow compatriots in
Cambridge.
With great difficulty and with support
from his compatriot Littlewood, Hardy manages to convince Ramanujan of his need
for a formal learning to be recognised in Cambridge. The scene in the film
beautifully depicts Hardy taking Ramanujan on a walk through the famous library
of Cambridge. He says to Ramanujan that if area of maths are to be challenged
you must give proof to the formulas and that too in a manner which are
understandable to the other mathematicians and for that structured learning in
maths is mandated for Ramanujan. The scene shows Hardy leading Ramanujan to the
hall where Newton’s Principia book is preserved. Hardy says Newton produced
this monumental findings, which took long time for the people to understand.
Hardy further says to Ramanujan that if you provide proof to your formulas,
which you have noted in your note books, the time will not be far when your
note books could find a place in this very precinct of Cambridge where Newton’s
book finds his place. Hardy’s prophetic vision has been proved right.
Ramanujan’s lost note books have finally made their way to Cambridge and are in
the prized possession Cambridge even today. The scene so beautifully and with
that artistic elegance, which beholds the Hollywood, depicts the dichotomy of
relationship that existed between Hardy and Ramanujan.
Ramanujan and Hardy had one of the most
productive collaborations ever and during his five years of stay in Cambridge,
Ramanujan wrote some 30 papers some on his own and some jointly with Hardy.
Most of his works transformed the field of mathematics. On his way to
achievements in maths, Ramanujan had to overcome severe hardships, sense of
racism, difficulties of the World War 1, effects on his uncompromising vegetarian
habits and many more. Hardy was no doubt a great mentor for Ramanujan but then
there is an argument that he may not have shown that much needed humanitarian
considerations for Ramanujan, which may have ensured a better living condition
for Ramanujan in England. But then these thoughts are an afterthought,
which have little or no meaning.
Ramanujan’s contributions were soon recognised by
his compatriots in Cambridge and he was befittingly elected ‘Fellow of Trinity
College’, Cambridge, even though he didn’t have a formal college degree. But
unfortunately his stay at Cambridge was the harshest for Ramanujan. He was a
strict vegetarian and he remained uncompromising about his dietetic observance.
The World War 1 too did not help his cause, which made availability of choice
materials for Ramanujan very scarce. Ramanujan always cooked his own food and
most often neglected his health. During his five years stay in England,
Ramanujan was constantly mentored by Hardy and he cemented a long outstanding
partnership with Hardy. It was during this stay in England that Ramanujan was
awarded a BSc (later renamed PhD). In 1918, Ramanujan was elected as a Fellow
of the Royal Society (FRS), as a Research student in Mathematics, Distinguished
as a pure mathematician, particularly for his investigations in elliptic
functions and the theory of numbers. A rare honour for the not so formally
educated, that too at a very young age. Ramanujan was also elected to the
Trinity College Fellowship, in Oct. 1918 (a prize fellowship worth 250 pounds a
year for six years with no duties or condition). Most unfortunately he was not
destined to make full use of this fellowship.
His health kept deteriorating in England
and Ramanujan was often times admitted to hospital. When his health improved
slightly he preferred to return back to India. Ramanujan returned to India in
1919, after World War I, as a highly acclaimed mathematician. But years of stay
in an unfamiliar climate in England clubbed with his uncompromising life style,
had taken a heavy toll on his health. On his return to India, Ramanujan
was diagnosed with tuberculosis and the disease had already taken a heavy toll
on his health and he was continuing to be very weak and fragile. All through
this period, Ramanujan continued to be associated with his works and exchanged
letters with Hardy. In January 1920, he wrote the last letter to Hardy about
his discovery of “Mock Theta functions” another master class contribution from
a man who was almost in his death bed. In his last letter to Hardy, Ramanujan speaks of a class of very interesting functions,
which he describes as "mock theta functions". In his quintessential
style Ramanujan gives 17 examples of the mick theta functions but provides no
precise definition. However, we now know that by "theta function"
Ramanujan actually meant what we call today a "modular form" and by
"mock" something "whimsical". His 17 functions, which he
described in his last letter, Ramanujan had described properties, which are
analogous to those of usual modular functions, which do not belong to any known
class. The mystery Ramanujan’s Mock Theta function was later solved in 2002 by
Sander Zwegers in his Ph.D.
Ramanujan succumbed to his long standing
ailments and died on 26th April, 1920. In his small lifetime Ramanujan compiled
more than 3,000 results on equations and identities, many of them have been posthumously
proven right. His ‘Lost Notebook’ was found in 1976 by Prof. George Andrews of
Pennsylvania State University, and its facsimile edition was brought out by
Narosa Publishing House in 1987, on the occasion of Ramanujan’s birth
centenary. Besides his published work, Ramanujan has left behind several
notebooks, which have been the subject of academic discussion and study even
today.
In his short life of little over 32 years,
Ramanujan scaled unimaginable heights. What is so very unique is that his
mathematics, done over a hundred years ago, finds applications today in areas
other than pure mathematics, which were not even established during his time.
Two among these are signal processing and Black Hole physics. What could have been the contributions of
Ramanujan if conditions were more conducive to him and if he had lived longer
is something which will continue to remain in the realms of speculation. It was
a befitting tribute that the legendary Ramanujan’s life and works were chosen
for making of the Dev Patel starrer Hollywood film ‘The Man Who Knew
Infinity’ (2015), which has perpetuated his genius to the world audience. May
he continue to inspire millions of Indians and global students.
Rest in Peace
Ramanujan in the heavenly abode of your Goddess Namagiri.
