Srinivasa Ramanujan (1887-1920) was one of India’s greatest
mathematical figures. He was a child prodigy, with a natural instinct
for mathematics. As though he could see how everything comes together.
He was particularly interested in modular functions and number theory.
This essay is divided into two sections. In the first section I will
report on Ramanujan’s life. I have decided to take a unique approach,
and instead of reporting solely based on biographies (and biographies
about Ramanujan are a-plenty) I will be analyzing letters he wrote.
These letters have been stored in the National Archives in Delhi, the
Archives of the State of Tamil Nadu, and in collections of various
mathematicians with whom Ramanujan corresponded. They were compiled by
Bruce Berndt and Robert Rankin in their book Ramanujan, Letters and
Commentary. I believe these letters will give me a “first source” not
only pertaining to Ramanujan’s life but also a more profound view of
his thoughts, his personality. The letters are my main source for this
essay, however I do use a couple of good biographies to enhance the
information I got from the letters and to fill in some gaps. And while
in this biographic section I will survey some of Ramanujan’s
mathematical ideas, I will not go very deep into them. This I will do
in the second section in which I will focus on a few of Ramanujan’s
mathematical ideas. In the last section, I will use Mathematica to
compute and verify some of Ramanujan’s theorems from the second section.
Ramanujan was born on the 22nd of December, 1887 in his Maternal
grandmother’s house in Erode. Erode is a small town approximately 250
miles south west of Madras (see map). At the age of 1, Ramanujan’s
mother took him to her home in Kumakonam (160 miles from Madras) where
her husband, Kuppuswamy Srinivasa Ayingar was a clerk in a cloth
merchant shop. Ramanujan inherited his first name, “Srinivasa” from his
father.1 I was curious as to what the name “Srinivasa” means. A quick
online search revealed that Srinivasa means “The Adobe of Sri”, the
Goddess of Fortune in Sanskrit.
From 1892 to 1898 Ramanujan studied in several different primary
schools in Kumbakonam, and in 1898 he enrolled in the town high school.
at that point he had begun to realize his passion for mathematics- he
became more and more absorbed in the world of numbers and mathematics.
This came at a price- Ramanujan started neglecting all other school
subjects, and in 1905 he dropped from school and ran away to
Vizagapatnam, a town 400 miles north of Madras.
The one book that really influenced Ramanujan and ignited his
mathematical flames was A Synopsis of Elementary Results in Pure and
Applied Mathematics by G. S. Carr. The book was mostly a collection of
approximately 6,000 theorems of Algebra, Calculus, Trigonometry and
Analytical Geometry. Most of the theorems had no proofs, they were just
listed “as-is”. Ramanujan proceeded to test many of those theorems,
and come up with new theorems of his own. One effect this book had on
Ramanujan is that it molded Ramanujan’s own style of listing theorems
without resorting to proving them. Contrast this to mathematicians such
as Euclid who were heavy on proofs.
In 1906, Ramanujan attempted to enter the University of Madras. He
took the “First Arts Examination” which was an entry exam which would
have allowed him to get enrolled in the University. However Ramanujan
failed the exam (all topics, except for mathematics) and was denied
admission. Ramanujan kept studying mathematics on his own, including
continuing to analyze the theorems from Carr’s book. In a letter
applying for a job as an accountant in the local port, Ramanujan
reports “I have passed the Matriculation Examination and studied up to
the F.A. but was prevented from pursuing my studies further… I have,
however, been devoting all my time to Mathematics…”
Ramanujan met with different Indian scholars and by 1912 the word
about him spread and his name started to become well know in the
mathematics field in India. In May 1913, with the help of Rao
Ramachandra, a famous Indian scholar and president of the Indian
Mathematics society, Ramanujan secured a research scholarship at the
University of Madras. In the same year, Ramanujan sends a letter to G.
H. Hardy, professor in the University of Cambridge and a famous British
Mathematician and a who has already established a name for himself in
the mathematics literature. This letter ignites an interesting
partnership between the two. In his letter, Ramanujan introduces
himself to Hardy. He tries hard to explain why Hardy should listen to
him despite Ramanujan not having University Education. Ramanujan
emphasizes that while he had no University education, he still had
undergone the ordinary school but then decided to strike a new path for
himself, rather than trod through the conventional university
curriculum. He mentions some theorems which he found himself that are
not taught in the conventional university coursework. He also stresses
how he has devoted himself to mathematics in all of his spare time. He
states that local mathematicians of whom he had shown his investigation
on divergent series had termed his results “startling”.
Ramanujan then proceeds to respond to a paper Hardy wrote titled Orders
of Infinity. In particular, Ramanujan refers to this problem: Given a
number x, how many prime numbers can be found that are less than x? For
example, if x = 20, the answer would be 7 since there are 7 prime
numbers less than 20: 2, 3, 7, 11, 13 and 17 and 19. According to
Hardy, an expression for x has yet to be found. Ramanujan, however,
managed to find a close enough approximation, and he asks Hardy to
review it and other theorems, and help Ramanujan get his theorems
published (Ramanujan mentions that he is poor and thus doesn’t have the
funds to get his works out). 7This letter caught Hardy’s attention,
however Hardy was still a bit skeptical at first since after all, how
could a poor man from India with no academic training show such
mathematical talent? Also, many of the theorems in the letter were
provided with no proof.
Hardy asks Ramanujan for more explanation and proofs. Ramanujan
responds with more explanations about his theorems, and asks Hardy to
verify the results. From the tone of the letter I believe Ramanujan
realizes that this is the opportunity he had been waiting for. Previous
letters Ramanujan sent to different British mathematicians he received
a negative answer, this was the only time in which a prestigious
British mathematician took interest in Ramanujan’s work and asked for
more information. Ramanujan asks Hardy for a chance, and also honestly
confides to Hardy that his main motivation for this letter was having
food on his plate, for if he is hungry it is harder for him to do the
math he loves.
The two letters convince Hardy. Hardy had also consulted with some
colleagues at his University and they were all impressed with
Ramanujan’s mathematical talent. Hardy writes to Ramanujan that even
though proofs are either missing, inaccurate or incomplete, his work
appears to be right and is very remarkable. Ramanujan writes to Hardy
that he sees him as a “sympathetic friend”. Hardy arranges to bring
Ramanujan to England under a scholarship from the University of
Over the course of the next 6 years, Hardy and Ramanujan collaborated
and published research together. They were a good complement to each
other, for Ramanujan was an “Intuitive” mathematician who liked to find
new theorems, while Hardy, according to his letters, cared a lot about
“Rigorous proofs”. Hardy helped Ramanujan expand and prove many of his
At that period it also became apparent that Ramanujan is ill. In many
of the letters which Ramanujan wrote during that period, he states that
he was not feeling so well. He was suffering from Tuberculosis, a lung
disease that was hard to treat at that period. Perhaps all of
Ramanujan’s efforts and over straining himself with conjuring
mathematical theorems catalyzed the deterioration of his health. On
1919 Ramanujan returned back to India/ on April 29, 1920 a letter was
sent from the registrar of the University of Madras to Hardy, informing
Hardy of Ramanujan’s death which occurred on April 26. Ramanujan had
succumbed to his illness and died on the young age of 33.
Ramanujan seemed to be interested in theorems rather then in proofs
thereof. He saw properties in numbers which others could not see. He
wrote his theorems moving from one to another without devoting much to
proofs. For example, in the first letter Ramanujan wrote to Hardy he
included 9 pages full of different theorems.
As shown in this essay, Ramanujan was a genius with a intuition and
imagination when it came to numbers. He was not heavy on proofs, but
rather enjoyed writing examples and theorems. I will end this essay
with a story told about him by Hardy. Ramanujan was lying sick in the
hospital bed, when Hardy dropped for a visit. Hardy told Ramanujan how
he took a cab to the hospital, and noticed that the cab number was
“1729”. Hardy tried hard to think about any interesting properties
about that number but could not do so, reaching the conclusion that
1729 is “rather a dull number” Ramanujan, immediately hearing that
number, even though he was not in good health, immediately managed to
come up with an interesting property for this number. He told hardy how
the 1729 was not a dull number but rather a very interesting one. It
is the smallest number which can be expressed as a sum of 2 cubes in 2
different ways. for example, is equal to 35, but there are no other
two numbers of which the sum of cubes equals 35. However, for the
number 1729, we have and both of which are equal to 1729. This is a
perfect example illustrating Ramanujan’s magic and intuition with
1.Brendt, Bruce C. and Rankin, Robert A., Ramanujan: Letters and Commentary, Rhode Island: American Mathematical Society, 1995.
2.Clawson, Calvin C., Mathematical Mysteries: The Beauty and Magic of Numbers, New York: Perseus Books, 1999.
3.Kanigel, Robert, The Man Who Knew Infinity: A Life of the Genius Ramanujan, New York: Charles Scribner’s Sons, 1991.
4.Ranganathan, S. R., Ramanujan: The Man and the Mathematician, London: Asia Publishing House, 1967.