Srinivasa Ramanujan

The Mathematical Genius

Prof. K. R RAJAGOPALAN

The time is sometime at the beginning of this century. In a class of eleven or twelve-year olds, the teacher is explaining the concept of division. “If three fruits are divided among three boys, each gets one fruit; if 5 fruits are divided among five, then each gets one. So, if a number is divided by itself, the quotient is always 1.” Quite ordinary! So far, yet. Now jumps up a dark-complexioned, stocky boy asking, “Sir, if zero mangoes are divided among zero boys, would each get one?” The teacher is upset, naturally, and admonishes him, “Do not ask silly questions; zero has no value and so no division of fruits is, possible.” The matter does not end there either. The Boy persists, “Sir, zero has value; when you put one zero, to the right of 3, we get 30; two zeroes gives us 300. So how can you say that zero has no value?” The discussion must have stopped there, as the teacher had no reply to that. This incident happened in Kumbhakonam, a pilgrim centre in Tamil Nadu, wherein there are a large number of ancient Siva and Vishnu temples; and a Kumbhamela called “Mahamakham” is held once in twelve years.

Another classroom scene at college level, fifteen years or so later. A British professor is working out a fairly tough problem on the blackboard. There is an Indian student who tells the teacher that he has committed an error; also adds that the problem could be worked out more easily. The professor is startled, asks superciliously the student to come and exhibit his “superior knowledge.” There are whispers and derisive jeers from the classmates also. But the dark, stodgy Indian, walks unruffled to the board as if he did not notice all these, and solves the problem in fewer steps!

In both cases, the student was Srinivasa Ramanujan–fondly called by his mother and friends as “Chinnaswamy.” Srinivasa Ramanujan, one of the greatest mathematicians that this country has produced, and – as has recently been stated at a seminar in honour of his centenary in U. S. – one of the great names among world mathematicians also, was born at Erode on December 22, 1887. His centenary year would hence be from 22-12-’86 to 21-12-’87.

The date 22, is the product of the only even prime number 2 and the first two-digit prime number 11. (Ramanujan’s special interest was in the theory of prime numbers.) He was born in the Tamil month of Margazhi, a month which is supposed to be the “most superior” of all the months. (Lord Krishna says in the Bhagavadgeetathat He is Margasirsha among months.) The year of birth, according to the Indian cycle of sixty years, was Sarvajit, which means “all-conquering”. The day was Thursday, “Guruwara.” (Guru, meaning the biggest. Jupiter (Guru) is the biggest of all planets.) Since the Visishtadwaitic philosopher Ramanujacharya was also born on a Thursday and since this family were Vaishnavaite by persuasion, the parents named the child as Ramanujan. His father’s name was Srinivasan. By standards of those years Ramanujan should be considered fairly tall, being five and half feet in height (165 cms).

If Ramanujan showed his precocity in mathematics at a fairly early age, Gauss (1777-1855) is known to have exhibited his extraordinary powers of calculation even at an earlier age of three! His father was working on the wage bill of the labourers under his charge. The child Gauss was looking over the figures and found a mistake in the calculations, which was duly rectified.

The influence of the father on Ramanujan appears to be very slight, while that of his mother was deep and abiding. She was herself quite knowledgeable in astrology and the Puranas; she was also a very devout Bhakta of Lord Narasimha of Namakkal and Namagiri Devi. Namakkal, a small town in Salem District, is famous for the Pallava style of rock-cut temple of Lord Narasimha, the fourth man-lion incarnation of Vishnu; and also for the larger-than-life statue of the supreme devotee Anjaneya who stands in mighty grandeur with folded hands looking at his Lord Narasimha, with the sky alone as roof. Chinnaswamy inherited his love of the epics and the devotion to the Lord from his mother and throughout his life was quite a religious person. Many times during dreams, he used to get visions of scrolls of mathematics before his eyes and would, on waking up, try to write them in his note-books.

He had few friends in his early life, being more interested in working out mathematics rather than playing with his classmates or children of the same age group. But he used to assist many of his fellow-pupils in getting their difficulties solved in tackling “hard” problems. When he was in the eighth class, he had mastered Loney’s Trigonometry which is taught at the undergraduate level even today. Some college students also used to seek his assistance for their doubts in mathematics. This type of giving “tuitions” stood him in good stead in later years when he had to make money to meet the Spartan needs of himself and family.

Ramanujan’s introduction to higher mathematics might have begun when he secured a copy of Carr’s “Synopsis of elementary results in pure and applied mathematics” published in 1880 in England. This is the first of Carr’s two books and contained merely the statements of more than five thousand results in many branches of mathematics. Ramanujan appears to have been influenced very much by this book, because, he followed the same pattern of writing only the results he found in his note-books. He has never given any explanation or hint as to how he arrived at them.

His educational career can be briefly given. He passed his matriculation examination in first class in 1903 from Kumbbakonam and so became eligible for a scholarship to join the F. A. class. He failed in that class, as he could not cope up with the history of Greece and Rome which was part of the course. After sometime he again tried the same examination at Pachaiappa’s College, Madras, and again failed in 1907. Here ended his formal education when he was twenty years old.

In the meantime, as was the custom of those days, Ramanujan was married to Janakiammal (11 years old). The wedding was arranged by the mother only, and the father refused to attend even the wedding ceremony.

For the next three or four years, Ramanujan tried his best to secure a decent job which would fetch him and his family a decent living. He went from place to place, to Madras and to the far off Visakhapatnam. But to little avail. He managed to give tuition to a few students and lived with his friends for sometime. 

During all this while, he kept up his interest in mathematics deriving more and more results and meticulously writing them in his note-books. He showed some of them to a number of professors of mathematics in the various colleges of Madras and other places also. He met Prof. Ross of Madras Christian College (it was situated then in the city of Madras) but nothing came out of this meeting too. He met the founder-president of the Indian Mathematical Society, who encouraged him to write out his results more neatly, giving all the intervening steps. His first research paper on “Some properties of Bernoulli’s numbers” was published in December 1911 in the Journal of the Indian Mathematical Society. Perhaps this would be the first case of a research paper in the 20th century in mathematics, written by a person who did not have even undergraduate education.

This paper, perhaps, formed a turning point in his life. This paper was noticed by some professors one of whom gave him a recommendation letter. With this assistance, Ramanujan was able to secure a temporary clerical post in the Accountant General’s Office, Madras, in 1912, where he worked for a month or two. Later, he got another clerical post in the Madras Port Trust, where he worked from March 1912 to April 1913. From May onwards, upto March of next year, he was the first research scholar in mathematics at the University of Madras, where he submitted three quarterly reports a needed by the university. He left for England at the invitation of Prof. G. H. Hardy on 17th March 1914 and spent the next five years there working with such well-known mathematicians as Hardy and Littlewood. He fell ill and, because of that, returned to Madras, India, in April 1919. He never recovered from that dogged illness and finally succumbed to the dreaded tuberculosis on 26th April 1920. He thus lived for the short span of 32 years 4 months and 4 days.

Troubles and Travails

Ramanujan did have a very hard time, finding it difficult even to make both ends meet on many an occasion. He met quite a few “mathematicians” but none really helped. In facing such situations, Ramanujan is not unique. Many a famous mathematician also has had to face similar, if not more, poignant situations. A few examples might suffice. Laplace (1749-1827) went to D’Alembert with a recommendation letter, but the latter would not even see him. Of course, Laplace wrote to him later giving him some of the results in mechanics which he had found. Now D’Alembert had no hesitation to offer the post of professor of mathematics at the Military School of Paris. Ramanujan also wrote a letter enclosing some of his results to Prof. Hardy, who strived his best to give Ramanujan a scholarship at the University of Madras itself, before he could arrange for Ramanujan to go over to England to work directly under him.

Both Legendre and Cauchy did not properly evaluate the work of Abel (1802-1829) and so he suffered from poverty. Later, another mathematician Hermite opined that “Abel has left some­thing for other mathematicians to keep them busy for 500 years.” Galois (1811-1832) was another to die young and in difficulties. He too failed in the entrance examination twice–a case of “a candidate of superior intelligence being lost with an examiner of inferior intelligence.” Cauchy lost the paper that Galois submitted and Poisson perfunctorily read through it and pro­nounced it as “being incomprehensible.” Galois died in a duel, absolutely frustrated with his life. He is reported to have written: “If a carcass is needed to arouse people, I will donate mine!” Both Abel and Galois died in their ’Twenties.

Just as Ramanujan, quite a few other mathematicians were also born of humble parents and had no mathematical “lineage” on either side. Mention could be made of Laplace and Gauss in this connection.

His Mathematical Output

In a general article like this, it would not be possible to mention higher mathematics. But without mathematics, Ramanujan would be nowhere. One could mention that he wrote in all twenty-eight papers in various journals in India and abroad over a period of seven years from 1911-1919. Apart from these, he submitted three quarterly reports to the Madras University which were “rediscovered” only recently. These saw the light of day when B. C. Berndt edited them with details and explana­tions and published them in 1985 under the title “Ramanujan’s note-books – Part I”. These contain results which have been proved in a fairly rigorous manner. Berndt remarks, “It is rather remarkable that Ramanujan’s formulas are invariably correct even though his methods are generally without a sound theoretical foundation.” His amazing insights enabled him to determine when his formal arguments led to bonafide formulas and when they did not.”

We shall only give the properties of Ramanujan’s number 1729.

1729 = 1³ + 12³ = 9³ + 10³
1+ 7+2+9 = 19 and 19 is a factor of 1729
1729 = 19 X 91 (note reversal of digits)
1729 = 10³ + 25² + 10² + 2²

Using only three of the four digits excluding 2, an even number, we get 17, 71, 79, 97 which are all prime numbers.

He is supposed to have left behind him three note-books, full of results (on the model of Carr’s book referred to earlier). He used to carry them wherever he went. Hardy sent one note­book of Ramanujan to the Madras University after his death. This was transcribed neatly by one T. A. Satakopan, lecturer in mathematics of Madras Christian College and was again sent to Prof. Watson in England who delivered a series of lectures on Ramanujan’s works at the London Mathematical Society. It contained 134 pages systematically arranged into sixteen chapters, and the rest of 80 pages contained miscellaneous heterogeneous material. Satakopan’s copy contained 356 foolscap pages. Another set of four volumes were copied by Watson into 288 pages of his close handwriting and contained 21 chapters systematically arrang­ed and some thirty pages of heterogeneous material. They, together, contained some three to four thousand theorems. Ramanujan’s note-books (two of them) were issued in a facsimile edition by the Tata Institute of Fundamental Research, Bombay, in 1957 under the editorship of K. Chandrasekharan. The third note-book remained untraced for a long time. Recently (in the ’Eighties), an American mathematician is reported to have come across this book among the effects of an English mathematician when, after his death, the books were being gone through.

Quite a prodigious effort considering that Ramanujan had to pass through trying times when he wrote these note-books. Many of his results are being worked upon even now, during this centenary year; more mathematicians would be engaged in this task.

From a quick: survey done upto 1940, it was found that 104 papers had been published on what Ramanujan had done in journals in India and abroad.

A Few Anecdotes

Ramanujan was quite fond of listening to religious discourses and witnessing street-dramas or Terukkoothus. He never bothered about walking a few miles for such purposes. He was a traditiona­list sporting the Kudumi or tuft, wearing the “tenkalai” Namam on his forehead and going to the Sarangapani (Lord Vishnu in the reclining form) temple and other temples.

He had to move the tuft of hair, and wear Western clothes including waist-coat, tie, socks, shoes and a hat. He waited till his wife and mother left for Kumbhakonam and then only (so as not to hurt his mother’s sentiments) he had his head cropped. When he sent a copy of his photograph in Western clothes to his mother, she could not recognise him at all at first.

He was quite fond punning on words. When he was shifted to Chetpet, a suburb of Madras, he is reported to have said, “Now everything will be over quickly” (Chetpat). He died in Chetpet. His relatives were thinking of taking him to Tanjavur for possible recovery, when Ramanujan refused to go there saying it was “tan+za+ur” i. e., a place where he would die. “Why bother about death; it has the same number of letters as birth–five letters each.” (In Tamil and in English both the words birth and death have five letters each.)

His wife, who was quite young, asked him in her naivete to take her with him to England so that she could cook his food and look after him. Ramanujan, with his tongue in the cheek, told her that it would be very difficult for him to look after her as she, with her good looks and young age, would be carried away by the Englishmen over there!

The final words of his mother to him as he made preparations to sail were – “Do not lean out of the windows. You may fall into the sea.” (There were no ‘bars’ in the windows of railway compartments then and so, such advice would invariably be given to youngsters when they boarded a train and were travelling without an elderly escort.)

The manager of the Madras Port Trust, Narayana Iyer, became quite fond of Ramanujan and the two spent many an evening and night together in his house in Triplicane. Narayana Iyer was quite fascinated with the slate on which Ramanujan worked out his problems. When Ramanujan left for England, Narayana Iyer requested (and got) permission to keep his slate as a memento. It appears that it is still preserved by the grandson, a namesake.

We shall end this short article with a few of the encomiums showered on him during his life and after his death by great mathematicians.

“... his name will become one of the greatest in the history of mathematics...” Prof. Neville in his letter to the Registrar recommending Ramanujan for a scholarship to go abroad.

Hardy had rated all mathematicians on the basis of pure talent on a scale from zero to hundred. Hardy gives himself a rating of 25. Littlewood (a fellow-collaborator of Hardy and Ramanujan) a rating of 30, Hilbert 80 and Ramanujan 100.

“Ramanujan is ... one superlatively great mathematician whom India has produced in the last 1000 years” –Neville in 1941 quoted in Nature (1942).

“Ramanujan is the greatest mathematician of this century.”  –Julian Huxley.
“I learnt more from him than he from me. I would equate his genius only with those of Jacobi and Euter.” – Hardy.

“Ramanujan was a gift of God to the mathematical world.” – E. T. Bell

We, the believers of God in India, would wholeheartedly endorse the last statement.