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V.S. Ravi

Jun 05, 2022, 03:57 PM | Updated 03:57 PM IST

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"*For sheer mathematical ability in tangled algebra, Euler and Jacobi have had no rival unless it be the Indian Mathematical genius, Sreenivasa Ramanujan in our own century*".

E.T.Bell in *Men of Mathematics*.

Even those who have no knowledge of mathematics would be thrilled to their core to know about the Indian mathematical genius Ramanujan, and his achievements.

Ramanujan was the greatest mathematician of the 20th century. His intellectual feats are the stuff of legend.

He was born to poor orthodox parents in the obscure town called Kumbakonam, in Tamil Nadu on 22 December 1887.

He was a brilliant student at school. When he passed out, his headmaster introduced him as an outstanding student who deserved scores higher than the maximum possible marks. He received a scholarship for college education.

But at college he lost interest in all subjects except mathematics. As a result he failed in most and lost his scholarship. He enrolled in other colleges but again he excelled in mathematics, but failed in most others. As a result, he could not get any ‘certificate’ in education beyond school.

Day and night he would do nothing but mathematics. He would forget to eat his meals, and would even forego his sleep in the pursuit of mathematics. His mother and wife had to feed him while he pursued his passion for that subject.

Being a married person, the responsibility of looking after his parents and his wife fell on him. He had to earn a living to sustain his family.

But who would give him a job? His work was of no value, and was incomprehensible even to Indian mathematicians. He became so poor that he had no paper to work on and would do all his math on a slate; he would only write down the results in paper.

His humility can be gauged from his letter to the Port Trust dated 9 February 1912, seeking employment as a clerk. 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.*"

He was appointed as a clerk.

Even as a clerk in the port trust Ramanujan indulged in his passion - mathematics.

With the help of his friends, Ramanujan wrote to two British mathematicians H.F.Baker and E.W.Hobson, expecting replies from then and a positive response of an audience with them. However, to Ramanujan's great disappointment, these two professors returned Ramanujan’s papers without any comments.

Then Ramanujan wrote to G. H. Hardy, the famous British mathematician at Cambridge university, England

And and this is how the letter began (*I have avoided mathematical symbols in a few places to make the letter easily understandable to readers not familiar with pure mathematics).*

This letter is now regarded as the most famous letter in the history of mathematics

Letter from S. Ramanujan to G.H. Hardy (16 January 1913)

Dear Sir,

*I beg to introduce myself to you as a clerk in the Accounts Department of the Port Trust Office at Madras on a salary of only £20 per annum. I am now about 23 years of age. I have had no University education but I have 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 the conventional regular course which is followed in a University course, 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 the local mathematicians as "startling"*

*Just as in elementary mathematics you give a meaning to n is negative and fractional to conform to the law which holds when n is a positive integer, similarly the whole of my investigations proceed on giving a meaning to Eulerian Second Integral for all values of n.*

*My friends who have gone through the regular course of University education tell me that [ eq1] is true only when n is positive. They say that this integral relation is not true when n is negative. Supposing this is true only for positive values of n and also supposing the definition [ eq. 2] to be universally true, I have given meanings to these integrals and under the conditions I state the integral is true for all values of n negative and fractional. My whole investigations are based upon this and I have been developing this to a remarkable extent so much so that the local mathematicians are not able to understand me in my higher flights. Very recently I came across a tract published by you styled Order of Infinity in page 36 of which I find a statement that no definite expression has been as yet found for the number of prime numbers less than any given number. I have found an expression which very nearly approximates to the real result, the error being negligible.*

*I would request you to go through the enclosed papers. Being poor, if you are convinced that there is anything of value I would like to have my theorems published. I have not given the actual investigations nor the expressions that I get but I have indicated the lines on which I proceed. Being inexperienced I would very highly value any advice you give me. Requesting to be excused for the trouble I give you.*

*I remain, Dear Sir, Yours truly,*

*S. Ramanujan*

*P.S. My address is S.Ramanujan, Clerk Accounts*

From Hardy's point of view, some insignificant clerk from India had approached him, someone who claimed to have had no university training but who asserted to have made startling new discoveries in mathematics. By the second paragraph of his letter, Ramanujan was insisting he could give meaning to negative values of the gamma function.

By the third he was disputing an assertion made by Hardy in his work called *Orders of Infinity*. Ramanujan claimed to have found an approximation for the prime counting function π(x) using which the error is negligible. In other words he was challenging the prime number theorem - the best approximation known then.

Now it may be mentioned that Hardy used to receive a large number of such letters from all over the world, almost every day. Therefore Hardy only gave Ramanujan's letter a cursory glance at first and did not pay much attention to the mathematical theorems it contained, thinking that the author of the letter was either a charlatan, or a lunatic.

But Hardy later recalled that when he had gone to play Tennis, the theorems haunted him. On returning home Hardy read the letter with attention. When he was half-way through, Hardy realised that he was staring at the theorems written down by an extraordinary genius.

The beauty of Ramanujan's theorems lay in the fact that he gave out the final conclusion of each theorem or equation by a stroke of genius without giving the preliminary steps. This was a feat beyond the comprehension of Hardy himself.

Hardy showed the letter to his colleague Littlewood who shared his views.

The first theorem had already been determined by G.Bauer in 1859. The second was new to Hardy, and was derived from a class of functions called hypo geometric series which had first been researched by Euler and Gauss. Hardy found these results "much more intriguing" than Gauss's work on integrals. After seeing Ramanujan's theorems on continued fractions from the last page of the manuscripts, Hardy said the theorems "defeated me completely; I had never seen anything in the least like them before",and that *they "must be true, because, if they were not true, no one would have the imagination to invent them. Hardy did not waste any time in inviting Ramanujan to Cambridge University,*

Even though his mathematical spirits ran high, Ramanujan’s mind and body were not at ease in England. He was an orthodox Hindu and could not adapt himself to foreign conditions - be it food, clothing or climate.

He was a strict vegetarian and would not eat in the college dining hall fearing that the cooks might cook meat and vegetables in the same vessels. He had no option but to cook his own food himself. But with his family members not being there he would cook at very odd times or skip meals altogether. Also the available food did not suit him.

In the biting cold weather, Ramanujan would bathe twice a day, he would start his day with his fastidious morning rituals in his traditional clothes and only then proceed to work. Once his friend Mahalanobis found him shivering in his room, and showed him how to use the electric blanket.

He was never comfortable in Western suits.

And in these matters nobody could understand or support him, not even Hardy.

His relationship with Hardy was just confined to mathematics, though in actual fact they had developed affection for each other.

To make thing worse, Ramanujan had arrived during the worst part of the century. The World War I raged on. Cambridge university had become more of a military ground. Many students and professors left to join the war, Littlewood was one of them. Ramanujan greatly missed the company.of his friends

A deeply religious Hindu , Ramanujan credited his substantial mathematical capacities to divinity and said the mathematical knowledge he displayed was revealed to him in his dreams by his family goddess Namagiri Thayar.

He once said, "An equation for me has no meaning unless it expresses a thought of God”. The Goddess Namagiri, whom he worshiped, gave him flashes of mathematical insights and without her counsel he could not take any step. He claimed to be just the agent, God the real doer.

The brilliant American mathematician Andrews of Pennsylvania State University accidentally discovered the" lost notebook" of Ramanujan.

In the spring of 1976, George Andrews of Pennsylvania State University visited the library at Trinity College, Cambridge, to examine the papers of the late G.N.Watson. Among these papers, Andrews discovered a sheaf of 138 pages in the handwriting of Srinivasa Ramanujan.

When he was at Pennsylvania University, I had an opportunity to meet him. My cousin Krishna, a brilliat mathematician who does research in pure mathematics introduced me to him. Thus I heard from the horse's mouth the story of the discovery of the lost notebook, one of the greatest moments in the history of mathematics.

Ramanujan's theorems have many practical uses e.g guided Missiles. Ramanujan developed an important mathematical concept called Mock theta function. Ramanujan wrote down ten mock theta functions of order 5 in his 1920 death-bed letter to Hardy, and stated some relations between them that were proved by Watson. In his "lost notebook" he stated some further identities relating these functions, equivalent to the mock theta conjectures that were proved by another mathematician Hickerson.

Andrews found representations of many of these functions as the quotient of an indefinite theta series by modular forms of weight 1/2. *These are extremely complex mathematical concepts, that require rigorous study of and insight into pure mathematics, which I must confess I don't possess.Therefore it is not unlikely that, here and there, I have committed a few errors , while outlining the concepts. However I would be more than happy if I have succeeded in transmitting the electromagnetic power of his equations*

In December 1917, Ramanujan was elected to the London Mathematical Society. He became a Fellow of the Royal Society in 1918. He was elected for his investigation in Elliptic functions and the Theory of Numbers. It was Hardy who nominated him.

On 13 October 1918, he became the first Indian to be elected a Fellow of Trinity College, Cambridge.

Cambridge mathematician Bela Bollobas noted, “I am sure there are quite a few people who could have played Hardy’s role. But Ramanujan's role in that particular partnership I don’t think could have been played by anybody else."

About his work, Littlewood had written to Hardy, “I can believe that he’s at least a Jacobi.” Hardy would rate him even higher: “I have never met his equal, I can compare him only with Euler or Jacobi".

On a scale out of 100 of natural mathematical ability, Hardy assigned himself 25, Littlewood 30, and to David Hilbert, the most eminent mathematician of his day, he assigned 80. To Ramanujan he assigned 100.

Ramanujan suffered from two major ailments. He seems to have picked up Tuberculosis in England and Hepatic Amoebiasis in India. In those days no cure was available for either of these diseases unless caught early. The mind boggles at the though of how much he would have achieved had he lived up to a ripe old age.

So Ramanujan was withering away with no hope of recovery. Once when Hardy had gone to see an ailing Ramanujan, just before his death, Hardy casually remarked that he had come by car. Ramanujan desired to know the number of the car. Hardy said it was a very insignificant number 1729. "No, no, Hardee " Ramanujan said, "It is a number which is unusually significant. It is the smallest number which can be expressed as the sum of two cubes, in two ways-- 12 3 +1 3 and again as 10 3 + 9 3 ". Hardy was flabbergasted and stunned.

Ramanujan's theorems produced by flashes of mathematical insight will keep mathematicians busy for centuries. As physicist and mathematician Freeman Dyson would say many years later, after being introduced to Ramanujan’s work on congruence properties of the partition function, “He [Ramanujan] discovered so much, and yet he left so much more in his garden for other people to discover. In the forty-four years since that happy day [introduction to Ramanujan’s work], I have intermittently been coming back to Ramanujan’s garden. Every time when I come back, I find fresh flowers”.

V.S.Ravi is a distinguished and highly decorated IPS officer having served both the Government of AP and the Government of India, for 35 years. He retired in 1998. He is a scion of the Alladi family, being a grandson of the Late Sir Alladi Krishnaswamy Iyer, one of the Chief architects of the Constitution . Sri Ravi is one of the foremost authorities on Shakespeare in the country. He has contributed articles on Shakespeare to the Hindu and News Time Now. He passed Physics (Hons) with distinction and he has kept himself in touch with the latest developments in science and technology.

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