Tuesday, July 18, 2017

Srinivasa Ramanujan, the Indian mathematician



Srinivasa Ramanujan was a mathematician par excellence. He was one superlative mathematician whom  India has produced in the last thousand years. His leaps of intuition confound mathematicians even today nine decades after his death. His papers are still plumbed for their secrets. He made significant contribution to the analytical theory of numbers and worked on elliptic functions, continued fractions, and infinite series. His theorems are applied in areas as polymer chemistry, computers, even cancer. He turned mathematics upside down and rose from utter obscurity to pinnacle of success in that field.


He was born in Erode in Tamil Nadu on December 22, 1887. His father was a clerk in a cloth shop. From his early childhood he showed evidence of being a prodigy. Senior students would have difficulties in mathematics solved by him. When he was 13 he got from a college library Loney's Trigonometry. He mastered this tough tome and began his research. He evolved mathematical theorems and formulae not given in the book. His life took a decisive turn after he obtained G.S.Carr's book titled " A Synopsis of Elementary Results in Pure and Applied Mathematics". This book triggered the mathematical genius in him. The book -- a compilation of thousands of mathematical results, set down with little or no indication of proof-- generated his interest in mathematics and he worked through the book's results and beyond. By 1904 Ramanujan had begun to undertake deep research. He investigated the series (1/n) and calculated Euler's constant to 15 decimal places. He began to study the Bernoulli numbers, although this was entirely his own independent discovery. He was given a scholarship to the Government College in Kumbakonam which he entered in 1904. Since he neglected his other subjects at the cost of mathematics he failed in college examination and dropped out of the college.


 When his father found the boy always scribbling numbers and not doing much else, he thought his son had gone mad. To set him right, he forced his son to marry a eight year old Janaki. Unkempt and uncouth, he would visit offices, showing his frayed notebooks and declaring his knowledge of mathematics. He begged for a clerical job. He found someone who was impressed by his  Notebooks on mathematics. Francis Spring, Director of the Madras Port Trust recognized the lad's worth and gave him a clerical job on a monthly salary of Rs.25/-In 1913. The University of Madras granted him a fellowship of Rs 75/- a month though he had no qualifying degree.


Then, things took a swift turn. Ramanujan had sent a letter to the great mathematician G.H.Hardy, of Cambridge University, in which he set out 120 theorems and formulae. He had seen a copy of Hardy's 1910 book "Orders of Infinity". Hardy, together with another mathematician Littlewood, studied the long list of unproved theorems which Ramanujan enclosed with his letter. Hardy was so much impressed that he brought Ramanujan , in 1914, to  Trinity  College,  Cambridge, to begin an extraordinary collaboration. This collaboration with Hardy led to important results.


 Ramanujan found himself a stranger at  Cambridge. The cold was unbearable and he had to cook his own food being a Brahmin and vegetarian. The outbreak of World War I made obtaining special items of food harder and it was not long before Ramanujan had serious health problems. However, he pursued his research in mathematics with determination. He played with numbers, as a child would with a toy. His innate genius led him to mathematical truths. The job of proving them, he left to lesser mortals. He independently discovered results of Gauss, Kummer and others on hypergeometric series. His own work on partial sums and products of hypergeometric series have led to major development in the topic.


 Ramanujan fell seriously ill in 1917 and his doctors feared that he would die. He did improve a little by September but spent most of his time in various nursing homes. On February 18, 1918 Ramanujan was elected a Fellow of the Cambridge Philosophical Society and later he was also elected as a Fellow of the Royal Society of London.


 Ramanujan sailed to India on 27 February 1919 arriving on 13 March. His health was very poor and, despite medical treatment, he died on April 6, 1920. 



Note on Prof.G.H. Hardy




                                 G.H.Hardy     S.Ramanujan  


Godfrey Harold ("G. H.") Hardy FRS ( 1877 –  1947) was an English  mathematician , known for his achievements in number theory  and mathematical analysis.


He is usually known by those outside the field of mathematics for his essay from 1940 on the aesthetics of mathematics, A Mathematician's Apology, which is often considered one of the best insights into the mind of a working mathematician written for the layman.



 In an interview by Paul Erdos, when Hardy was asked what his greatest contribution to mathematics was, Hardy unhesitatingly replied that it was the discovery of Ramanujan . He called their collaboration "the one romantic incident in my life."


                                           A  true story.


                1729 is the natural number  following 1728 and preceding 1730. 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 Srinivasa 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 unfavourable 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 these:

1729 = (12 x 12 x 12) + (1 x 1 x 1)

as well as

(9 x 9 x 9) + (10 x 10 x 10)




His companionship with Hardy was all about contradictions. Hardy was an atheist, who followed logic and sense, while Ramanujan was religious and intuitive. Many a times, Hardy was unable to figure out how Ramanujan, a quiet, pleasant and dignified man, could rely on intuition to come up with theories. Often,Ramanujan would attribute his abilities to Goddess Mahalakshmi, saying that he would see visions of complex equations on a scroll in his dreams. Together, Hardy and Ramanujan had many accomplishments that changed the world of mathematics.


Years after his death, his family home in Sarangapani Street, Chennai, was turned into a museum. An international journal called  The Ramanujan Journal was launched, to publish all research and work related to his findings.

The Man Who Knew Infinity – Robert Kanigel---is one of the best books on Ramanujan. Kanigel explains in two pages how to make a South Indian Dosai.


There is a Hollywood movie " Man Who Knew Infinity" which  stars Dev Patel as Ramanujan and Jeremy Irons as G. H. Hardy.Worth seeing.There is an Indian film too but it has not reached Mumbai. Request Chennai friends to enlighten us.