Srinivasa Aiyangar Ramanujan (Tamil: ஸ்ரீனிவாஸ ஐயங்கார் ராமானுஜன்) (December 22, 1887 – April 26, 1920) was a groundbreaking Indian mathematician. A child prodigy, he was largely self-taught in mathematics.
Ramanujan mainly worked in analytical number theory and is famous for many summation formulas involving constants such as π, prime numbers and the partition function. Often, his formulae were stated without proof and were only later proven to be true.
Almost a century after his death, it was said of him, "Ramanujan was a mathematician so great that his name transcends jealousies, the one superlatively great mathematician whom India has produced in the last thousand years. His leaps of intuition confound mathematicians even today, seven decades after his death. His papers are still plumbed for their secrets. His theorems are being applied in areas scarcely imaginable during his lifetime." (quoted from Kanigel's biography, "The Man who knew Infinity", p.3)
In 1997 the Ramanujan Journal was launched to publish work "in areas of mathematics influenced by Ramanujan".
Childhood and early life
Born in Erode, Tamil Nadu, India, by the age of twelve Ramanujan had mastered trigonometry so completely that he was discovering sophisticated theorems that astonished his teachers. In 1898 at age 10, he entered the Town High School in Kumbakonam, where he appears to have first encountered formal mathematics.
At 11 he had mastered the mathematical knowledge of the lodgers at his home, both students at the Goverment College, and was loaned books on advanced trigonometry, which he mastered by 13. His biographer reports that by 14 his genius was beginning to show, not only did he achieve merit certificates and academic awards throughout his school years, he was assisting the school in the logistics of assigning its 1200 students (each with their own needs) to its 35-odd teachers, completing exams in half the alloted time, was already showing his familiarity with infinite series; his peers at the time later commented "We, including teachers, rarely understood him," and "stood in respectful awe" of him.
Adulthood in India
Ramanujan sought to obtain recognition which he hoped would free him from work to focus on his mathematics. He doggedly solicited support, in the form of well placed individuals, attempting to get work and if possible some kind of sponsorship or allowance.
Whilst working, he published several papers in Indian mathematical journals. Rejected several times in his search for financial support, for producing results that nobody he contacted in India could fully understand, he finally wrote in 1913 letter to a famous mathematician, G. H. Hardy, a Fellow of Trinity College, Cambridge in England, enclosing a long list of theorems of hitherto unheard of complexity, which he stated he could prove. In fact, Ramanujan wrote to three Cambridge academics. The other letters, to H. F. Baker and E. W. Hobson, received no reply.
Hardy, together with his colleague Littlewood, analysed this unsolicited missive, and later commented that of the theorems in this initial letter, discovered and stated as results by an unknown and untrained Indian mathematician, "not one could have been set in the most advanced mathematical examination in the world". Although Hardy was one of the pre-eminent mathematicians of the day and an expert in several of the fields Ramanujan was writing about, he added that many of them "defeated me completely; I had never seen anything in the least like them before."
Life in England
After some initial skepticism, Hardy replied with comments, requesting proofs for some of the discoveries, and began to make plans to bring Ramanujan to England. As an orthodox Brahmin, Ramanujan consulted the astrological data for his journey, because of religious concerns that he would lose his caste by traveling to foreign shores. Ramanujan's mother had a dream in which the family Goddess told her not to stand in the way of her son's travel, and so he made plans accordingly, although he took pains to keep a proper Brahmin lifestyle as far as he could, when he did.
A fruitful collaboration ensued, which Hardy described as "the one romantic incident in my life". Hardy said of Ramanujan's formulas, some of which he could not initially understand, that "a single look at them is enough to show that they could only be written down by a mathematician of the highest class. They must be true, for if they were not true, no one would have had the imagination to invent them." Hardy stated in an interview by Paul Erdős that his own greatest contribution to mathematics was the discovery of Ramanujan, and compared Ramanujan at least to the mathematical giants Euler and Jacobi in terms of genius. Ramanujan was later appointed a Fellow of Trinity, and the highest level of honor in science, a Fellow of the Royal Society (FRS).
Illness and return to India
Plagued by health problems all his life, in a country far from home, and obsessively involved with his studies, Ramanujan's health worsened in England, perhaps exacerbated by stress, and by the scarcity of vegetarian food during the First World War. He was diagnosed with tuberculosis (Henderson, 1996) and a severe vitamin deficiency, though a 1994 analysis of Ramanujan's medical records and symptoms by Dr. D.A.B Young concluded that it was much more likely he had hepatic amoebiasis, a parasitic infection of the liver. This is also supported by the fact that Ramanujan spent time in Madras, a coastal city where the disease was widespread. It was a difficult disease to diagnose, but once diagnosed was readily curable (Berndt, 1998). He returned to India in 1919 and died soon after in Kumbakonam, his final gift to the world being the discovery of 'mock Theta functions'. His wife S. Janaki Ammal lived outside Chennai (formerly Madras) until her death in 1994. Janaki had been nine when they were married, a fairly common practice in India at the time. (Henderson, 1996)
Ramanujan lived as a Brahmin all his life. Views of his actual beliefs vary: his first Indian biographers described him as rigorously orthodox, whereas G. H. Hardy (a militant atheist) believed him to be essentially agnostic as far as metaphysical matters were concerned.
Hardy reported a statement of Ramanujan's to the effect that all religions are equally correct. Kanigel's biography states that Ramanujan would probably not have shown Hardy his religious side anyway; on the other hand Kanigel paints a generally negative picture of Hardy.
Ramanujan credited his understanding to his family Goddess, Namagiri , and looked to her for inspiration in his work. He often said, "An equation for me has no meaning, unless it represents a thought of God."
In mathematics, there is a distinction between having an insight and having a proof. Ramanujan's talent suggested a plethora of formulae that could then be investigated in depth later. As a byproduct, new directions of research were opened up.
Hardy wrote of Ramanujan:
- "The limitations of his knowledge were as startling as its profundity. Here was a man who could work out modular equations and theorems... to orders unheard of, whose mastery of continued fractions was... beyond that of any mathematician in the world, who had found for himself the functional equation of the Zeta function and the dominant terms of many of the most famous problems in the analytic theory of numbers; and yet he had never heard of a doubly periodic function or of Cauchy's theorem, and had indeed but the vaguest idea of what a function of a complex variable was..."
Theorems and discoveries
These include both Ramanujan's own discoveries, and those developed or proven in collaboration with Hardy.
He also made major breakthroughs and discoveries in the areas of:
It is said his discoveries were unusually rich; that is, in many of them there was far more than initially met the eye.
The Ramanujan conjecture and its role
Although there are numerous statements that could bear the name Ramanujan conjecture, there is one in particular that was very influential on later work. That Ramanujan conjecture is an assertion on the size of the coefficients of the tau-function, a typical cusp form in the theory of modular forms. It was finally proved as a consequence of the proof of the Weil conjectures some decades later; the reduction step is complicated.
While he was still in India, Ramanujan recorded many results in three notebooks of loose leaf paper. Results were written up, without their derivations.
This style of working may have been for several reasons. Since paper was very expensive, Ramanujan would do most of his work and perhaps his proofs on slate, and then transfer just the results to paper. Using a slate was common for mathematics students in India at the time. He was also quite likely to have been influenced by the style of one of the books he had learned much of his advanced mathematics from G. S. Carr's Synopsis of Pure and Applied Mathematics, used by Carr in his tutoring. It summarised several thousand results, stating them without proofs. Finally, it is possible that Ramanujan considered his workings to be for his personal interest alone; and therefore only recorded the results. (Berndt, 1998)
The first notebook was 351 pages with 16 somewhat organized chapters and some unorganized material. The second notebook had 256 pages in 21 chapters and 100 unorganized pages, with the third notebook containing 33 unorganized pages. The results in his notebooks inspired numerous papers by later mathematicians trying to prove what he had found. Hardy himself created papers exploring material from Ramanujan's work as did G. N. Watson, B. M. Wilson, and Bruce Berndt. (Berndt, 1998)
An overview of Ramanujan's notebooks by Bruce C. Berndt, in Charlemagne and His Heritage: 1200 Years of Civilization and Science in Europe, Volume 2: Mathematical Arts, P. L. Butzer, H. Th. Jongen, and W. Oberschelp, editors, Brepols, Turnhout, 1998, pp. 119-146, (22 pg. pdf file)
- Modern Mathematicians, Harry Henderson, Facts on File Inc., 1996
Last updated: 05-21-2005 19:36:32