Most famous mathematicians were educated at renowned centers of learning and were taught by inspiring teachers, if not by distinguished research mathematicians. The one exception to this rule is Ramanujan, born on December 22, 1887 in Erode in the southern Indian State of Tamil Nadu. He lived most of his life in Kumbakonam, located to the east of Erode and about 250 kilometers south-southwest of Madras (Chennai). At an early age, he won prizes for his mathematical prowess—not mathematics books as one might surmise, but books of English poetry, reflecting British colonial rule at that time. At the age of about 15, he borrowed a copy of G. S. Carr’s Synopsis of Pure and Applied Mathematics, which was the most influential book in his development. Carr was a tutor and compiled this compendium of approximately 4,000-5,000 results (with very few proofs) to facilitate his tutoring at Cambridge and London. One or two years later, Ramanujan entered the Government College of Kumbakonam, often called “the Cambridge of South India,” because of its excellent academic standards. By this time, Ramanujan was consumed by mathematics and would not seriously study any other subject. Consequently, he failed his examinations at the end of the first year and lost his scholarship. Because his family was poor, Ramanujan was forced to terminate his formal education.
At about the time Ramanujan entered college, he began to record his mathematical discoveries in notebooks. Living in poverty with no means of financial support, suffering at times from serious illnesses (including two long bouts of dysentery), and working in isolation, Ramanujan devoted all of his efforts in the next five years to mathematics, while continuing to record his discoveries without proofs in notebooks.

In 1909, Ramanujan married S. Janaki, who was only nine years old. More pressure was therefore put upon him to find a job, and so in 1910 he arranged a meeting with V. Ramaswami Ayyar, who had founded the Indian Mathematical Society three years earlier and who was working as a deputy collector. After Ramanujan showed V. R. Ayyar his notebooks, the latter contacted R. Ramachandra Rao, collector in the town of Nellore, north of Madras, who agreed to provide Ramanujan with a monthly stipend so that he could continue to work unabatedly on mathematics and not worry about having a job.

In 1910, with the financial support of Rao, Ramanujan moved to Madras. For reasons that are unclear, after 15 months, Ramanujan declined further support and subsequently became a clerk in the Madras Port Trust Office, where he was encouraged, especially, by Sir Francis Spring and S. Narayana Aiyar, chairman and chief accountant, respectively. They persuaded Ramanujan to write English mathematicians about his mathematical discoveries. One of them, G. H. Hardy, professor of mathematics at Cambridge University and one of the foremost analysts and number theorists in the 20th century, responded encouragingly and invited Ramanujan to come to Cambridge to develop his mathematical gifts. Ramanujan’s family was Aiyangar, a conservative orthodox branch in the Brahmin tradition, and his mother especially was adamantly opposed to her son’s “crossing the seas” and thereby becoming “unclean.” After overcoming family reluctance, Ramanujan boarded a passenger ship for England on March 17, 1914.

At about this time, Ramanujan evidently stopped recording his theorems in notebooks. That Ramanujan no longer concentrated on logging entries in his notebooks is evident from two letters that he wrote friends in Madras during his first year in England. In a letter of November 13, 1914 to his friend R. Krishna Rao, Ramanujan confided, “I have changed my plan of publishing my results. I am not going to publish any of the old results in my notebooks till the war is over.” And in a letter dated January 7, 1915 to S. M. Subramanian, Ramanujan admitted, “I am doing my work very slowly. My notebook is sleeping in a corner for these four or five months. I am publishing only my present researches as I have not yet proved the results in my notebooks rigorously.’’

On March 24, 1915, near the end of his first winter in Cambridge, Ramanujan wrote his friend E. Vinayaka Row in Madras, “I was not well till the beginning of this term owing to the weather and consequently I couldn’t publish anything for about five months.’’ By the end of his third year in England, Ramanujan was critically ill, and, for the next two years, he was confined to nursing homes. After World War I ended, Ramanujan returned home in March 1919, but his health continued to deteriorate, and on April 26, 1920 Ramanujan died at the age of 32.

In both England and India, Ramanujan was treated for tuberculosis, but his symptoms did not match those of the disease. More recently, an English physician, D. A. B. Young, carefully examined all extant records and symptoms of Ramanujan’s illness and convincingly concluded that Ramanujan suffered from hepatic amoebiasis, a parasitic infection of the liver. Amoebiasis is a protozoal infection of the large intestine that gives rise to dysentery. Relapses occur when the host-parasite relationship is disturbed, which likely happened when Ramanujan entered a colder climate. The illness is very difficult to diagnose, but once diagnosed, it can be cured.

Despite being confined to nursing homes for two of his five years in England, Ramanujan made enormously important contributions to mathematics, several in collaboration with Hardy, which, although they won him immediate and lasting fame, are probably recognized and appreciated more so today than they were at that time. Most of Ramanujan’s discoveries lie in the areas of (primarily) number theory, analysis, and combinatorics, (the arrangement of or operation on discrete mathematical elements belonging to finite sets or making up geometric configurations, Source: Merriam-Webster), but they influence many modern branches of both mathematics and physics. After Ramanujan died, Hardy strongly urged that Ramanujan’s notebooks be edited and published. By “editing,’’ Hardy meant that each claim made by Ramanujan in his notebooks should be examined and proved, if a proof did not already exist. Of the three notebooks that Ramanujan left us, the second is the most extensive. The notebooks contain about 3,200–3,300 entries, almost all of them without proofs and most of them not rediscoveries, despite working without contact with other mathematicians before leaving for Cambridge.

At the University of Madras, various papers and handwritten copies of all three notebooks were sent to Hardy in 1923 with the intent of bringing together all of Ramanujan’s work for publication. Ramanujan’s Collected Papers were published in 1927, but his notebooks and other manuscripts were not published.

Sometime in the late 1920s, two English mathematicians, G. N. Watson and B. M. Wilson, undertook the task of editing Ramanujan’s notebooks. Wilson died prematurely in 1935, and although Watson worked for 10-15 years on the task and wrote over 30 papers inspired by Ramanujan’s mathematics, the work was never completed.

It was not until 1957 that the notebooks were made available to the public when the Tata Institute of Fundamental Research in Bombay published a photocopy edition, but no editing was undertaken.

While residing for a year at the Institute for Advanced Study in Princeton, on a cold winter day in early February 1974, I was reading two papers by Emil Grosswald, in which some formulas from the notebooks were proved. I observed that I could prove these formulas by using a theorem I had proved two years earlier and so was naturally curious to determine if there were other formulas in the notebooks that I could prove employing my theorem. Fortunately, the library at Princeton University had a copy of the Tata Institute’s edition, and, indeed, I found a few more formulas of the same sort that I could prove. In the next three years, I divided my time between Ramanujan’s notebooks and other ongoing research.

All of the aforementioned entries can be found in Chapter 14 of Ramanujan’s second notebook. After the spring semester at the University of Illinois ended in May 1977, I decided to attempt to find proofs for all 87 formulas in Chapter 14. After I worked on this project for nearly a year, George Andrews from Pennsylvania State University visited the University of Illinois and informed me that on a visit to Trinity College Library at Cambridge two years earlier, he learned that Watson and Wilson’s efforts in editing the notebooks were preserved there. The librarian kindly sent me a copy of the notes, and so with their help on certain chapters, I began to devote all of my research time toward proving the theorems stated by Ramanujan in his three notebooks. With the assistance of several mathematicians, I completed the task in five volumes over a period of 21 years.

When Andrews visited Trinity College Library in 1976, he also discovered Ramanujan’s “lost notebook,’’ which was undoubtedly sent to Hardy in the aforementioned shipment of papers in 1923. Hardy likely kept it in his possession until possibly the late 1930s or early 1940s when he passed it to Watson, who, by that time, had lost his passion for Ramanujan’s work. The “lost notebook’’ was found among Watson’s papers after his death in 1965 and was sent by Rankin to Trinity College Library on December 26, 1968, where it resided until it was rediscovered by Andrews less than eight years later. The lost notebook, actually a sheaf of 138 disparate pages, contains the statements of approximately 650 theorems, all without proofs, and clearly emanates from the last year of Ramanujan’s life. The excitement among mathematicians caused by Andrews’s discovery can be compared to that which would arise in the music world from the finding of Beethoven’s tenth symphony. Along with other manuscripts and letters by Ramanujan found in the libraries at both Cambridge and Oxford, the lost notebook was finally published in 1988. In the mid-90s, I began to work with Andrews on the task of providing proofs for all the entries in this volume. Our second book on this project will be submitted to Springer in the late summer or early fall of 2007. Two or three further volumes remain to be written.

Why did Ramanujan not record any of his proofs in the three earlier notebooks and lost notebook? There are perhaps several reasons. First, Ramanujan was perhaps influenced by the style of Carr’s book in which one theorem after another is stated without proof. Second, like most Indian students in his time, Ramanujan worked primarily on a slate. Paper was expensive. Thus, after rubbing out his proofs with his sleeve, Ramanujan recorded only the final results in his notebooks. Third, Ramanujan never intended that his notebooks be made available to the mathematical public. They were his own personal compilation of what he had discovered. If someone had asked him how to prove a particular result in the notebooks, undoubtedly Ramanujan felt he could remember his proof.

Speculations about Ramanujan’s methods are plentiful. Many have suggested that he discovered his results by “intuition,” or by making deductions from numerous calculations, or by inspiration from Goddess Namagiri. Although any or all of these considerations may have some merit, none offer much insight. Moreover, assessments focusing on Eastern mysticism are worthless. As Ramanujan himself admitted, some of his proofs may not have been rigorous by contemporary standards. Nonetheless, despite the lack of rigor at times, Ramanujan undoubtedly thought and devised proofs like any other mathematician, but with insights that surpass all but a few of the greatest mathematicians.

Andrews, the author, and others have been struggling to devise proofs of Ramanujan’s discoveries for over three decades now. But even if we are successful in finding proofs, considerable efforts still remain, as we try to uncover the veil of mist enveloping Ramanujan’s ideas, insights, and methods.