Two famous mathematicians [who] became closely associated with one another as “teacher” and “student.” The reason for the quotation marks here will soon be obvious, because the teacher later claimed that he learned more from the student than the student had learned from him. G. H. Hardy was England’s leading mathematician, a professor at Cambridge University, a Fellow of the Royal Society, and the recipient of an honorary degree from Harvard. Remarkably precocious in early childhood, especially in mathematics, he became an exceptionally brilliant student, winning one scholarship after another. He was acknowledged the star graduate in mathematics at Cambridge, where he remained to become a professor o f mathematics. He also became a world-class mathematician. His longtime friend C. R Snow relates that Hardy, at the peak of his career, ranked himself fifth among the most important mathematicians of his day, and it should be pointed out that Hardy’s colleagues regarded him as an overly modest man (Snow, 1967). If the Study of Mathematically Precocious Youth (SMPY) had been in existence when Hardy was a schoolboy, he would have been a most prized and promising student in the program.
One day Hardy received a strange-looking letter from Madras, India. It was full of mathematical formulations written in a quite unconventional—one might even say bizarre—form. The writer seemed almost mathematically illiterate by Cambridge standards. It was signed “Srinivasa Ramanujan.” At first glance, Hardy thought it might even be some kind of fraud. Puzzling over this letter with its abstruse formulations, he surmised it was written either by some trickster or by someone sincere but poorly educated in mathematics. Hardy sought the opinion of his most highly esteemed colleague, J. E. Littlewood, the other famous mathematician at Cambridge. After the two of them had spent several hours studying the strange letter, they finally realized, with excitement and absolute certainty, that they had “discovered” a major mathematical genius. The weird-looking formulas, it turned out, revealed profound mathematical insights of a kind that are never created by ordinarily gifted mathematicians. Hardy regarded this “discovery” as the single most important event in his life. Here was the prospect of fulfilling what, until then, had been for him only an improbable dream: of ever knowing in person a mathematician possibly of Gauss’s caliber.
The above is an excerpt from the book Intellectual Talent: Psychometric and Social Issues by the renowned Psychologist Dr. Arthur R. Jensen. The book continues:
A colleague in Hardy’s department then traveled to India and persuaded Ramanujan to go to Cambridge, with all his expenses and a salary paid by the university. When the youth arrived from India, it was evident that, by ordinary standards, his educational background was meager and his almost entirely self-taught knowledge of math was full of gaps. He had not been at all successful in school, from which he had flunked out twice, and was never graduated. To say, however, that he was obsessed by mathematics is an understatement. As a boy in Madras, he was too poor to buy paper on which to work out his math problems. He did his prodigious mathematical work on a slate, copying his final results with red ink on old, discarded newspapers.
This may be the most intriguing, from the same book:
At Cambridge, Ramanujan was not required to take courses or exams. That would have been almost an insult and a sure waste of time. He learned some essential things from Hardy, but what excited Hardy the most had nothing to do with Ramanujan’s great facility in learning the most advanced concepts and technical skills of mathematical analysis. Hardy himself had that kind of facility. What so impressed him was Ramanujan’s uncanny mathematical intuition and capacity for inventing incredibly original and profound theorems. That, of course, is what real mathematical genius is all about. Facility in solving textbook problems and in passing difficult tests is utterly trivial when discussing genius. Although working out the proof of a theorem, unlike discovering a theorem, may take immense technical skill and assiduous effort, it is not itself a hallmark of genius. Indeed, Ramanujan seldom bothered to prove his own theorems; proof was a technical feat that could be left to lesser geniuses. Moreover, in some cases, because of his spotty mathematical education, he probably would have been unable to produce a formal proof even if he had wanted to. But a great many important theorems were generated in his obsessively active brain. Often he seemed to be in another world.
Ramanujan's Wikipedia page says this about him:
During his short life, Ramanujan independently compiled nearly 3900 results (mostly identities and equations). Nearly all his claims have now been proven correct, although a small number of these results were actually false and some were already known. He stated results that were both original and highly unconventional, such as the Ramanujan prime and the Ramanujan theta function, and these have inspired a vast amount of further research. The Ramanujan Journal, an international publication, was launched to publish work in all areas of mathematics influenced by his work.
And finally, from the same source:
Ramanujan was awarded a Bachelor of Science degree by research (this degree was later renamed PhD) in March 1916 for his work on highly composite numbers, the first part of which was published as a paper in the Proceedings of the London Mathematical Society. The paper was over 50 pages with different properties of such numbers proven. Hardy remarked that this was one of the most unusual papers seen in mathematical research at that time and that Ramanujan showed extraordinary ingenuity in handling it.
