Ramanujan mathematician

Ramanujan was a child prodigy, and a mathematical genius. The greatest thing about him probably is how he turned out to be this amazing mathematician despite having little or no formal outside exposure to advanced mathematics. He was a self-taught genius from very humble origins, completely disconnected from the world of other excelling mathematicians and largely worked out of his own, in utter isolation (and often in poverty). Ramanujan possessed an incredibly amazing intuition for numbers, fractions and infinite series, possibly like no other mathematician ever did. He churned out a huge number of significant and complex results, largely based on 'intuition' mingled with argument and induction, and some sort of innate insight that only he seemed to possess, often without formal proofs and coherent accounts, and at times, without the formal background knowledge of related fields in mathematics that are often used to arrive at such results.

Quoting Hardy's observation on 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"

Hardy would compare Ramanujan to the likes of geniuses like Jacobi and Euler, and often mentioned that, he had never met his equal.

Ramanujan had an untimely death at a young age of 32, but by then, he had developed an unparalleled intuition for continued fractions and series, like no other known mathematician. He left behind a 'notebook' with merely summaries and results in it, with little or no proofs - his personal notebook. It seemed that, poor Ramanujan, often used to derive his results on a 'slate' (due to lack of paper), and just jot down his result. He had often derived existing classic results and at times, his own. This notebook later inspired a lot of work, in attempts to prove some of the results, and also led to fields such as 'highly composite numbers'. Ramanujan suggested a huge plethora of formulae based on sheer intuition, that could all then be investigated in depth later. It is said that Ramanujan's discoveries are unusually rich and that there is often more to them than initially meets the eye.

As a young man, he failed to get a degree, as he did not clear his fine arts courses, although he always performed exceptionally well in mathematics. His peers rarely understood him at school and was always in awe of his mathematical acumen. Ramanujan had mastered a book on trigonometry at the age of 13 and produced pretty sophisticated results right then. He finished his mathematics exams in half the time, and at graduation was even awarded more than the maximum possible marks, as a recognition for his exceptional performance. He had independently developed and investigated Bernoulli numbers in great detail, and had also derived the Euler's constant all at a very young age, in utter isolation from the rest of the world.

A very shy, quiet and deeply religious man, with pleasant manners, his talent was recognized in stages, by mathematicians in India first, and later by Hardy in Cambridge, who was simply astounded when he came across the many fascinating and complex results that this hitherto unknown young man out of nowhere had churned out by sheer intuition.

There is only one thing I'd say, as I end this answer:

Ramanujan suffered illnesses before and after his marriage to Janaki (1909) and before his departure to England. From all accounts available it appears that his health was reasonably good during the first three years of his stay in Cambridge, despite his strict vegetarian diet, the food shortages and his own 'cooking only once a day or two'. This was also his most productive period in Cambridge. From May 1917, when he was first admitted to the Nursing Hostel in Cambridge for 5 months, he seems to have been in and out of TB Sanatoria - Mendip Hills in Somerset (2-3 weeks in Oct. 1917), Matlock House in Derbyshire (Nov. 1917 - June 1918), Fitzroy House in London (June - Dec. 1918) and Colinette House, Putney (end of Dec. 1918) - until his departure to India in March 1919. Symptoms of improvement showed, after considerable treatment for tuberculosis, in the autumn of 1918. He was able to meet all his medical expenses incurred during his illness out of his earnings accumulated through frugal living. Ramachandra Rao was the first person to offer a salary of Rs.20 per month. He however insisted on Ramanujan going back to Madras where a comparatively better academic atmosphere conducive to research prevailed. Ramanujan was living in the ‘Summer House’, in Sami Pillai Street, Triplicane, Chennai, reluctantly accepting the dole from Mr. Ramachandra Rao. Número de Ramanujan: -Se denomina número de Hardy-Ramanujan a todo entero natural que se puede expresar como la suma de dos cubos de dos maneras diferentes, como el conocido número 1729, protagonista de la anécdota entre Hardy y Ramanujan: 1^3 + 12^3 = 9^3 + 10^3 = 1.729 - Otros números que poseen esta propiedad habían sido descubiertos por el matemático francés Bernard Frénicle de Bessy (1602-1675): 2^3 + 16^3 = 9^3 + 15^3 = 4.104 10^3 + 27^3 = 19^3 + 24^3 = 20.683 2^3 + 34^3 = 15^3 + 33^3 = 39.312 9^3 + 34^3 = 16^3 + 33^3 = 40.033 - El más pequeño de los números descomponibles de dos maneras diferentes en suma de dos potencias a la cuarta es 635.318.657, y fue descubierto por Euler (1707-1763): 158^4 + 59^4 = 133^4 + 134^4 = 635.318.657 Número Taxicab: -Se denomina nésimo número taxicab, denotado como Ta(n) o Taxicab(n), al más pequeño número que puede ser expresado como una suma de dos cubos positivos no nulos de n maneras distintas (sin contar variaciones del orden de los operandos). Así: Ta(1) = 2 = 1^3 + 1^3 Ta(2) = 1.729 = 1^3 + 12^3 = 9^3 + 10^3 Ta(3) = 87.539.319 = 167^3 + 436^3 = 228^3 + 423^3 =255^3 + 414^3 -Variante del taxicab es el cabtaxi (un número cabtaxi es definido como el número entero más pequeño que se puede escribir de n maneras diferentes (sin contar variaciones del orden de los operandos) como suma de dos cubos positivos no nulos o negativos). Por ejemplo: CTa(2) = 91 = 3^3 + 4^3 = 6^3 - 5^3 El número 1729 se conoce como el número de Hardy-Ramanujan por una famosa anécdota del matemático británico G. H. Hardy en relación con una visita al hospital para ver a Ramanujan. En palabras de Hardy:91 Recuerdo una vez que fui a verle cuando estaba enfermo en Putney. Había viajado en el taxi número 1729 y remarqué que me parecía un número intrascendente, y esperaba de él que no hiciera sino un gesto desdeñoso. "No", me respondió, "es un número muy interesante; es el número más pequeño expresable como la suma de dos cubos de dos maneras diferentes". En efecto, la cifra tiene dos descomposiciones diferentes: Las generalizaciones de esta idea han creado la noción de "número taxicab" y de "número cabtaxi". Coincidentemente, 1729 es también un número de Carmichael.