Biography hardy ramanujan partitions

Theorems formulated by him are to date studied by students across the world and within very few years of his lifespan, he made some exceptional discoveries in mathematics. His biography and achievements prove a lot about him and his struggles to contribute to the field of this subject. All this is also an essential part of the syllabus for aspirants preparing for the upcoming IAS Exam.

The facts, achievements and contributions presented by Srinivasa Ramanujan have not just been acknowledged within India, but also globally by leading mathematicians. Aspirants can also learn about other Indian mathematicians and their contributionsby visiting the linked article. His geniuses of the 20th century are still giving shape to 21st-century mathematics.

Discussed below is the history, achievements, contributions, etc. Actually, and if we skip the details of convergence, the formula is not difficult to prove.

Biography hardy ramanujan partitions

If the reader is interested in a more up-to-date treatment, it can be found in [1]. Euler, Introductio in analysin infinitorumcap. Euler, Introductio in analysin infinitorum, Introduction to the analysis of the infiniteedited by A. I am now about 23 years of age. I have had no University education but I have undergone the ordinary school course.

After leaving school I have been employing the spare time at my disposal to work at Mathematics. I have not trodden through the conventional regular course which is followed in a University course, but I am striking out a new path for myself. I would request you to go through the enclosed papers. Being poor, if you are convinced that there is anything of value I would like to have my theorems published.

I have not given the actual investigations nor the expressions that I get but I have indicated the lines on which I proceed. Being inexperienced I would very highly value any advice you give me. Requesting to be excused for the trouble I give you. Yours very sincerely S. A deeply religious Hindu, Ramanujan credited his substantial mathematical capacities to divinity, and stated that the mathematical knowledge he displayed was revealed to him by his family goddess.

From the time he travelled to Trinity College, Cambridge, to work with the mathematician and tutor Godfrey Harold Hardy and John Edensor Littlewood on March 17, to when ill health compelled Ramanujan's return to India where he died in at the age of 32 he had achieved his lifelong ambition; to have his ideas, his maths by be validated by established mathematicians.

Although he was extremely keen to have his work published as indicated by his early letters, he was also the first to acknowledge his limitations. The two vessels used then for preparing hot water are alone still with me; these remind me often of those days. Eventually, though, the frustrated Ramanujan spiralled into depression and illness, even attempting suicide at one time.

His last letters to Hardy, written Januaryshow that he was still continuing to produce new ideas and theorems. His "lost notebook", really a collection of loose and unordered sheets of paper — more than one hundred pages written on sides in Ramanujan's distinctive handwriting were rediscovered by George Andrews, the American mathematician, in It took almost 50 years to discover these papers because they were in a box of effects of G.

Andrews was due to attend a European conference in Strasbourg, when he obtained permission and support from the Trinity College library and from his professor, Ben Noble, to visit Cambridge after the conference, in order to investigate the unpublished writings of Watson et al. Noble agreed and with that commenced a remarkable tale. Although not labelled as such, the identity of the papers was settled because Ramanujan's final letters to Hardy had referred to the discovery of what Ramanujan called mock theta functions, although without great detail, and the manuscript included what appeared to be his full notes on these.

The majority of the formulas are about q-series and mock theta functions, about a third are about modular equations and singular moduli, and the remaining formulas are mainly about integrals, Dirichlet series, congruences, and asymptotics. Particularly knowing that Hardy was only a few years older than Ramanujan. Here was a man who could work out modular equations, and theorems of complex multiplication, to orders unheard of, whose mastery of continued fractions was, on the formal side at any rate, beyond that of any mathematician in the world … It was impossible to ask such a man to submit to systematic instruction, to try to learn mathematics from the beginning once more.

When asked in an interview what his greatest contribution to mathematics was, Hardy unhesitatingly replied that it was the discovery of Ramanujan, and even called their collaboration; "the one romantic incident in my life. However, Hardy too became depressed later in life and attempted suicide by an overdose at one point. Some have blamed the Riemann Hypothesis for Ramanujan and Hardy's instabilities, giving it something of the reputation of a curse.

Flanked by a seaport township and linked by The Madras Railway Company to Mumba in the west and the tea plantations of Bengaluru in the South. In Moses and Company, the tailors on Mount Road, were advertising their woollen suits and woollen underwear for Europe-bound students. Madras Corporation was debating the closure of a road in Santhome, others, many more likely preoccupied by the Golu or festive display of dolls in the late autumn year.

Travelers, merchants, mostly British, mostly men of the empire would take tea at the old Spencer's hotel. Under lethargic, unhurried fans, birds chirping in the shade of the Fish Tail palms, affairs of the subcontinent would be conferred. Outside, women in their cool saris and children ribboned were on their way to school. Schools, at least according to finicky British-India records, were to be benefiting from a [Christian] missionary British education.

This fact has some bearing on how some students integrated their faith, undoubtedly Hindi, with Christian teachings, as well as how this formed their outlook on science and mathematics. I also saw how much faith we place in the field of science and its theories. Madras may well have been in good condition but that was really a matter of perspective.

Almost half-million people, many of them Hindu, nearly all unskilled earned their living as street hawkers, street sellers, taxi drivers, mechanics and other manual or industrial trades. Bare-bodied men straining with heavily-laden carts, some pulling from the front, others pushing from behind, up the steps of the bridges in front of the Central Station and near Stanley Hospital, in the scorching mid-day heat.

Some of the handcart pullers would tie a sack-cloth around their feet. In those days, the poor from neighbouring villages migrated to the city in search of work. They came with nothing from their homes - they bathed, ate and slept on the railway platforms, some on the pavements. The whole under light of the most dazzling colours, of mirrored bangles that glistened from rickety market stalls gilded with flags that blew calmly in the warm Indian breeze.

Fragrant sacks stacked high in winding narrow alleyways, spice lanes that slapped the air with aromatic flavours never less evocative and never less visceral. Poor, but here were a people maintained in deeply rooted conviction for the interconnectedness of life and the sanctity of nature. A faith tempered in ceremony, prayer and sacrifice.

Their temples blessed and among the holiest place, for the people a way of looking at biography hardy ramanujan partitions almost metonymically, a love for the city shouldered on the shape of idols, temples, and shrines. These had, after all, received a visit by the Saivite saints Nayanmars. The 7th century CE young Saiva poet-saint, Sambandar, would speak of the Region as a place of beautiful groves, with waves that creep up to the shore and then dance on it.

As do the fisherfolk who spear the many fish in the waters, and peacocks in its plenty to celebrate the Thiruvadhirai or the "sacred big wave" using which this universe was created over trillion years in the past. In this year, a young year-old Janakiammal from Rajendram, a village close to Marudur Railway Station was such a patnee hona, or a wife-to-be.

The Ramanujan-Janakiammal wedding was a five-day ceremony and it took place along with the wedding of another sister of Janakiammal. In this year the yesr-old Ramanujan joined the University of Madras as its first research scholar. His wife and mother lived with him before Ramanujan left for England, on March 17, Janakiammal joined him in Madras and nursed him biography hardy ramanujan partitions his untimely death on April 26, Primes The theory of primes, as we know, goes back to the Rhind Mathematical Papyrus, from around BC has Egyptian fraction expansions of different forms for prime and composite numbers.

Euclid c. Even after years it stands as an excellent model of reasoning. Here it is: If the list of primes were finite, then by multiplying them together and adding 1 we would get a new number which is not divisible by any prime on our list - a contradiction. This is demonstrated trivially; Suppose p 1Viz-a-viz the unique factorization theorem this can be uniquely written as a product of a prime.

Thus, none of the prime divisors of N appears in the list of primes. This contradicts the assumption that the list contained all primes. The conclusion is that no such list is complete, and the number of primes must be infinite. Today, prime numbers are widely studied in the field of number theory. One approach to investigate prime numbers is to study numbers of a certain form.

The greatest common divisor gcd of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For example, the greatest common divisor of 8 and 12 is 4. Each of these relations can be proved by mathematical induction. Inwith limited options available, he sought government unemployment benefits.

However, his determination paid off, and inhe published a significant paper on Bernoulli numbers in the Journal of the Indian Mathematical Society. His talent eventually led to a position as a shipping clerk at the Madras Port Trust, allowing him to continue nurturing his mathematical aspirations while building a local reputation. Srinivasa Ramanujan made significant contributions to various branches of biography hardy ramanujan partitions, particularly in number theory.

His collaborations with the esteemed British mathematician G. Hardy led to the development of groundbreaking concepts such as the "circle method," which provides an exact formula for the number of integer partitions of a number, denoted as p n. For instance, Ramanujan demonstrated that p 5 equals 7, providing a framework that has had lasting implications in analytic number theory.

His unique insights into the divisibility properties of p n led to discoveries that catalyzed advancements in modular forms, a field that plays a critical role in modern mathematics. Ramanujan's legacy extends beyond published papers and theorems; he also left behind a treasure trove of ideas in his remarkable manuscripts. Notably, he introduced mock theta functions in his final year, a concept that remains poorly understood even today but possesses significant ramifications in both mathematics and theoretical physics, including applications in black hole theory.

The sheer volume of his claims—approximately 4, without proofs—has inspired mathematicians to explore and validate his conjectures, affirming his status as one of the most innovative mathematicians of the 20th century. Srinivasa Ramanujan's life took a significant turn when he began corresponding with the renowned British mathematician G. Hardy in Initially, Hardy suspected that Ramanujan's extraordinary mathematical claims could be a hoax.

However, after reviewing his work, Hardy recognized Ramanujan's genius and arranged for him to study at Cambridge University. This mentorship blossomed into a fruitful collaboration that lasted for five years, during which Ramanujan published over 20 papers, many of which were groundbreaking in the fields of number theory and continued to influence mathematics long after his passing.

At Cambridge, Ramanujan thrived under Hardy's guidance, where his intuitive understanding of mathematics was sculpted through formal techniques. Together, they ventured into the circle method, a powerful analytical technique that enabled them to derive precise formulas for partition numbers. Their work together reflected a symbiotic relationship, blending Ramanujan's raw intuition with Hardy's structured approach, ultimately leading to significant advancements within analytic number theory.

Despite their successful partnership, Ramanujan's time in England was fraught with challenges, particularly his struggle with health issues exacerbated by the English climate. InRamanujan contracted tuberculosis, which increasingly limited his ability to work. Nonetheless, he continued to make groundbreaking discoveries, including the innovative concept of mock theta functions, even as his health deteriorated.

This period at Cambridge not only marked a pinnacle in his career but also laid the groundwork for future explorations of his increasingly complex and visionary ideas in mathematics.