# Hardy and ramanujan relationship

### Who Was Ramanujan?—Stephen Wolfram Blog

Godfrey Harold Hardy FRS (7 February – 1 December ) was an English Hardy almost immediately recognised Ramanujan's extraordinary albeit . years and his relationship with John Edensor Littlewood and Ramanujan. Hardy is. The one major difference that I would like to point out is that Hardy was more inclined while what Ramanujan had was raw intellect, pure intellectual horsepower, so Was Ramanujan in a relationship with G. H. Hardy?. Srinivasa Ramanujan FRS was an Indian mathematician who lived during the British Rule in Recognizing the extraordinary work sent to him as samples, Hardy arranged travel for Ramanujan to . He had a close relationship with her.

In he married a nine-year-old girl, but failed to secure any steady income until the beginning ofwhen he became a clerk in the Madras Port Trust office on a meagre salary. All this time, Ramanujan remained obsessed with mathematics and kept working on continued fractions, divergent series, elliptic integrals, hypergeometric series and the distribution of primes.

ByRamanujan was desperate to gain recognition from leading mathematicians, especially those in England. So, at the beginning ofwhen he was just past 25, he dispatched a letter to Hardy in Cambridge with a long list of his discoveries —- a letter which changed both their lives.

The mathematical scene in England in the first half of the 20th century was dominated by Hardy and another titan of Trinity College, J.

### Srinivasa Ramanujan - Wikipedia

The two formed a legendary partnership, unique to this day, writing an astounding joint papers. They were instrumental in turning England into a superpower in mathematics, especially in number theory and analysis.

Hardy was not the first mathematician to whom Ramanujan had sent his results, however the first two to whom he had written judged him to be a crank. But Hardy was not only an outstanding mathematician, he was also a wonderful teacher, eager to nurture talent. Genius unknown After dinner in Trinity one evening, some of the fellows adjourned to the combination room. Over their claret and port Hardy mentioned to Littlewood some of the claims he had received in the mail from an unknown Indian.

Some assertions they knew well, others they could prove, others they could disprove, but many they found not only fascinating and unusual but also impossible to resolve. Wikimedia This toing and froing between Hardy and Littlewood continued the next day and beyond, and soon they were convinced that their correspondent was a genius.

So Hardy sent an encouraging reply to Ramanujan, which led to a frequent exchange of letters. It was clear to Hardy that Ramanujan was totally exceptional: Hardy knew that if Ramanujan was to fulfil his potential, he had to have a solid foundation in mathematics, at least as much as the best Cambridge graduates.

As a BrahminRamanujan was not allowed to cross the ocean and his mother was totally opposed to the idea of the voyage. H Neville, another fellow of Trinity Collegewho was on a serendipitous trip to Madras, to secure Ramanujan a scholarship from the University of Madras. Fearless mentoring I cannot but admire Hardy for his care in mentoring Ramanujan.

Ramanujan is evidently a man with a taste for Mathematics, and with some ability, but he has got on to wrong lines. Neither were particularly good choices: Baker worked on algebraic geometry and Hobson on mathematical analysis, both subjects fairly far from what Ramanujan was doing. But in any event, neither of them responded. And so it was that on Thursday, January 16,Ramanujan sent his letter to G. Hardy was born in to schoolteacher parents based about 30 miles south of London.

He was from the beginning a top student, particularly in mathematics. Even when I was growing up in England in the early s, it was typical for such students to go to Winchester for high school and Cambridge for college. The other, slightly more famous, track—less austere and less mathematically oriented—was Eton and Oxfordwhich happens to be where I went.

Hardy thought he should have been top, but actually came in 4th, and decided that what he really liked was the somewhat more rigorous and formal approach to mathematics that was then becoming popular in Continental Europe. Hardy was at Trinity College —the largest and most scientifically distinguished college at Cambridge University—and when he graduated inhe was duly elected to a college fellowship. For a decade Hardy basically worked on the finer points of calculus, figuring out how to do different kinds of integrals and sums, and injecting greater rigor into issues like convergence and the interchange of limits.

By or so, Hardy had pretty much settled into a routine of life as a Cambridge professor, pursuing a steady program of academic work. But then he met John Littlewood. Littlewood had grown up in South Africa and was eight years younger than Hardy, a recent Senior Wrangler, and in many ways much more adventurous.

And in Hardy—who had previously always worked on his own—began a collaboration with Littlewood that ultimately lasted the rest of his life. As a person, Hardy gives me the impression of a good schoolboy who never fully grew up. He seemed to like living in a structured environment, concentrating on his math exercises, and displaying cleverness whenever he could. He could be very nerdy—whether about cricket scores, proving the non-existence of God, or writing down rules for his collaboration with Littlewood.

### The man who taught infinity: how GH Hardy tamed Srinivasa Ramanujan's genius

So in early there was Hardy: But then he received the letter from Ramanujan. Again, they began unpromisingly, with rather vague statements about having a method to count the number of primes up to a given size. But by page 3, there were definite formulas for sums and integrals and things. But some were definitely more exotic. Their general texture, though, was typical of these types of math formulas.

At least two pages of the original letter have gone missing.

## G. H. Hardy

First he consulted Littlewood. Was it perhaps a practical joke? Were these formulas all already known, or perhaps completely wrong? Some they recognized, and knew were correct.

But many they did not. It took him a week to actually reply to Ramanujan, opening with a certain measured and precisely expressed excitement: Of course, Hardy could have taken it upon himself to find his own proofs. But I think part of it was that he wanted to get an idea of how Ramanujan thought—and what level of mathematician he really was.

So how was he getting his results? But he was certainly doing all sorts of calculations with numbers and formulas—in effect doing experiments.

And presumably he was looking at the results of these calculations to get an idea of what might be true. But presumably he used some mixture of traditional mathematical proof, calculational evidence, and lots of intuition. Instead, he just started conducting a correspondence about the details of the results, and the fragments of proofs he was able to give.

Well, that has to do with the Riemann zeta function as well. But not as crazy as it might at first seem. But back to the story.

Stealing was a major issue in academia then as it is now. A Way of Doing Mathematics By the time he got his scholarship, Ramanujan had started writing more papers, and publishing them in the Journal of the Indian Mathematical Society.

Compared to his big claims about primes and divergent series, the topics of these papers were quite tame. But the papers were remarkable nevertheless. For Ramanujan, though, complicated formulas were often what really told the story. As an aside, back in the late s I started writing papers that involved formulas generated by computer.

And in one particular paperthe formulas happened to have lots of occurrences of the number 9. People tend to think of working with algebraic formulas as an exact process—generating, for example, coefficients that are exactly 16, not just roughly But for Ramanujan, approximations were routinely part of the story, even when the final results were exact.

- *The Romance of a Mathematician’s Life: The Ramanujan-Hardy Conundrum
- Who Was Ramanujan?
- Srinivasa Ramanujan

We can start doing all sorts of transformations among square roots, and trying to derive theorems from them. Or we can just evaluate each expression numerically, and find that the first one 2. In the mathematical tradition of someone like Hardy—or, for that matter, in a typical modern calculus course—such a direct calculational way of answering the question seems somehow inappropriate and improper. And of course if the numbers are very close one has to be careful about numerical round-off and so on.

But for example in Mathematica and the Wolfram Language today—particularly with their built-in precision tracking for numbers—we often use numerical approximations internally as part of deriving exact results, much like Ramanujan did. When Hardy asked Ramanujan for proofs, part of what he wanted was to get a kind of story for each result that explained why it was true.

And the same happens whenever a key part of a result comes from pure computation of complicated formulas, or in modern times, from automated theorem proving. But Ramanujan was different. For him, just the object itself would tell a story.

Ramanujan was of course generating all these things by his own calculational efforts. But back in the late s and early s I had the experience of starting to generate lots of complicated results automatically by computer. I just had an intuition about, for example, what functions might appear in the result. And given this, I could then get the computer to go in and fill in the details—and check that the result was correct.

And this often happens for example when there are infinite or infinitesimal quantities or limits involved. And one of the things Hardy had specialized in was giving proofs that were careful in handling such things.

In particular, in a kind of algebraic analog of the theory of transfinite numbers, he talked about comparing growth rates of things like nested exponential functions—and we even make some use of what are now called Hardy fields in dealing with generalizations of power series in the Wolfram Language.

But his greatest skill was, I think, something in a sense more mysterious: For example, he noted: For Ramanujan was in some fundamental sense an experimental mathematician: Hardy on the other hand worked like a traditional mathematician, progressively extending the narrative of existing mathematics. Most of his papers begin—explicitly or implicitly—by quoting some result from the mathematical literature, and then proceed by telling the story of how this result can be extended by a series of rigorous steps.

There are no sudden empirical discoveries—and no seemingly inexplicable jumps based on intuition from them.

**Srinivasa Ramanujan genius of India - ramanujan biography in hindi - Indian mathematician**

A century later this is still the way almost all pure mathematics is done. Strange structures that arise when one successively adds numbers to their digit reversals. Bizarre nested recurrence relations that generate primes. Peculiar representations of integers using trees of bitwise xors. For many mathematicians—like Hardy—the process of proof is the core of mathematical activity. Particularly today, as we start to be able to automate more and more proofs, they can seem a bit like mundane manual labor, where the outcome may be interesting but the process of getting there is not.

But proofs can also be illuminating. They can in effect be stories that introduce new abstract concepts that transcend the particulars of a given proof, and provide raw material to understand many other mathematical results. For Ramanujan, though, I suspect it was facts and results that were the center of his mathematical thinking, and proofs felt a bit like some strange European custom necessary to take his results out of his particular context, and convince European mathematicians that they were correct.

In the early part ofHardy and Ramanujan continued to exchange letters. Ramanujan described results; Hardy critiqued what Ramanujan said, and pushed for proofs and traditional mathematical presentation. Hardy and Littlewood offered to put up some of the money themselves. There are strange little notes in the bureaucratic record, like on February Before leaving India, Ramanujan had prepared for European life by getting Western clothes, and learning things like how to eat with a knife and fork, and how to tie a tie.

Many Indian students had come to England before, and there was a whole procedure for them. Ramanujan, of Madras, whose work in the higher mathematics has excited the wonder of Cambridge, is now in residence at Trinity. Ramanujan in Cambridge What was the Ramanujan who arrived in Cambridge like? He was described as enthusiastic and eager, though diffident. He made jokes, sometimes at his own expense. He could talk about politics and philosophy as well as mathematics.

He was never particularly introspective. In official settings he was polite and deferential and tried to follow local customs. His native language was Tamiland earlier in his life he had failed English exams, but by the time he arrived in England, his English was excellent. He liked to hang out with other Indian students, sometimes going to musical events, or boating on the river.