For a long time after his death, Ramanujan was not taken seriously by the general public until 2015, when his story was made into the movie "The Man Who Knew Infinity", and Ramanujan's name entered the field of mass culture, and he was known for his legendary talent for number theory.
Today marks the 132nd anniversary of Ramanujan's birth. Nearly a century after his death, people still enjoy his mathematical legacy and look forward to the discovery of more wonders.
Commemorative stamp of India's National Mathematical Day 2012 (Source: Wikipedia)
Written by | Yang Owl
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In the face of genius, we have only one possibility to defend, and that is to love him.
- Goethe
Mathematics is inseparable from proof, but one mathematician often skips this crucial step. In 1913, he gave G. Hardy, who was in Cambridge, a young man. H. Hardy), sent a bunch of formulas with little proof, such as:
This is a formula for solving integrals. Although there are hundreds of methods for solving calculus in textbooks, it is possible that a function given arbitrarily may not be able to be cumulative. The formula on this letter corresponds to a shortcut to a solution. Hardy took a lot of effort to prove this formula.
On top of that, Hardy noticed that there were more formulas on the letter that made him exclaim, "Completely beat me" and "I've never seen anything like this." Who is it that can make Hardy, who is known as the "representative of the British school of analysis in the 20th century", express such admiration?
The masters of these complicated formulas are called ஸ்ரீனிவாஸராமானுஜன்ஐயங்கார் (Srinivasa Ramanujan), an Indian Tamil. For some time after his death, he remained unappreciated by the general public until 2015, when his story was made into the movie The Man Who Knew Infinity, and Ramanujan's name entered the field of popular culture, and he became known for his legendary talent for number theory.
In India
Ramanujan was born on 22 December 1887 in Chennai (formerly known as Madras) in tamil Nadu, India, and grew up in a devout Hindu family. At the age of 11, by borrowing higher mathematics books from tenants, he acquired all the knowledge of university mathematics at that time. At the age of 16, he obtained from a friend a library copy of A Synopsis of Elementary Results in Pure and Applied Mathematics, which contained 5,000 theorems. The book awakened Ramanujan's mathematical talents and led him to scribble in notebooks, recording the inspiration given to him by the goddess of mathematics.
However, his unique talent did not allow him to receive a comprehensive reward. Whether it was a scholarship exam or a college graduation exam, he could only pass the part about mathematics. With no degree and no guidance, he had to study mathematics on his own in private. The Ramanujan family was an orthodox Brahmin, a sacrificial nobleman under the Hindu caste system, and the family, though spiritually rich, was materially poor. Ramanujan spent the next few years destitute and even starving.
In 1910, Ramanujan met V. Ramanujan, the founder of the Indian Mathematical Association. Ramaswamy Aiyer）。 In order to be able to find a job in the tax department to make ends meet, he showed Yair his mathematical notes. As a result, Yar became Ramanujan's first Bole. With Yair's help, Ramanujan became acquainted with prominent local mathematicians and was recommended to publish his mathematical results in the Journal of the Indian Mathematical Society.
At the beginning of each issue, magazines always ask challenging questions about entertaining readers, and Ramanujan asks one such question:
Ramanujan expected someone to reply to him, but it turned out that the genius's question might only be answered by the genius himself, and he gave the solution after six months, for which he provided a creative formula,
By substituting x=2, n=1, a=0, the above problem can be solved.
This equation states that any number, divided into x, n, and a parts, can be represented by an infinitely nested square root. Ramanujan was extremely fond of "infinity", and he studied not only the square roots nested in infinity, but also the fractions (fractions of fractions ...). ), and infinite series. Mathematicians once commented that "infinite series is Ramanujan's first love".
That's why biographer Robert Kanigel called him "the One Who Knows No End."
Ramanujan's first paper was on Bernoulli numbers, which are numbers defined in infinite series. In his 17-page paper, Some Properties of Bernoulli's Numbers, Ramanujan gives three proofs, two inferences, and three conjectures. His work initially had many flaws, as its editor put it: "Mr. Ramanujan's approach is so concise and novel, and his expression is so lacking in clarity and precision that it is difficult for the average mathematical reader, not accustomed to this kind of intellectual gymnastics, to keep up with his train of thought." ”
In 1913, with the help of the Indian Mathematical Association, Ramanujan sent the study to a British mathematician. Although he was snubbed by several mathematicians for the confusion of the proof, Hardy, the top British mathematician at the time, found that the person who wrote the letter was a genius.
In his letter, Ramanujan mentions a formula that at first glance looks a lot like an elementary school math problem:
This formula is very novel to Hardy, and it is derived from a class of supergeometric series functions that were originally developed by Euler and Gauss. Hardy found these results "more interesting" than Gauss's work on integrals. After seeing his theorems about continuous fractions on the last page of the manuscript, Hardy says that these theorems "must be true, because, if they are not true, no one has the imagination to create them."
Mathematics in the 20th century was already highly systematic, and Hardy could hardly imagine an untrained man doing such a genius job.
As the article begins to describe, Hardy praised Ramanujan. After several correspondences, Hardy invited Ramanujan to work at Cambridge.
Go to Cambridge
Ramanujan's journey to Cambridge took a lot of trouble. Since his family was orthodox Brahmins, and for religious reasons, going to a foreign country could lose his caste, Ramanujan initially refused to travel to England. Later, his mother had a dream in which the goddess Namagiri ordered her not to hinder her son from pursuing his ideals any longer. Ramanujan's trip to Cambridge was made.
After nearly a month at sea, Ramanujan came to Cambridge. Then, Hardy and J. Littlewood E. Littlewood) set out to study his notebook. While Hardy has received 120 theorems in previous letters, there are many more in Ramanujan's notebook. Hardy found that some of these theorems had already been discovered, but Ramanujan didn't know it yet, but some were real breakthroughs.
Hardy and Ramanujan have very different personalities. Hardy was an atheist, an advocate of proof and mathematical rigor; Ramanujan was a religious man who relied heavily on his intuition and insight. Hardy did everything he could to fill gaps in each other's education and guide him when formal evidence was needed to support the findings.
They worked together in Cambridge for five years. Later, when Hardy was asked what was his greatest contribution to mathematics, he replied without thinking that he had discovered Ramanujan. He called their partnership "theone romantic incident in my life."
In March 1916, Ramanujan received a Bachelor of Science degree (later renamed Ph.D.) for his work on highly synthetic numbers, and the first part of the results of his studies was published in the Proceedings of the London Mathematical Society. The more than 50-page paper, which proved the various properties of these numbers, was one of the most unusual papers in mathematical research at the time—Ramanujan showed extraordinary ingenuity in dealing with it.
Ramanujan proposed a remarkable set of propositions, but because his mathematical expressions were often different from those of other mathematicians, many of them were not proven and were called Ramanujan's conjectures. The most famous of these is that he asserted the size of the Ramanujan τ function.
In 1916, Ramanujan published an article on an arithmetic function that wanted to claim a solution, i.e. the sum of the s powers of the factors of n. This problem leads to the τ function, which he discovered some properties of, but failed to give explicitly. It was not until 1974 that the Belgian mathematician Pierre Deligne proved this conjecture and won the Fields Medal.
On 6 December 1917, Ramanujan was elected to the London Mathematical Society. In 1918, he was elected a Fellow of the Royal Society. He is the second Indian to be inducted and one of the youngest members in the history of the Royal Society. On 13 October 1918, he became the first Indian to be elected a Fellow of Trinity College, Cambridge.
Back home
Despite the accolades, Ramanujan's physical condition deteriorated. In fact, he has been plagued by health problems all his life.
In his early years in India, long-term poverty and hunger caused Ramanujan to suffer from illness. After marriage, he suffered from hydrocele of the testicles and had to undergo surgery, but the family could not afford the operation until a doctor volunteered to treat him free of charge. However, this was only the beginning of his frailty.
After arriving in Britain, Ramanujan's health deteriorated due to the rainy climate and chronic food shortages caused by rations during the First World War, and he was diagnosed with tuberculosis and severe vitamin deficiencies and placed in a nursing home.
In 1919, he returned to India with achievements and illness, but the climate and diet of his hometown were also unable to return to heaven.
Ramanujan died of illness in 1920 at the age of 32.
On his deathbed, Ramanujan wrote a letter to Hardy. The letter describes several new functions that behave differently from the known modular form (an analytic function), but closely mimic them. He speculates that his analog mold corresponds to the normal mold form previously identified by Carl Jacobi, and that both forms will eventually receive similar outputs.
Unfortunately, no one at the time understood what Ramanujan was talking about.
In 2012, on the 125th anniversary of Ramanujan's birth, several mathematicians announced that they had solved the riddle that Ramanujan had left behind. Using a modern mathematical tool that had not yet been developed at the time, they proved that analog mold forms could be calculated as Ramanujan predicted.
Modular unfolding is one of the basic tools for calculating the entropy of black holes. While some black holes are not modular, a new formula based on Ramanujan's vision could allow physicists to calculate the entropy of a black hole in the same way that they calculate the entropy of ordinary physical systems.
However, Ramanujan, who often skips even proofs, probably doesn't care about the practical application of these formulas. He used to say, "An equation has no meaning to me unless it represents the mind of God." Ramanujan used his extraordinary insight in an innocent way, simply to appreciate the beauty of mathematics.
After Ramanujan's death, his brother compiled his handwritten notes, which contained formulas for strange modulus, hypergeometric series, and even fractions, waiting to be discovered.
Google's homepage graffiti commemorating the 125th anniversary of Ramanujan's birth (Source: Google)
Note: A highly synthetic number is when any natural number smaller than it has fewer factors than the number of factors, such as 1,2, 4, 6, 12, 24, 36, 48, 60, 120, 180...
Resources
[1] Kanigel, Robert (1991). The Man Who KnewInfinity: a Life of the Genius Ramanujan. New York: Charles Scribner's Sons.
[2] https://www.newscientist.com/article/mg21628904-200-mathematical-proof-reveals-magic-of-ramanujans-genius/
[3] https://arxiv.org/pdf/1905.04060.pdf
[4] https://www.sciencedaily.com/releases/2012/12/121217091604.htm
[5] https://zh.wikipedia.org/wiki/%E6%88%88%E5%BC%97%E9%9B%B7%C2%B7%E5%93%88%E7%BD%97%E5%BE%B7%C2%B7%E5%93%88%E4%BB%A3
[6] https://zh.wikipedia.org/wiki/%E6%96%AF%E9%87%8C%E5%B0%BC%E7%93%A6%E7%91%9F%C2%B7%E6%8B%89%E9%A9%AC%E5%8A%AA%E9%87%91