"Ramanujan is the strangest person in the entire mathematics world, and perhaps even in the entire history of science. He is like a supernova that exploded, illuminating the darkest and deepest corners of mathematics, and then was unfortunately knocked down by tuberculosis at the age of 33, just like his predecessor Riemann."
Ramanujan is known as a mathematical genius who "talked to God". In 1923, the mathematics community released an important news. Someone successfully proved that all odd numbers large enough can be written as the sum of three prime numbers. And this is of great significance to the cracking of the Riemann conjecture, and Ramanujan has contributed a lot to the development of this method.
The Riemann hypothesis, that is, the unsolved puzzle of prime numbers, is regarded as the "Mount Everest" of mathematical research, which has attracted a generation of mathematicians to devote themselves to the research of number theory, including many famous figures in the history of mathematics. Ramanujan is one of them.
"Melodious Prime Numbers: Two Hundred Years of Mathematical Swan Song Riemann Hypothesis", [English] Marcus du Sautoy (Marcus du Sautoy).
Professor of Mathematics at Oxford University and researcher of the Royal Society Marcus Du Sotoy Cope's masterpiece, left the footprints of famous figures in the history of mathematics on the way Riemann assumed this "Mount of Mathematical Everest". With the development of computers and the addition of particle physics, the story of the conquest of the Riemann Hypothesis opened a new chapter.
Can you solve the following equation?
This was the first question Ramanujan posed publicly in 1911 , before he was famous. The answer will be posted in the "Math" section below, so if you want to try it out, remember to get the answer before you swipe there.
Srinivasa Ramanujan was one of the greatest mathematicians of all time. Mathematicians today do not fully understand his work. His work was not only applied to mathematics, but was later used in physics such as black holes and string theory.
Renowned mathematician Ken Ono wrote: "Ramanujan had more wisdom than even all the top mathematicians in the world combined."
JE Littlewood, a great mathematician who knew Ramanujan, said that "all natural numbers are close friends of Ramanujan", as if he knew the personality of each number.
Freeman Dyson (Freeman Dyson) commented on him: "I think Ramanujan is more like Mozart. In his eyes, mathematics is a natural and obvious knowledge. His talent is a miracle that we cannot understand."
A more famous anecdote:
The famous mathematician Hardy took a taxi to visit Ramanujan in the hospital. To help him out, Hardy told Ramanujan that the taxi's number was 1729, "a number that doesn't look very interesting". Ramanujan said: "No, this is a very interesting number. Among all the numbers that can be written in two ways as the sum of the cubes of two natural numbers, it is the smallest one." Ramanujan meant that we now call this number a taxi number .
Despite his extraordinary achievements, Ramanujan was largely self-taught , with no formal education in mathematics.
He made major contributions to the fields of analysis , number theory , infinite series , and continued fractions , including solving mathematical problems that were considered unsolvable at the time. In fact, it is estimated that during Ramanujan's short life, he discovered some 3,900 meaningful formulas!
Ramanujan's story is one of the greatest stories in mathematics. From the inspiration and research of a thin book, Ramanujan was able to formulate thousands of theorems. Some of these were rediscoveries of results that had been proven and known to the mathematical community (he independently derived many of the results that Euler, Gauss, etc. had discovered before him), and some were new discoveries that had not been seen before.
Here we will try to understand Ramanujan's life and mathematical talent. We'll also see some of his amazing achievements, how mathematicians don't know how he achieved them until today.
self-taught prodigy
Ramanujan was gifted from an early age.
Ramanujan was born in India in 1887. The whole family lives in a small traditional house. His mother is a housewife and sometimes goes to the local temple to pray. At about ten years old, Ramanujan graduated from primary school with the highest grades in the district. After entering middle school, he was exposed to formal mathematics for the first time.
At 13, he borrowed a book on advanced trigonometry and mastered the field. At this age, he had independently discovered some complicated theorems. At the age of 14, he helped teachers assign 1,200 students to 35 teachers.
In mathematics, Ramanujan could answer the questions in half the time, and he was also very familiar with geometry and infinite series. In 1902, Ramanujan had mastered how to solve cubic equations . Later, he developed his own unique method of solving quartic equations .
The turning point that really awakened Ramanujan's talent was in 1903. At the age of 16, Ramanujan got a copy of "Summary of Basic Results of Pure and Applied Mathematics " by GS Carr. Written in 1886, this book summarizes most of the mathematical results known at the time. It is said that Ramanujan carefully studied 5,000 mathematical theorems in this book.
The following year, Ramanujan discovered and studied Bernoulli numbers and calculated the Euler-Mascheroni constant to 15 decimal places —a landmark achievement. When Ramanujan graduated from secondary school in 1904, the principal said he was an excellent student who deserved more than the perfect mark.
Then in college, only mathematics could keep Ramanujan focused. This caused him some trouble as he struggled to pass other classes. He ended up dropping out of school and didn't find a job. He was plunged into extreme poverty, often without enough to eat. However, he still does everything possible to conduct independent research.
At that time, Ramanujan was plagued by illness, and by the second half of 1910, he was already very weak. He gave the notebook to a friend and told him that if something happened to him, he would send the notebook to other mathematicians. Fortunately, Ramanujan eventually recovered and got his notebook back. In that same year, other mathematicians became aware of Ramanujan's brilliance. Eventually, some of his work was published in the Proceedings of the Indian Mathematical Society.
Ramanujan and Hardy
On January 16, 1913, Ramanujan wrote a letter to the then top mathematician GH Hardy. At the beginning of the letter, Ramanujan mentioned that he was self-taught. He wrote: "I'm a clerk... I don't have a college education...but I'm forging a new path for myself..."
What follows are several pages of mathematical derivations, some of which have been derived before and some of which are not quite correct, but there are three remarkable formulas at the end of the letter. Hardy was shocked by these formulas, and he wrote back:
These formulas totally won me over. I've never seen anything like it before... whoever can write them must be a top mathematician. These formulas must be right, because no one would have the imagination to invent them.
— Hardy
Hardy asked his colleague Littlewood to review the papers. Littlewood was equally struck by Ramanujan's genius. After discussing the papers with Littlewood, Hardy concluded that Ramanujan must have been "the finest mathematician, a man of extraordinary ingenuity and power"
Then Hardy wrote back to Ramanujan, inviting him to study in England. But Hardy also emphasized the importance of evidence, "It is very necessary for me to see the proof of the assertion you have written."
In the letter, Ramanujan simply wrote down the formula inspired by the book, but did not write a proof at all. But for a mathematician, a proposition without a proof is no better than a conjecture. They soon established a correspondence link, and Ramanujan sent Hardy more mathematics.
In just the first two letters, Ramanujan gave Hardy 120 theorems! What Hardy didn't know was that Ramanujan had two thick notebooks filled with beautiful mathematical propositions, the second of which had 21 chapters. After reviewing part of the notebook with Littlewood, Hardy remarked, "Only Euler and Jacobi stand alongside Ramanujan."
Ramanujan worked with Hardy and Littlewood at Cambridge for almost five years. Although Ramanujan's talent is visible to the naked eye, the cooperation between them is not easy. Hardy pursued rigorous proofs, but Ramanujan had a completely different belief and was very firm. He believed the goddess gave him mathematical intuition in a dream.
Hardy and Ramanujan set out to prove some of Ramanujan's propositions together. Hardy did his best to educate and guide Ramanujan in the way of modern mathematics.
During World War I, Ramanujan's situation in Britain was very difficult. He is vegetarian but can hardly get vegetables. As an Indian, he faced harsh racial discrimination in England and was very lonely (despite his infinite number of natural number friends). Fortunately, Ramanujan eventually overcame these obstacles.
On May 2, 1918, Ramanujan was elected a member of the Royal Society for his "contributions to elliptic functions and logarithm" . He is also 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 University.
Mathematics of Ramanujan
Every article about Ramanujan includes some of his mathematical work, so here we list some of his discoveries. But it needs to be emphasized that it is impossible for this article to contain all the formulas he found, because there are 3900 in total !
Hardy once said that Ramanujan's discoveries were extraordinarily rich and often richer than they first appeared. Another distinctive feature of Ramanujan's discoveries is that they usually involve very large numbers or very complex expressions, which can almost only be calculated by computers. Of course, at the beginning of the 20th century, there were no such computers.
In fact, we don't know exactly how Ramanujan did it. This is the biggest mystery in the whole story.
Ramanujan's π expression
This part is interesting, Ramanujan discovered a new expression for 1/π , which is an infinite series.
This series converges so fast that the algorithms people now use to calculate π are based on it. The first term of this series gives the first seven significant figures of π, and the first two terms give the first 15 significant figures. In fact, the series gives 8 more significant figures for each additional term calculated!
Divergent series
Ramanujan studied infinite divergent series - which, according to Abel, are the work of the devil and untouchable by mortals. However, Ramanujan could assign some mathematical meaning to the different divergent sums. In fact, a divergent series that later shocked the Internet was discovered by Ramanujan:
We can make this formula work by giving special meaning to the equals sign. In fact, such a result agrees with some experimental phenomena in physics, such as the Casimir force between two metal plates .
infinitely nested square roots
Back in India, Ramanujan challenged the mathematical community to find the value of the following infinitely nested square root .
Ramanujan waited six months, but no one had an answer. So he directly published the answer, the answer is not complicated, it is 3.
almost integer
No one knows how Ramanujan did it, but he did find out such a few powers that are very close to integers :
In fact, the square root in the above formula is related to the so-called " class number " through L function, elliptic curve and modular form theory.
Unlimited multiplication points
In the first letter sent to Hardy, Ramanujan wrote a heaven-defying point at the end of the third page:
It holds for 0 < a < b + 1/2.
Generally speaking, it is almost impossible to obtain a closed and beautiful solution by multiplying and integrating such unlucky functions. It's hard to believe.
Number of divisions
In set theory, the set of non-empty subsets into which a set is divided is called a partition . For an n-element set, the number of different partitions is called the number of partitions p(n ).
Ramanujan derived the following formula for calculating the number of divisions
in,
The first term of this series is a good approximation to p(n), and Ramanujan found that
last comment
Ramanujan's formula is beautiful, ingenious and unique. Some formulas are like miracles, and until today we still don't know how Ramanujan got them, which is a great mystery.
Ramanujan's achievements mainly include:
Solved an ancient math riddle that no one thought had a solution
Inspired and led many research areas in modern mathematics , such as the importance of modular forms in number theory, Weil conjectures, L-functions, and the Langlands program
Physical theories such as black hole physics, theta-like functions, string theory, and quantum gravity theory
Ramanujan's health has been poor, and eventually died of illness in 1920. His dying work later became the mathematical basis for complex physical theories such as black holes and string theory. Of course, Ramanujan couldn't imagine such a thing as a black hole, but this Chacha highlights his powerful intuition and talent.
The famous physicist Kaku Michio commented on Ramanujan: "Ramanujan is the strangest person in the entire mathematics world, and may even be the strangest person in the entire history of science. He is like a supernova that exploded, illuminating the darkest and deepest corners of mathematics, and then was unfortunately knocked down by tuberculosis at the age of 33, just like his predecessor Riemann."
Ramanujan's story has been adapted into novels and films, and it is hoped that his story will continue to inspire young people to pursue mathematics—not just as a subject, but as an art.
This article is authorized to be reproduced from the WeChat public account "Institute of Physics, Chinese Academy of Sciences". The original link: Ramanujan: The Greatest Mathematical .
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