This article on Ramanujan is a guest article by Radhika Bangar from Maharashtra, India.
“An equation means nothing to me unless it expresses a thought of God.”
In the history of Mathematics, this name is still applauded with great respect and honor. Srinivasa Ramanujan, an Indian Mathematician, was a child prodigy and genius. Irrespective of having little or no access to having exposure to advance mathematics, he turned out to be genius as a kid! His life story, with humble and sometimes difficult beginnings, is as interesting in its own right as his astonishing work was. He had stunned the West with his intuitive theorems in Mathematics, with over 3,900 theorems and results to his credit. His infinite series of pi was among his most celebrated findings.
Srinivasa Ramanujan was born on December 22, 1887, in Erode, in Tamil Nadu, India. His father was K. Srinivasa Iyengar, an accounting clerk for a clothing merchant. His mother was Komalatammal, who earned a small amount of money each month as a local temple singer. Born in a Hindu Brahmin family, his mother ensured the boy was in tune with Brahmin traditions and culture. Although his family was of high caste, they were destitute.
His interest and devotion to mathematics were to the point of obsession. He ignored everything else and would play with numbers day and night on a slate and in his mind. A book unlocked his interest in mathematics. It wasn’t by a famous mathematician, and it wasn’t full of the most up-to-date work, either. The book was A Synopsis of Elementary Results in Pure and Applied Mathematics (1880, revised in 1886), by George Shoobridge Carr. The book consists solely of thousands of theorems, many presented without proofs, and those with proofs only have the briefest.
Ramanujan encountered this book in 1903 when he was 15 years old. The book was not an orderly procession of theorems all tied up with tidy proofs, and this encouraged Ramanujan to jump in and make connections on his own. However, since the proofs included were often just one-liners, Ramanujan had a false impression of the rigor required in mathematics. Ramanujan made the book his constant companion and improved it further on his own.
His obsession and preoccupation with mathematics did not allow him to pass his intermediate examination despite three attempts. He could not even score the minimum passing marks in other subjects. Despite being a prodigy in mathematics, Ramanujan did not have an auspicious start to his career. He obtained a college scholarship in 1904, but he quickly lost it by failing in nonmathematical subjects. Another try at college in Madras (now Chennai) also ended poorly when he failed his first Arts exam. It was around this time that he began his famous notebooks.
JOURNEY TO WEST
In 1907 when Ramanujan started thinking of a career in Mathematics, he was poor, had no formal college education, and desperately needed a benefactor. Seshu Aiyar, a professor at Presidency College, Madras, suggested Ramanujan write letters to G.H. Hardy, a Fellow of the Royal Society and Cayley Lecturer in Mathematics at Cambridge, a celebrated mathematician who was 10 years senior to Ramanujan.
Prof. Hardy puzzled over nine pages of mathematical notes that Ramanujan had sent. They seemed rather incredible. He reviewed the papers with J. E. Littlewood, another eminent Cambridge mathematician, telling Littlewood that they had been written by either a crank or a genius, but he wasn’t sure which. After spending two and a half hours poring over the outlandishly original work, the mathematicians concluded. They were looking at the papers of a mathematical genius!
- The genius of Carl Jacobi, one of the greatest mathematicians in history
- The life and work of Leibniz, the father of integral calculus
- The mathematician who set the stage for Albert Einstein
Hardy, intrigued by Ramanujan’s letter and notes, took them to Cambridge’s colleagues. And here began a different era in mathematics which showed the world the beauty of mathematics, with an example of everlasting friendship proving how a passion can break all the barriers of ethical, cultural, religious differences.
“It was his insight into algebraic formulae, transformations of infinite series, and so forth that was most amazing. On this side most certainly I have never met his equal, and I can compare him only with Euler or Jacobi.”G.H. Hardy
Hardy was eager for Ramanujan as he arrived in April 1914, three months before World War I. Within days, he began his work with Hardy and Littlewood. Two years later, he was awarded the equivalent of a Ph.D. for his work. Ramanujan’s prodigious mathematical output amazed Hardy and Littlewood. The notebooks he brought from India were filled with thousands of identities, equations, and theorems he discovered for himself in 1903-1914. Earlier mathematicians had discovered some; some, through inexperience, were mistaken; many were entirely new.
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“I had never seen anything in the least like them before. A single look at them is enough to show that could only be written by a mathematician of the highest class. They must be true because, if they were not true, no one would have the imagination to invent them.G.H. Hardy
RAMANUJAN’S EXTRAORDINARY MATHEMATICAL OUTPUT
Ramanujan’s had very little formal training in mathematics, and indeed large areas of mathematics were unknown to him. He had a strong belief in God. He said the Hindu Goddess Namagiri had appeared in one of his dreams. He quoted:
“While asleep, I had an unusual experience. There was a red screen formed by flowing blood, as it were. I was observing it. Suddenly a hand began to write on the screen. I became all attention. That hand wrote several elliptic integrals. They stuck to my mind. As soon as I woke up, I committed them to writings.”
Prof. Bruce Berndt is an analytic number theorist who, since 1977, has spent decades researching Ramanujan’s theorems. He has published several books about them. He was told an interesting story by Hungarian mathematician Paul Erdos about something G. H. Hardy had once told him:
“Suppose that we rate mathematicians on the basis of pure talent on a scale from 0 to 100. Hardy gave himself a score of 25, Littlewood 30, Hilbert 80 and Ramanujan 100”Paul Erdos
Number theory and String theory
In 1918 Ramanujan became the first Indian mathematician to be elected a Fellow of the British Royal Society. In his short lifetime, he produced almost 4000 proofs, identities, conjectures, and equations. His Theta function lies at the heart of String theory in physics.
“….each of the 24 modes in the Ramanujan function corresponds to a physical vibration of a string. Whenever the string executes its complex motions in space-time by splitting and recombining, a large part of highly sophisticated mathematical identities must be satisfied.These are precisely the mathematical identities discovered by Ramanujan.”Michio Kaku
Get pi fast
In his notebooks, Ramanujan wrote down 17 ways to represent 1/pi as an infinite series. Series representations have been known for centuries. For example, the Gregory-Leibniz series, discovered in the 17th century, is pi/4 = 1 – 1/3 + 1/5 -1/7 + … However, this series converges extremely slowly; it takes more than 600 terms to settle down at 3.14, let alone the rest of the number. Ramanujan came up with something much more elaborate that got to 1/pi faster: 1/pi = (sqrt(8)/9801) * (1103 + 659832/24591257856 + …). This series gets you to 3.141592 after the first term and adds 8 correct digits per term thereafter. This series was used in 1985 to calculate pi to more than 17 million digits even though it hasn’t yet been proved!
The Taxicab Number
In a famous anecdote, Hardy took a cab to visit Ramanujan. When he got there, he told Ramanujan that the cab’s number 1729 was “rather a dull one.” Ramanujan said, “No, it is an exciting number. It is the smallest number expressible as a sum of two cubes in two different ways. That is, 1729 = 1^3 + 12^3 = 9^3 + 10^3.” This number is now called the Hardy-Ramanujan number, and the smallest numbers that can be expressed as the sum of two cubes in n different ways have been dubbed taxicab numbers. The next number in the sequence, the smallest number that can be expressed as the sum of two cubes in three different ways, is 87,539,319.
Ramanujan died aged 32 in Madras on April 26, 1920, due to hepatic amoebiasis caused by liver parasites. Hardy and Ramanujan’s bond was firm. Hardy served as a father-figure to Ramanujan, a distant, impersonal father who was the ideal taskmaster and had high expectations of Ramanujan.
“For my part, it is difficult for me to say what I owe to Ramanujan- his originality has been a constant source of suggestion to me ever since I knew him, and his death is one of the worst blows I have ever had.”
–G. H. Hardy