Srinivasa Ramanujan

The Extraordinary Life And Work Of Ramanujan

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
Courtesy:Wikipedia
Srinivasa Ramanujan
Image Courtesy:Wikipedia

Srinivasa Ramanujan was born on December 22, 1887 in the town of 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 singer at the local temple. Born in Hindu Brahmin family, his mother ensured the boy was in tune with Brahmin traditions and culture. Although his family were high caste, they were very poor.

EARLY MATHEMATICS

His interest and devotion to mathematics was to the point of obsession. He ignored everything else and would play with numbers day and night on a slate and in his mind. His interest in mathematics was unlocked by a book. 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 rigour required in mathematics. Ramanujan made the book his constant companion and improved it further on his own. 

EARLY FAILURES

His obsession and preoccupation with mathematics did not allow him to pass his intermediate examination in spite of 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 scholarship to college in 1904, but he quickly lost it by failing in non mathematical 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. 

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JOURNEY TO WEST

In year 1907 when Ramanujan started thinking of a career in Mathematics, he was poor, had no formal college education and desperately needed a benefactor. It was Seshu Aiyar, a professor at Presidency College, Madras, who suggested Ramanujan to 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 quite sure which. After spending two and half hours poring over the outlandishly original work, the mathematicians came to a conclusion. They were looking at the papers of a mathematical genius!

Also watch: The Genius of Ramanujan

Hardy intrigued by Ramanujan’s letter and notes, is said to have taken them to his colleagues in Cambridge. And here began a different era in mathematics which showed the world the beauty of mathematics, with 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
Ramanujan (centre) and his colleague G. H. Hardy (extreme right), with other scientists, outside the Senate House, Cambridge,
Image Courtesy:Wikipedia

Hardy was eager for Ramanujan as he arrived in April 1914, three months before the outbreak of World War I. Within days, he began his work with Hardy and Littlewood. Two years later, he was awarded equivalent of a PhD 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 the years 1903-1914. Some had been discovered by earlier mathematicians; 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 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 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 screen. I became all attention. That hand wrote a number of elliptic integrals. They stuck to my mind. As soon as I woke up, I committed them to writings.”

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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 selected 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 – ? + ? -1/7 + … 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!

Also read: All the articles of Basics of Astrophysics series

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 a very interesting 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.

Hardy-Ramanujan “taxicab numbers”
Image Courtesy:www.storyofmathematics.com
Hardy-Ramanujan “taxicab numbers”
Image Courtesy:www.storyofmathematics.com

S Ramanujan died aged 32 in Madras on April 26,1920 due to hepatic amoebiasis caused by liver parasites. Hardy and Ramanujan’s bond was very strong, 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

Also Read:5 Important Lessons That Stephen Hawking Taught Us About Life

4 thoughts on “The Extraordinary Life And Work Of Ramanujan”

  1. We have been told that without the power of numbers from 0, nothing will exist on earth and the Universe. Numbers exist from the Unseen. O or zero is the Unseen.

  2. Pingback: 20 Most Inspirational Marie Curie Quotes

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