Srinivasa Ramanujan – Who discovered Infinity

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Born

Ramanujan was born on 22 December 1887 in ErodeMadras Presidency (now Pallipalayam, Erode, Tamil Nadu), at the residence of his maternal  grandparents.

Family

Ramanujan's Home
Ramanujan’s Home, Kumbakonam (Source-www.ramanujanmuseum.org)

His father, K. Srinivasa Iyengar, worked as a clerk in a sari shop and hailed from the district of Thanjavur. His mother, Komalatammal, was a housewife and also sang at a local temple. From his mother, he learned about tradition and puranas. He learned to sing religious songs, to attend pujas at the temple as Ramanujan was very spiritual as her mother.

They used to live in Sarangapani Street in a traditional home in the town of Kumbakonam. The family home is now a museum. On 14 July 1909, Ramanujan was married to Janakiammal (21 March 1899 – 13 April 1994) when she was just ten-year old bride. She belonged from Rajendram, a village close to Marudur (Karur district) Railway Station.

Education & Career

Ramanujan moved with his mother to her parents’ house in Kanchipuram, near Madras (now Chennai) due to the dieses of smallpox. On 1 October 1892, Ramanujan took admission at the local school.  Soon after, Ramanujan and his mother moved back to Kumbakonam  as his maternal grandfather lost his job as a court official in Kanchipuram and Ramanujan was enrolled in the Kangayan Primary School.

Soon his paternal grandfather died and he was again sent back to his maternal grandparents, in Madras. But he did not like school in Madras, many a times he attempted to avoid attending classes. However his family hired a local constable to make sure he attended school but within six months, Ramanujan left to Kumbakonam.

Finally at the Kangayan Primary School, Ramanujan did well in studies. In November 1897, he passed his primary examinations and stood first in the district. In the same year, Ramanujan entered Town Higher Secondary School where he studied formal mathematics for the first time.

By age of 11, he gathered quite of the mathematical knowledge and latter on he borrowed a book on advanced trigonometry written by S. L. Loney. He completely mastered this book by the age of 13 and discovered sophisticated theorems on his own. By 14, he received merit certificates and academic awards which continued throughout his school career.  He was so brilliant in mathematics that he completed mathematical exams in half the allotted time. He showed great deal of interest and familiarity with geometry and infinite series.

In 1930, at the age of 16 Ramanujan started reading book on pure and applied mathematics having huge collections of theorems, which became the key element in awakening the genius. The very next year, he had independently developed and investigated the Bernoulli numbers and had calculated the Euler–Mascheroni constant up to 15 decimal places on his own.

At the age 21, he graduated from Town Higher Secondary School in 1904, with the K. Ranganatha Rao prize for his brilliance in mathematics by the school’s headmaster, Krishnaswami Iyer.  He also received a scholarship to study at Government Arts College, Kumbakonam. But Ramanujan was so intent on studying mathematics that he could not focus on any other subjects and failed most of them, losing his scholarship. In August 1905, he ran away from home, to Visakhapatnam.  After a month’s struggle he enrolled at Pachaiyappa’s College in Madras. Once again he showed his extraordinary talent  in mathematics but performed poorly in other subjects. Ramanujan failed for two consecutive years in his Fellow of Arts exam in 1906 and finally left the college without a degree. Later on he initiated to pursue independent research in mathematics while living in extreme poverty and was often on the brink of starvation.

Once Ramanujan met the deputy collector V. Ramaswamy Aiyer, who was the founder of the Indian Mathematical Society, for a job at the revenue department.

But after having seen his mathematical work, Ramaswamy Aiyer sent Ramanujan, with letters of introduction, to his mathematician friend  R. Ramachandra Rao, the district collector for Nellore and the secretary of the Indian Mathematical Society, in Madras.

Ramachandra Rao was impressed by Ramanujan’s research but doubted that initially. But latter on he Started his mathematical research with Rao’s financial aid taking care of his daily needs. Ramanujan, with the help of Ramaswamy Aiyer, had his work published in the Journal of the Indian Mathematical Society.

Ghhardy@72
G. H. Hardy

In 1913, Ramanujan wrote letters  about his work to few British mathematicians like  M. J. M. Hill ,  H. F. Baker,  E. W. Hobson, and G. H. Hardy. Everyone rejected his except G.H Hardy who recognised some of Ramanujan’s formulae. English mathematician G. H. Hardy received a strange letter from Ramanujan. The ten-page letter contained about 120 statements of theorems on infinite series, improper integrals, continued fractions, and number theory. After having seen Rumanujan’s work Hardy commented that “these theorems defeated me completely; I had never seen anything in the least like them before“. Some of them are:Capture3

(valid for 0 < a < b + 1/2)

Every prominent mathematician gets letters from cranks, and at first glance Hardy no doubt put this letter in that class. But something about the formulas made him take a second look, and show it to his collaborator J. E. Littlewood. After a few hours, they concluded that the results “must be true because, if they were not true, no one would have had the imagination to invent them”. One of the theorems Hardy found scarcely possible to believe which isCapture4

The first result had already been determined by a mathematician named Bauer. The second one was new to Hardy, and was derived from a class of functions called a hypergeometric series which had first been researched by Leonhard Euler and Carl Friedrich Gauss.

On 8 February 1913, Hardy wrote a letter to Ramanujan, expressing his interest for his work. Ramanujan boarded the S.S. Nevasa on 17 March 1914, and at 10 o’clock in the morning, the ship departed from Madras. He arrived in London on 14 April and immediately began his work with Littlewood and Hardy.  Ramanujan left a deep impression on Hardy and Littlewood. Littlewood once commented, “I can believe that he’s at least a Jacobi”, while Hardy said “he can compare him only with Leonhard Euler or Jacobi”. Ramanujan spent nearly five years in Cambridge collaborating with Hardy and Littlewood and published a part of his findings there.

Ramanujan was awarded a Bachelor of Science degree by research (this degree was later renamed PhD) in March 1916 for his work on highly composite numbers, the first part of which was published as a paper in the Proceedings of the London Mathematical Society.

On 6 December 1917, he was elected to the London Mathematical Society. He became a Fellow of the Royal Society in 1918, becoming the second Indian to do so, following Ardaseer Cursetjee in 1841, and he was one of the youngest Fellows 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. 

Known For

Ramanujan had remarkable capabilities of giving rapid solution for problems. One examples of his brilliance is when he was sharing a room with P. C. Mahalanobis who once posted a problem to Ramanujan, which is

Imagine that you are on a street with houses marked 1 through n. There is a house in between (x) such that the sum of the house numbers to left of it equals the sum of the house numbers to its right. If n is between 50 and 500, what are n and x?” This is a bi-variate problem with multiple solutions. Ramanujan gave the answer with a twist: He gave a continued fraction. Mahalanobis was amazed  and asked how he did it. “It is simple. The minute I heard the problem, I knew that the answer was a continued fraction. Which continued fraction, I asked myself. Then the answer came to my mind, Ramanujan replied. 

Ramanujan’s talent suggested a plethora of formulae, out of which the most interesting of these include the intriguing infinite series for π, one of which is given belowFormula

This result is based on the negative fundamental discriminant d = −4×58 = −232 with class number h(d) = 2 (note that 5×7×13×58 = 26390 and that 9801=99×99; 396=4×99).

His intuition also led him to derive some previously unknown identities, such asFormala1

for all theta .

In 1918, Hardy and Ramanujan studied the partition function P(n) extensively and gave a non-convergent asymptotic series that permits exact computation of the number of partitions of an integer. Hans Rademacher, in 1937, was able to refine their formula to find an exact convergent series solution to this problem. One remarkable result of the Hardy-Ramanujan collaboration was a formula for the number p(n) of partitions of a number n. A partition of a positive integer n is just an expression for n as a sum of positive integers, regardless of order. Thus p(4) = 5 because 4 can be written as 1+1+1+1, 1+1+2, 2+2, 1+3, or 4. Ramanujan and Hardy’s work in this area gave rise to a powerful new method for finding asymptotic formulae, called the circle method.

Ramanujan’s other works are Landau–Ramanujan constant, Ramanujan theta function, Ramanujan tau function, Ramanujan’s sum, Ramanujan’s master theorem and  in the end  Mock theta function

He discovered mock theta functions in the last year of his life. The first examples of mock theta functions were described by Srinivasa Ramanujan in his last 1920 letter to G. H. Hardy and in his lost notebook. For many years these functions were a mystery.

Died

Ramanujan returned to Kumbakonam, Madras Presidency in 1919 and died soon thereafter at the age of 32.  He was diagnosed with tuberculosis and a severe vitamin deficiency and was confined to a sanatorium.

Honor

In December 2011, in recognition of his contribution to mathematics, the Government of India declared that Ramanujan’s birth date (22 December) would be celebrated every year as National Mathematics Day and declared 2012 the National Mathematics Year.