Srinivasa Ramanujan And His Inventions

Mathematics is a very difficult but interesting subject with lots of rules, theorems to solve questions in practical life, and whenever one will have to name The biggest mathematician the first name would be Srinivasa Ramanujan for his incredible contribution in the history of maths.

Ramanujan was an incredible and great mathematician. His life started from a small village, in the southern part of the country, Erode in Madras on 22 December 1887. His full name was Srinivasa Iyengar Ramanujan. When he was born, he was considered as the blessing of the goddess Namakkal, a local God of the region. Shortly after his birth, his family shifted to Kumbakonam where his father worked as a clerk in a cloth shop. He went to an elementary school at age of 5 and stood first in the primary school in the whole district.

Early Life and Education

S. Ramanujan’s childhood differentiated him from his peers. Unlike many other children of the same age, he was not naughty. On the contrary, he exhibited an unusual calm and pensive manner along with a phenomenal memory.

He was a self-willed child and carried out the activities as he liked within his time frame. What set Ramanujan apart was not only his passionate love of mathematics but also the intrinsic curiosity for the world that he had from an early age. From early childhood, he acquired the habit to question his mother on various specifics of the world.

Ramanujan’s prompt into mathematics is traced to a critical instance when he came home sobbing following the defeat in math by his classmate. At first, he attended to all subjects in his early education but as years passed, the precociousness of mathematics began emerging.

He usually solved math problems mentally even before the teacher would write them on a blackboard. He also gave easier and prompt approaches to teachers that served as an alternative method of solving problems. Whenever he wrote the mathematics paper during examinations, he would always finish in half of the time allotted.

The teachers at S. Ramanujan’s school loved him for his special nature and outstanding talent, they praised him in the classroom openly. As a child, he struggled with deep questions like what is the absolute truth in mathematics and often interrogated his maths teachers about these topics.

In January of 1898, he entered the Town High School in Kumbhakonam where his illustriousness suddenly outshone during a division mathematics class. This was where Ramanujan’s life-long passion for zero and infinity started, learning that anything divided by zero results to infinity.

He could memorize the values of pi (n), e, square root of two and others to any number places anytime. Apart from his mathematical explorations, Ramanujan turned to the ancient Indian philosophical books: Vedas, Upanishads and Hindu scriptures. He showed his ability to recite shlokas from these scriptures and also comment on the meaning of what he said.

When he was 15 he obtained a book named "A Synopsis of Elementary Results in Pure and Applied Mathematics" which was written by George Schoobridge Carr. In 1903 he passed his class 11th, Matriculate Exam, and won the 'Subramanyam Scholarship' for his English and Maths.

Ramanujan's greatest asset also proved to be his loss as he lost his scholarship of both the government college and later at the University of Madras because of his devotion to maths which led to the fall of his other subjects. In 1911, Ramanujan published a 17-page paper on Bernoulli numbers in the Journal of Indian Mathematical Society. He joined a clerical job in the Taluk office after marrying a girl named Janaki.

Journey into Mathematics:

Earning early life recognition, S. Ramanujan won the famous K. Rangnanatha Rao Prize for attaining first position in Mathematics during a school exam After this accomplishment, he went on to excel in Mathematics and English by winning the Junior Subrahmanyam Scholarship for his splendid performance following a school competitive examination series.

In the course of this time, one boarder in Ramanujan’s house made him aware to book on Trigonometry by S. L. Being a suggested textbook in colleges at this time, the book had an important role to play for Ramanujan’s perception of the topic. Even though he was thirteen years old, it did not take him long to learn the intricacies of Trigonometry. Nonetheless, when he climbed the ranks into higher classes problems emerged in his studies of; English, Greek and Roman History as well as Physiology. His incredible love for Mathematics stole his away from other subjects, making him do poorly in them.

When he shifted to F.A. (First Arts) in the Government College, Kumbhakonam, Ramanujan’s attraction towards Mathematics grew stronger. Mathematics came to him as a deluge and wasted in his studies of other subjects much time. His passion for Mathematics grew even wider when he read George Shoobridge Carr’s book- A Synopsis of Elementary Results in Pure and Applied Mathematics, published in 1886. It did not include discussions or step-by-step proofs but consisted of about six thousand problems in algebra, trigonometry calculus and analytical geometry. It gave clues to help sort out the issues. This book changed Ramanujan’s outlook on life.

All of the problems in Carr’s work became Ramanujans research projects. Having received no high-level math training, he tackled these problems with a ‘novel’ way. Though some of the problems had been resolved by European mathematicians over a century and half ago, Ramanujan cleverly dealt with them in his own style.

Sometimes he proposed new ways and explored the hitherto uncharted territories of mathematics, which was ahead of European mathematicians. Unaware he made amazing discoveries through this process. Ramanujans’ calculations which he found to be already a part of some higher mathematics text, were habitually concealed by him.

Ramanujan met the founder of the Indian Mathematical Society and showed his work to him who suggested him to go to P.V. Seshu Iyer who sent him to Diwan Bahadur at Nellore district. After that, he wrote an article to G.H. Hardy on 16 January 1930. This letter is known as the most important letter in Maths History.

The Most Important Letter

In a crucial 1913 letter to Hardy, Ramanujan furnished one hundred and twenty theorems as evidence of his work. Some of these formulas were known to Hardy from his studies, others original.

Hardy took more than two hours to examine the letter thoroughly and, he discussed his suspicions with J.E Littlewood of Trinity College an esteemed colleague. After three more hours of deliberation, they agreed that the work was indeed created by a mathematician genius. This convinced Hardy, who stated that “They must be true because if they were not so no one would have had the audacity to invent them,” and thus Ramanujan was invited on entering Trinity College alongside with him.

He improved his work by working with G.H. Hardy and J.E. Littlewood who were two of the great mathematicians of that time. In England, he was elected to the London Mathematical Society and he became the first Indian to become a "Fellow of the Royal Society".

Inventions Of Ramanujan

It is clear that being such a great and incredible genius he would have invented many formulas and theorems. Ramanujan had really invented n number of formulas but we will have a look at some Srinivasa Ramanujan inventions and contributions.

1.  Landau- Ramanujan constant
2.  Mock Theta Functions
3.  Ramanujan Conjecture
4.  Ramanujan Prime
5.  More than 3900 identities
6.  Number Theory
7. Ramanujan- Soldner Constant
8.  Ramanujan- Theta Function
9.  Ramanujan's Sum
10.  Ramanujan's Magic Square
11.  Ramanujan's Number (1792)
12.  Rogers- Ramanujan's Identities
13.  Ramanujan's Master Theorem
14.  Ramanujan- Sato Series
15. Complex Analysis  
16. Infinite Series  
17. Continued Fractions

Ramanujan gathered about 3900 results, including equations and identities. It was also his finite series for the calculation of pi that formed a basis for many algorithms used today. He offered some interesting formulas for the calculation of pi digits via alternative methods.

Exploring different problems in maths, he came up with several revolutionary solutions that played a vital role in the development of game theory. Intuition and instinct naturally propelled his contributions to this field, unmatched even today.

Ramanujan intimately described the mock theory function, an idea in modular forms of mathematics. Initially viewed as a mystery, it is now accepted as holomorphic parts of mass forms.

George Andrews found Ramanujan’s notebook in the Trinity College library in 1976 and published it as a book.

The 1729 is unique as the Ramanujan number. It is the total of two numbers in cube form ten and eleven respectively. In particular, totaling up to1729 are the sum of cube values for both 10 and its product with a value [multiplication][operation] nine. Interestingly, 1729 is the smallest number that can be written in this way as a sum of two cubes. Interestingly it follows the natural order, though coming after 1728 and before 1730.

Inventions of Ramanujan are not limited to complex analysis but also cover number theory, infinite series and continued fractions.

Collaboration with Hardy:

Apart from his work in number theory and mathematical analysis, G.H Hardy is known for the fact that he was associated with Srinivasa Ramanujan- a self-taught Indian maths prodigal.

A mathematical genius himself, Hardy was a human calculator from an early age. He not only had the math skills but also graduated with honors at Cambridge University where then launched his academic career.

Hardy was a central figure in the reform of British mathematics as early 20th century, substituting Continental rigor with traditional applied mathematics. Being a pacifist, he proudly announced that the work had no commercial or military value.

In an overly dramatic act that bought mathematical news prior to World War I, Hardy claimed he had proved the Riemann Hypothesis. Despite the fact that he showed many zeroes on critical line, he failed to prove the presence of others.

Hardy brought Ramanujan to Cambridge, recognizing his genius and working with him on many mathematical issues – but the Riemann Hypothesis remained a dreadlocked mystery.

This is reflected in the unusual stories from Ramanujan’s life, for instance, a well-known taxi number incident when Hardy was transported by cab 1729 which he regarded as trivial. The expression was instantly recognized by Ramanujan as the smallest number that is representable in two ways as a sum of cubes, terming it taxicab numbers.

Although Srinivasa Ramanujan invention was incredibly brilliant, he fell to depression and illness committing suicide at one point past his 32 years of age.

Having lived another 27 years, Hardy deemed the discovery of Ramanujan as his greatest contribution to maths and even referred to their collaboration a romantic incident. Two of their destinies being intertwined, the Riemann Hypothesis has come to be associated with both as something quasi-cursed.

In one of his last letters to Hardy, written on a ‘deathbed’, Ramanujan described the theory known as “mock theta functions”, which has set fire in 21st Century mathematics once again.

Despite the fact that for most of 20th century Ramanujan’s definition was neglected because it is excessively clear, today gain prominence with recent developments in research into mock modular forms.

Ramanujan’s land mark paper of 1916 laid the foundation for the theory modular forms. But with hindsight, it was his last letter that opened the way for more daring mathematical ventures in the twenty-first century.

The mock theta functions initially disregarded and treated at intervals was problematic because Ramanujan had no well-defined definition. Instead, he furnished 17 typical instances complete with hypotheses that kept mathematicians guessing for generations.

Throughout the 20th century, mathematicians failed to achieve these goals without ultimate unifying conceptual framework and achieved success on a major scale. But the breakthrough was achieved in 2002, when S. Zwegers completed his doctoral thesis under D. Zagier’s supervision introducing the idea of mock modular forms and creating a necessary framework

Mock modular forms have been identified as a special case of a larger theory including the classical theory of Hecke. In the recent past, works by mathematicians such as J. Bruinier, J. Funke and K Bringman have looked into this rich theory thus providing new information in mathematics This is demonstrated through the algebraic formula for partition function by Bruinier and Ono based on mock modular forms theory.

Foreshadow

Freeman Dyson, a physicist in 1987 predicted that such mock theta functions might make grand synthesis. In fact, the recent breakthroughs confirm Dyson’s prophesy which shows that mock modular forms can potentially fit into the greater Langlands program.

Finally, Hardy attests to Ramanujan’s unique vision of algebraic formulae and infinite series transformations. Inventions of Ramanujan involved laboring through a lot of numerical examples, from which properties were discovered by induction.

However, Hardy emphasized Ramanujan’s remarkable memory with patience for calculations and generalization ability along with the rapid flexibility of hypotheses. He compared him to Euler or Jacobi.

It is Ramanujan’s original pieces, his last writings that continue to inspire mathematicians and they shape the mathematical horizon of today.

Legacy

Nevertheless, the cultural heritage left by Srinivasa Ramanujan invention is much more than his mathematical work. A legend of an India on the path to liberation during British colonialism, he has emerged as a quintessential representation of rising great power in 20th century.

 To meet the challenges of an independent India, this transformative period required renewal in science and education. The comeback was spearheaded by Ramanujan but as eloquently stated in his tribute on the centenary of Ramanujans birth, Novel laureate Subramanyam Chandrasekhar.

Chandrasekhar remembered the day he heard about Ramanujan’s death in 1920 and commented on how happy it made him to know that someone from a background similar his had attained international recognition.

 Ramanujan’s travel, from a scientifically limited setting in India to collaboration with leading English mathematicians at Cambridge served as an inspiration. Though he came back to India returning with a promise of recognition as trailblazing mathematician, Ramanujan’s story was inspiring other Indian students who could take the shine off intellectual shackles and become free like him.

Chandrasekhar, who had the same background as that of Ramanujan studied at Cambridge and became a top astrophysicist winning Nobel Prize in 1983. His trajectory mirrored Ramanujan's ascent.

While science knows no borders, Ramanujan’s influence spans the world. Scientists everywhere claim to be inspired by his biography.

Ramanujan represents the genius of human mind to create ideas and use equations, formulas and symbols as instruments for thinking in order to plunge into mysteries of universe or one’s own life.

The legacy of Ramanujan will live on so long as the spirit to investigate prevails, which would be passed from one generation to another.

Conclusion

Two decades have already passed after his death, but Ramanujan’s mathematical discoveries live on and continue to develop. Ramanujan invention is historically important not only as a mathematician but also because he demonstrates the power of man’s mind.”

People like this are very rare, and to lose one representation in such individuals has vast implications. Genius can arise as easily anywhere in the world, and happily, he was among us. Sadly, very little is known of Ramanujan’s life and complex array of work which makes it unfamiliar even though esoteric.

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