The Indian government celebrated the 138th birth anniversary of Srinivasa Ramanujan, a renowned mathematician, on December 22. Despite having little formal schooling and living in abject poverty, he rose to become one of India’s greatest mathematicians. His mathematical concepts influenced 20th century mathematics and continue to inspire 21st century mathematicians today. Srinivasa Ramanujan made significant contributions to the analytical theory of numbers, focusing on elliptic functions, continuous fractions, and infinite series. He was a brilliant mathematician who rose to global prominence at the age of 26. He was born into a humble family with no notable professional accomplishments, but his mathematical concepts altered and molded century mathematics and continue to inspire modern mathematicians. Srinivasa Ramanujan was recognized as a mathematical genius, comparable to Leonhard Euler and Carl Jacobi. Despite having no formal mathematical instruction, Ramanujan’s mathematical understanding was astounding. Despite his lack of information about recent breakthroughs in the area, he easily calculated the Riemann series, elliptic integrals, hyper geometric series, and zeta function functional equations. Ramanujan, one of India’s greatest mathematicians, was born in Erode on December 22, 1887. Erode was a small village (at the time) about 400 kilometers from Tamil Nadu’s current metropolis, Chennai. His father was a clerk in Kumbkonam. Srinivasa Ramanujan began attending school as a pupil when he was five years old. It was only a matter of time before he was recognized for his amazing talent. He displayed bursts of brilliance that were unusual for a child his age. He finished his primary education in a few years and then transferred to Town High School for additional study. He showed an outstanding interest in mathematics. When he was still in school, he computed the approximate length of the earth’s equator. He was quite familiar with the values of the square root of two and the pie value. At the age of 16, he was awarded a scholarship. However, his enthusiasm for mathematics cost him the scholarship because he disregarded and failed other subjects. He was distraught after losing his scholarship. He could not afford to study on his own. He had to get work and cease studying permanently. He got a job as an accounting clerk at the Madras Port Trust. Despite being denied twice, his work was recognized by G.H. Hardy and J. E. Littlewood, and he traveled to England in 1914. Cambridge University awarded him a B.Sc. (later Ph.D.) degree in 1916 for his research on highly composite numbers. In 1916, when he was at his peak while working with his colleagues Hardy and Littlewood, he developed health concerns. He was hospitalized in Cambridge and diagnosed with tuberculosis and vitamin deficiencies. In 1919, after two years of struggle, he showed signs of improvement and chose to return to India. However, the relief was only brief; after arriving in Bombay, his health deteriorated further, and he died on April 26, 1920. His major contributions to mathematics are in the fields of analysis, infinite series, number theory, and game theory. His genius was in discovering his own theorems. Because of his outstanding contributions in mathematics, the Indian government declared his birthday, December 22nd, as Mathematics Day. ISTE, New Delhi, and NBHM, Mumbai, have taken the initiative to host a mathematical competition for students and professors of colleges in the name of Srinivasa Ramanujan from 2012 to the present so that students and teachers in India become aware of the legacy of such a great mathematician.
Ramanujan Summation is a mathematical technique used to assign finite values to divergent infinite series, most famously suggesting that the sum of all natural numbers is −1/12. While mathematically unconventional compared to traditional addition, this method is essential in modern physics—particularly string theory—to resolve infinite values into functional, finite results.
In this article we introduce some of Srinivasa Ramanujan’s contributions to mathematics.
Goldbach’s Conjecture : Goldbach’s conjecture is one of the important illustrations of Ramanujan contribution towards the proof of the conjecture. The statement is every even integer > 2 is the sum of two primes, that is, 6 = 3 + 3. Ramanujan and his associates had shown that every large integer could add sum of at most four prime numbers. (For example: 43 = 2 + 5 + 17 + 19).
Theory Of Equations: Ramanujan was taught how to solve cubic equations in 1902, and he went on to develop his own method for solving quadratics. He developed the formula for solving biquadratic equations. The following year, he attempted to present the formula for solving quintic, but was unable to do so since he was unaware that quintic could not be solved using radicals.
Magic Squares: As a young student, Srinivasa Ramanujan was interested in magic squares. The simplest magic square problem is to fill up the cells in a square with 3 rows and 3 columns with the numbers 1,2,3··· 9 such that each row sum = each column sum = each diagonal sum. One can find the following solution:
4 9 2
3 5 7
8 1 6
In this magic square, each row sum = each column sum = each diagonal sum = 15. This magic square requires an understanding of how 15 can be written as a sum of 3 non-repeated numbers.
Hardy-Ramanujan Number: Hardy once paid a visit to Putney, where Ramanujan was in the hospital. He arrived there in a taxi cab bearing the number 1729. Hardy was very superstitious due to his such nature, when he entered Ramanujan’s room, he said that he had just come in a taxi cab with number 1729, which seemed to him an unlucky number, but at that time, he prayed that his perception may go wrong as he wanted that his friend would get well soon, but Ramanujan promptly replied that this was a very interesting number as it is the smallest number which can be expressed as the sum of cubes of two numbers in two different ways: 1729 = 93+103 and 1729 = 13 + 123.
Ramanujan Summation: Ramanujan Summation is a technique developed by the brilliant Indian mathematician Srinivasa Ramanujan to assign finite values to divergent infinite series. He is most known for asserting that 1 + 2 + 3 + 4 + = – 1 12, which is a value derived from deep series manipulation rather than a traditional sum. This technique is used in physics, such as string theory, to resolve infinities and make sense of otherwise nonsensical results.
(The authors are freelancers. The views, opinions and conclusions expressed in this article are those of the authors and aren’t necessarily in accord with the views of “Kashmir Horizon”)
Dr. Mohsin Ahmad, Dr. Rasekh Ali, Farhana Rehman
