Srinivasa Ramanujan's Contributions in Mathematics

 

 Ramanujan, one of the elegant Mathematician of India was born in Erode on 22nd December 1887. Erode is a small village (in that time), 400 Km away from Tamilnadu’s present capital Chennai. His father was a clerk in Kumbkonam.At the age of five, Srinivasa Ramanujan made his first appearance in school as a student. It was only a matter of time before it came to be known that he had extraordinary talent. He showed flashes of brilliance which were not to be seen in any ordinary kid at that age. He completed his primary education in a couple of years and then went to Town High School for further studies. He showed extraordinary liking for mathematics. When he was yet in school, he mathematically calculated the approximate length of earth's equator. He very clearly knew the values of the square root of two and the pie value. At the age of 16, he got scholarship. But his love only for mathematics cost him the scholarship as he neglected and failed in other subjects. His loss of scholarship was a great blow to him. He could not afford to study on his own. He had to find work and leave studies for good. He found a job of an accounts clerk in the office of the Madras Port Trust. Despite being rejected two times, his work was recognized by both G. H. Hardy and J. E. Littlewood and he went to England in 1914. In 1916,he was awarded with a degree of B.Sc. (later named Ph.D.) by Cambridge University for his work on highly composite number. In 1916, when he was at his best while working with his colleagues Hardy & Littlewood, he met with health problems. He was hospitalized in Cambridge and was diagnosed with T.B. and vitamin deficiency. After two years struggle, in 1919, he showed some recovery and he decided to return back to India. However, the improvement was temporary and after his arrival at Bombay, his health deteriorated again and finally he passed away on 26th April, 1920.His main contributions in mathematics lie in the field of Analysis, Infinite Series, Number Theory & Game Theory. His geniusness was that he discovered his own theorems. Due to his great achievements in the field of Mathematics, Indian govt. decided to celebrated his birthday 22nd December as Mathematics Day. ISTE, New Delhi and NBHM, Mumbai have taken initiative to hold Mathematical competition for students as well as teachers of colleges on the name of Srinivasa Ramanujan from 2012 to till date so that students and teachers of India know about the legacy of such great mathematician of India.

I. Hardy-Ramanujan Number

Once Hardy visited to Putney where Ramanujan was hospitalized. He visited there in a taxi cab having number 1729. Hardy was very superstitious due to his such nature when he entered into Ramanujan’s room, he quoted that he had just came in a taxi cab having 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 as given below:

1729 =1³+12³=10³+9³     

Later some theorems were established in theory of elliptic curves which involves this fascinating number.

II. 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 be written as the sum of at most four (Example: 43 = 2+5+17+19).

III. Theory of Equations

Ramanujan was shown how to solve cubic equations in 1902 and he went on to find his own method to solve the quadratic. He derived the formula to solve biquadratic equations. The following year, he tried to provide the formula for solving quintic but he couldn’t as he was not aware of the fact that quintic could not be solved by radicals.

IV. Ramanujan-Hardy Asymptotic Formula

Ramanujan’s one of the major work was in the partition of numbers. By using partition function p(𝑛), he derived a number of formulae in order to calculate the partition of numbers. In a joint paper with Hardy, Ramanujan gave an asymptotic formulas for p(𝑛). In fact, a careful analysis of the generating function for p(𝑛) leads to the Hardy-Ramanujan asymptotic formula given by

                                            p(n) ~ 1/(4n√3) e^π√(2n/3) ,  n→∞

In their proof, they discovered a new method called the ‘circle method’ which made fundamental use of the modular property of the Dedekind η-function. We see from the Hardy-Ramanujan formula that p(𝑛) has exponential growth. It had the remarkable property that it appeared to give the correct value of p(𝑛) and this was later proved by Rademacher using special functions and then Kenono gave the algebraic formula to calculate partition function for any natural number 𝑛.

 V. Ramanujan’s Congruences

Ramanujan’s congruences are some remarkable congruences for the partition function. He discovered the congruences

                        (5𝑛+4) ≡0 (𝑚𝑜𝑑 5)

                          (7𝑛+5) ≡0 (𝑚𝑜𝑑7)

                         (11𝑛+6) ≡0(𝑚𝑜𝑑 11), ∀𝑛∈𝑁.

In his 1919 paper, he gave proof for the first two congruences using the following identities using Pochhammer symbol notation. After the death of Ramanujan, in 1920, the proof of all above congruences extracted from his unpublished work.

VI. Highly Composite Numbers

A natural number n is said to be highly composite number if it has more divisors than any smaller natural number. If we denote the number of divisors of n by d(n), then we say 𝑛 є N is called a highly composite if 𝑑 (𝑚) <𝑑(𝑛) ∀𝑚 < 𝑛 where 𝑚, n∊ N. For example, 𝑛 = 36 is highly composite because it has (36) = 9 and smaller natural numbers have less number of divisors. If

                       n = 2^k₂  3^k₃…p^k_p    (by Fundamental theorem of Arithmetic)

is the prime factorization of a highly composite number 𝑛then the primes 2,3,..., 𝑝 form a chain of consecutive primes where the sequence of exponents is decreasing; i.e.𝑘₂ ≥ 𝑘₃ ≥.....≥ k_p and the final exponent 𝑘_p is 1, except for 𝑛 = 4 and 𝑛 = 36.

VII. Some Other Contributions

Apart from the contributions mentioned above, he worked in some other areas of mathematics such as hypogeometric series, Bernoulli numbers, Fermat’s last theorem. He focused mainly on developing the relationship between partial sums and products of hyper-geometric series. He independently discovered Bernoulli numbers and using these numbers, he formulated the value of Euler’s constant up to 15 decimal places. He nearly verified Fermat’s last theorem which states that no three natural number 𝑥, 𝑦 and 𝑧 satisfy the equation 𝑥ⁿ +𝑦ⁿ  = 𝑧ⁿ for any integer 𝑛 > 2.

PUZZLE

 

Top 10 Mathematical Innovations

 

1. Arabic numerals

 Great advances in Western European science followed the introduction of Arabic numerals by the Italian mathematician Fibonacci in the early 13th century. He learned them from conducting business in Africa and the Middle East. Of course, they should really be called Hindu numerals because the Arabs got them from the Hindus. In any case, mathematics would be stuck in the dark ages without such versatile numerals.

2. Calculus (Isaac Newton, Gottfried Leibniz)

Newton gets all the credit, even though Leibniz invented calculus at about the same time, and with more convenient notation (still used today). In any event, calculus made all sorts of science possible that couldn’t have happened without its calculational powers. Today everything from architecture and astronomy to neuroscience and thermodynamics depends on calculus.

4. Zero and 3. Negative numbers (Brahmagupta)

Brahmagupta, a seventh-century Hindu astronomer, was not the first to discuss negative numbers, but he was the first to make sense of them. It’s not a coincidence that he also had to figure out the concept of zero to make negative numbers make sense. Zero was not just nothingness, but a meaningful number, the number you get by subtracting a number from itself. Zero was not just a placeholder, writes Joseph Mazur in his new book Enlightening Symbols. For what may have been the first time ever, there was a number to represent nothing.

5. Decimal fractions (Simon Stevin, Abu’l Hasan Al-Uqlidisi)

Stevin introduced the idea of decimal fractions to a European audience in a pamphlet published in 1585, promising to teach how all Computations that are met in Business may be performed by Integers alone without the aid of Fractions. He thought his decimal fraction approach would be of value not only to merchants but also to astrologers, surveyors and measures of tapestry. But long before Stevin, the basic idea of decimals had been applied in limited contexts. In the mid-10th century, al-Uqlidisi, in Damascus, wrote a treatise on Arabic (Hindu) numerals in which he dealt with decimal fractions, although historians differ on whether he understood them thoroughly or not.

6. Binary logic (George Boole)

Boole was interested in developing a mathematical representation of the “laws of thought,” which led to using symbols (such as x) to stand for concepts (such as Irish mathematicians). He hit a snag when he realized that his system required x times x to be equal to x. That requirement pretty much rules out most of mathematics, but Boole noticed that x squared does equal x for two numbers: 0 and 1. In 1854 he wrote a whole book based on doing logic with 0s and 1s — a book that was well-known to the founders of modern computer languages.

7. Non-Euclidean geometry (Carl Gauss, Nikolai Lobachevsky, János Bolyai, Bernhard Riemann)

Gauss, in the early 19th century, was probably the first to figure out an alternative to Euclid’s traditional geometry, but Gauss was a perfectionist, and perfection is the enemy of publication. So Lobachevsky and Bolyai get the credit for originating one non-Euclidean approach to space, while Riemann, much later, produced the non-Euclidean geometry that was most helpful for Einstein in articulating general relativity. The best thing about non-Euclidean geometry was that it demolished the dumb idea that some knowledge is known to be true a priori, without any need to check it out by real-world observations and experiments. Immanuel Kant thought Euclidean space was the exemplar of a priori knowledge. But not only is it not a priori, it’s not even right.

8. Complex numbers (Girolamo Cardano, Rafael Bombelli)

Before Cardano, square roots of negative numbers had shown up in various equations, but nobody took them very seriously, regarding them as meaningless. Cardano played around with them, but it was Bombelli in the mid-16th century who worked out the details of calculating with complex numbers, which combine ordinary numbers with roots of negative numbers. A century later John Wallis made the first serious case that the square roots of negative numbers were actually physically meaningful.

9. Matrix algebra (Arthur Cayley)

An ancient Chinese math text included matrix-like calculations, but their modern form was established in the mid-19th century by Cayley. (Several others, including Jacques Binet, had explored aspects of matrix multiplication before then.) Besides their many other applications, matrices became extremely useful for quantum mechanics. In fact, in 1925 Werner Heisenberg reinvented a system identical to matrix multiplication to do quantum calculations without even knowing that matrix algebra already existed.

10. Logarithms (John Napier, Joost Bürgi, Henry Briggs)

A great aid to anybody who multiplied or messed with powers and roots, logarithms made slide rules possible and clarified all sorts of mathematical relationships in various fields. Napier and Bürgi both had the basic idea in the late 16th century, but both spent a couple of decades calculating log tables before publishing them. Napier’s came first, in 1614. Briggs made them popular, though, by recasting Napier’s version into something closer to the modern base-10 form.

NEW TRENDS IN MATHEMATICS

 New trends in mathematics are emerging, driven by advancements in technology and shifting educational paradigms. 

Here are some of the latest developments:

Computer Algebra Systems: These systems are revolutionizing mathematics education by presenting new challenges and opportunities in teaching mathematical sciences.

Data-Driven Mathematics: With the increasing availability of data, mathematics is becoming more data-driven, enabling new insights and applications in fields like science, engineering, and economics.

Artificial Intelligence and Machine Learning: AI and ML are being applied to mathematical problems, leading to breakthroughs in areas like optimization, numerical analysis, and algebraic geometry.

Mathematics Education 4.0: This trend focuses on transforming mathematics education using technology, such as virtual and augmented reality, to create immersive learning experiences.

Interdisciplinary Mathematics: Mathematics is being applied to diverse fields like biology, medicine, and social sciences, leading to new mathematical models and techniques.

Outcome of new trends in Mathematics

1. Faster and More Accurate Problem Solving

AI-assisted tools like automated theorem provers help verify complex proofs, saving time and reducing errors.

Real-world problems (in physics, biology, etc.) are solved more effectively using computational models.

2. Improved Data Analysis and Decision-Making

Mathematics of data science leads to better algorithms in healthcare, finance, marketing, and more.

Topological data analysis helps find hidden patterns in high-dimensional data, like in genetics or brain scans.

3. Enhanced Cybersecurity

Post-quantum cryptography is preparing systems to stay secure even in the age of quantum computers.

New mathematical encryption methods are being integrated into digital communications.

4. Interdisciplinary Breakthroughs

Mathematics is now central in climate modeling, epidemic forecasting, neuroscience, and social dynamics.

More collaboration between mathematicians, scientists, and engineers.

5. Better Understanding of Complex Systems

Advanced modeling techniques help simulate ecosystems, weather systems, and neural networks.

Leads to better prediction and control strategies.

6. Revolution in Education and Research

Interactive tools and AI tutors are making advanced mathematics more accessible to students.