Srinivasa Ramanujan: Contributions, Formulas, and Legacy

Who Was Srinivasa Ramanujan? 

Indian mathematicians have made lasting contributions to mathematics through early work in topics like algebra, trigonometry, geometry, statistics, number theory, and calculus. These contributions were instrumental in establishing the foundation of modern mathematics. 

Srinivasa Ramanujan, the genius who redefined numbers, was a famous Indian mathematician who made extensive contributions to mathematical analysis and continued fractions. 

Srinivasa Ramanujan made substantial contributions to mathematical analysis, number theory, infinite series, and continued fractions. Some of his key contributions included discovering the Ramanujan-Hardy asymptotic formula for partition numbers, proving properties of the partition function, and deriving infinite series for pi. His mathematical methods influenced later research in number theory. He also did important work related to highly composite numbers, elliptic curves, and hypergeometric series.

In this blog, let us discover the impactful contributions of renowned Indian mathematician Srinivasa Ramanujan

Table of Contents

• Timeline of Srinivasa Ramanujan Iyengar (800 BCE to Present)
• Early Life and Education of Srinivasa Ramanujan
• Contributions of Indian Mathematician – Srinivasa Ramanujan
• What Was Ramanujan’s Lost Notebook?
• Legacy of Indian Mathematician – Srinivasa Ramanujan
• Interesting Facts About Srinivasa Ramanujan
• Key Takeaways: Indian Mathematicians and Their Contributions
• FAQs

Timeline of Srinivasa Ramanujan Iyengar  (800 BCE to Present) 

Early Life and Education of Srinivasa Ramanujan

Srinivasa Ramanujan was born on December 22, 1887, in Erode, Tamil Nadu, to a Tamil Brahmin Iyengar family. From his early age, this genius who redefined numbers showed an extraordinary aptitude for mathematics. 

His mother was a housewife, and his father worked as a clerk in a sari shop. In 1892, at the age of five, Ramanujan was enrolled at the local school and later attended Kangayam Primary School. 

Srinivasa Ramanujan grew up in financially challenging circumstances that affected his early education and living conditions. In 1897, he passed his primary examinations in English, Tamil, arithmetic and geography with the best scores. Ramanujan was largely self-taught in advanced mathematics. 

Later on, after his admission into Town's higher secondary school, Ramanujan encountered formal mathematics for the first time

• By the age of 11, Srinivasa Ramanujan had already shown remarkable mathematical talent and a profound understanding of advanced concepts. 
• By the age of 12, he had mastered an advanced trigonometry book. 
• By the age of 14, he had already received merit certificates and academic awards, which he continued to earn throughout his academic life.

In 1904, Ramanujan graduated from the town's higher secondary school, and the school's headmaster awarded him the Ranganatha Rao prize for mathematics. He also received a scholarship to study at Government Arts College, Kumbakonam. However, Ramanujan could not focus on any other subjects and failed most of them, losing his scholarship in the process.

He later enrolled at Pachaiyappa's College in Madras. He again excelled in mathematics but performed poorly in other subjects. Without a degree, he left college and continued to pursue independent research in mathematics.

Contributions of Indian Mathematician – Srinivasa Ramanujan

India's remarkable mathematical genius made substantial contributions to the analytical theory of numbers, elliptic functions, continued fractions, and infinite series. 

His contributions include the following:

Ramanujan's Hardy Number: 1729

Once Hardy visited Putney, where Ramanujan was hospitalised. He visited there in a taxicab having the number 1729.

Hardy was very superstitious due to his nature; when he entered into Ramanujan’s room, he quoted that he had just come in a taxicab having number 1729 which seemed to him an unlucky number

At that time, Ramanujan promptly replied that this was a very interesting number, as it is the smallest number that can be expressed as the sum of the cubes of two numbers in two different ways:

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

Infinite Series for π:

Srinivasa Ramanujan also discovered some remarkable infinite series for π around 1910. 

The series, 

Computes a further eight decimal places of π with each term in the series. Later on, number theorists developed several efficient algorithms using the infinite series of π given by Ramanujan. 

Goldbach’s Conjecture: 

Srinivasa Ramanujan did not directly solve problems such as Goldbach’s conjecture, but his analytical methods and discoveries in number theory later influenced research in these areas. 

Goldbach's Conjecture is one of the important illustrations of Ramanujan's contributions to the proof of the conjecture. 

The statement is that every even integer greater than 2 is the sum of two primes, such as 6 = 3 + 3. Ramanujan and his associates showed that every large integer can be expressed as the sum of at most four. 

Theory of Equations: 

In 1902, Ramanujan was shown how to solve cubic equations, and he later developed his own method for solving quadratic equations. He derived the formula to solve the biquadratic equations. 

Ramanujan - Hardy Asymptotic Formula:

Ramanujan’s one of the major works was in the partition of numbers. 

In a joint paper with Hardy, Ramanujan gave 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, 

In their proof, they discovered a new method called ‘circle method', which made 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 result was later proved by Rademacher using special functions, and then Ken one gave the algebraic formula to calculate the partition function for any natural number n.

Ramanujan’s Congruences: 

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

Where:

• ∀ = “for all”
• N = a variable
• ∈ = “belongs to”
• N = the set of natural numbers (1, 2, 3, …)

In his 1919 paper, he gave proof for the first 2 congruences using the following identities and the Pochhammer symbol Notation. After the death of Ramanujan, in 1920, the proof of all the above congruences was extracted from his unpublished work. 

For example,

n = 36 is highly composite because it has d(36) = 9, and smaller natural numbers have fewer divisors.

If n =  2^K₂    3^K₃      4^K₄     5^K₅      P^K_P  [by the fundamental theorem of arithmetic]

is the prime factorization of a highly composite number n, then the primes 2, 3, …, p form a chain of consecutive primes where the sequences of exponents is decreasing, i.e., 

K₂ ≥  K₃ ≥  K₄ ≥ …………………. K_p  and the final exponent is 1, except for n = 4 and n = 36 

Highly Composite Numbers: 

A natural number n is said to be a 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 is called a highly composite 

Here,

• ∀\forall∀ = “for all” 
• d(n) represents the number of positive divisors of n. 

This notation means that the number nnn has more positive divisors than every natural number mmm that is smaller than nnn. This property is used to define a highly composite number. 

Ramanujan’s research on highly composite numbers significantly influenced later work in number theory and divisor functions. 

Examples of Highly Composite Numbers

1, 2, 4, 6, 12, 24, 36, 48, 60

Ramanujan’s research on highly composite numbers significantly influenced later work in number theory and divisor functions.

Some other contributions of Ramanujan's: 

Apart from the contributions mentioned above, Ramanujan worked in some other areas of mathematics, such as hypergeometric series, Bernoulli numbers, and Fermat’s last theorem.

He focused mainly on developing the relationship between partial sums and products of hypergeometric 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 numbers x, y, and z satisfy the equations.

What Was Ramanujan’s Lost Notebook?

The book examines Ramanujan's Lost Notebook, which was discovered in 1976 and contains nearly 650 unpublished claims by Ramanujan. Andrews and Berndt have taken on the task of sorting the notebook, numbering the claims, and providing proofs for each one. The reviewed book is the first volume and examines 442 entries, focusing on continued fractions and q-series. It provides valuable insights into Ramanujan's genius and unlocks new mathematical discoveries within his lost work.

Here is a more detailed view of this "Lost Notebook"

Ramanujan had noted down the results of his research, without proofs, in three notebooks, between the years 1903 and 1914, before he left for England.

The first notebook has 16 chapters in 134 pages

The second is a revised, enlarged version of the first, containing 21 chapters in 252 pages

The third notebook contains 33 pages of unorganised material (included at the end of volume 2 of the facsimile edition [VII]).

Ramanujan took these notebooks with him to Cambridge. In one of his letters to a friend, he wrote that he had no time to look into them, and most probably he did not put them to use during his five-year stay abroad.

Sixty-seven years after the death of Ramanujan, due to the discovery of the ‘lost’ notebook by Prof. George Andrews in 1976, and the editing of the three Notebooks of Ramanujan by Prof. Bruce Berndt, there has been a resurgence of interest in the work of Ramanujan. These notebooks of Ramanujan have formed the basis for numerous papers by many mathematicians, who gave proofs of the theorems and conjectures of Ramanujan that he obtained through his intuition and sheer brilliance. This resurgence is a singular and unparalleled phenomenon in the annals of mathematics. 

Legacy of Indian Mathematician – Srinivasa Ramanujan

Ramanujan, the genius who redefined numbers, made deep and lasting contributions. His work was often intuitive; he claimed many formulas came to him through inspiration, which he later verified mathematically. 

Some of his lasting contributions include:

• Number Theory: Work on partitions and prime numbers
• Infinite Series: Developed highly original formulas, including rapidly converging series for π (pi)
• Ramanujan Prime & Tau Function: Advanced understanding of number patterns
• Mock Theta Functions: Concepts that puzzled mathematicians for decades and later found applications in physics

Some of the other lasting contributions of Ramanujan include the following:

• Ramanujan's sum
• Ramanujan prime
• Ramanujan constant
• Mock theta functions
• Ramanujan conjecture
• Ramanujan identities
• Ramanujan theta function
• Ramanujan's master theorem

Ramanujan has left a number of theorems and his notebooks, which researchers are still working on.

The Prime Minister of India, Dr Manmohan Singh, has declared the year 2012 as the “National Mathematical Year", and the date December 22, being the birthday of Srinivasa Ramanujan, has been declared as the “National Mathematics Day” to be celebrated every year.

Interesting Facts About Srinivasa Ramanujan

Srinivasa Ramanujan was a renowned Indian mathematician who made extraordinary contributions to mathematical analysis, number theory, infinite series, and continued fractions despite having no formal training in pure mathematics.

Here are some remarkable facts about Srinivasa Ramanujan, the genius who redefined numbers

• Ramanujan discovered mock theta functions in his last year of his life

• By the age of 12, Srinivasa Ramanujan had independently studied advanced trigonometry using books written by S. L. Loney. 

• Ramanujan carried out much of his early mathematical learning independently using the book A Synopsis of Elementary Results in Pure Mathematics by George Shoobridge Carr. 

• Ramanujan independently developed several advanced mathematical ideas, many of which were highly original and were later recognised by professional mathematicians for their depth and significance. 

• It is said that Ramanujan's discoveries are unusually rich as a by-product, new directions of research were opened
• Ramanujan was a religious man. He often said, 'An equation for me has no meaning unless it represents a thought of God.' He credited his success to his family Goddess, Namagiri of Namakkal

• “I had never seen anything in the least like them before.” — G. H. Hardy. 

This quote is about the mathematical genius of Srinivasa Ramanujan’s formulas, written by G.H. Hardy

G.H. Hardy, who was known as a great mathematician and one of Ramanujan’s academic advisors with J.E. Littlewood, said only a few giant mathematicians like Euler, Gauss, and Newton had the same talent which Ramanujan had

• “His insight into algebraical formulae… was quite uncanny.” — G. H. Hardy. 

This quote is about the mathematical genius of Srinivasa Ramanujan, written by G. H. Hardy, one of the most influential mathematicians of the 20th century.

In simpler terms, Hardy was trying to say:

"Ramanujan had an astonishing natural gift for discovering mathematical formulas, so remarkable that it seemed almost supernatural."

Famous History:

A popular mythical figure associated with Srinivasa Ramanujan describes his early curiosity about mathematics during primary school. 

According to the story, 

One day, a  primary school teacher of the third class was explaining to his students,

“If three fruits are divided among three persons, each would get one." 

Thus, it was generally accepted that any number divided by itself equals one. 

Ramanujan questioned what would happen if zero were divided by zero.

Although the story is personally based and not historically verified, it is often used to illustrate his exceptional curiosity and independent approach to mathematical thinking from a young age.

Why Did Srinivasa Ramanujan Die So Young?

Srinivasa Ramanujan died on 26 April 1920 in Kumbakonam, at the age of 32.

The exact cause of the death of Srinivasa Ramanujan remains uncertain. However, historians and medical researchers generally agree that a combination of illness, malnutrition, and difficult living conditions contributed to his early death.

The mystery of Srinivasa Ramanujan's illness:

Key Takeaways: Ramanujan and Their Contributions

In memory of Srinivasa Ramanujan, several books and films have been created to highlight his life and mathematical contributions. 

One notable example is "The Man Who Knew Infinity", which was adapted from the biography “The Man Who Knew Infinity” by Robert Kanigel.

The big mathematicians and specialists of that time noted that Ramanujan’s talent reminded them of Gauss, Jacobi, and Euler.

Stay tuned to JAIN PU College blogs and learn more about the famous Indian Mathematicians, their work on pi and series, mathematical research and other related articles.