Ramanujan : a great mathematician

Dec 22, 1887 – Apr 26, 1920 (32 yrs)

See also his Magic Square

Srinivasa Ramanujan, FRS, was an Indian mathematician and autodidact who, with almost no formal training in pure mathematics, made extraordinary contributions to mathematical analysis, number theory, infinite series, and continued fractions. As he had no access to large community of mathematicians, he developed his own mathematical research in isolation. As a result, he sometimes rediscovered known theorems in addition to producing new work. Ramanujan was said to be a natural genius by the English mathematician GH Hardy.

Ramanujan was born in a poor Braahman family. His father’s name was K Srinivasa Iyengar and his mother’s name was Kamalatammal. His father was a clerk at a Saree shop and his mother used to sing at a temple. They lived on Sarangpani Street in Kumbakonam, Tamil Naadu. He was divinely inspired. When his grand-mother Rangammal, an ardent devotee of Goddess Namagiri (the Goddess of their family deity Lord Narasinh Temple in Namakkal, Tamil Naadu) went to the Temple, she went into a trance and supposedly the Goddess spoke to her and said that – “The Goddess herself will speak to the world through her daughter’s son”.

He was formally introduced to Mathematics at the age of 10. He mastered the books on advanced trigonometry written by SL Loney by the age of 12. He even discovered theorems of his own, and re-discovered Euler’s identity independently. He demonstrated unusual mathematical skills at school, winning accolades and awards. By 17, Ramanujan had conducted his own mathematical research on Bernoulli numbers and the Euler–Mascheroni constant.

Ramanujan received a scholarship to study at Government College in Kumbakonam, but lost it when he failed his non-mathematical coursework. He joined another college to pursue independent mathematical research, working as a clerk in the Accountant-General’s office at the Madras Port Trust Office to support himself. In 1909, he was married to a 9-year old girl – Janaki Ammal, at the age of 21 years.

Ramanujan in England

In 1912–1913, he sent samples of his theorems to three academics at the University of Cambridge. One mathematician MJM Hill told that he lacked educational background and foundation to be needed by mathematicians. Still he gave him professional advice on whose basis he wrote to three professors in the University of Cambridge. HF Baker and EW Hobson returned his work. Then he wrote to GH Hardy with his 9 pages of work which he thought he was a fraud. But some of his formulae got his attention and recognizing the brilliance of his work, invited him to visit and work with him at Cambridge. He worked with him for nearly 5 years. Here it was a clash of different cultures, beliefs and working styles. Hardy was an atheist and an apostle of proof and mathematical rigor, whereas Ramanujan was a deeply religious man and relied very strongly on his intuition.

He got a BA degree, which was later converted in a PhD degree, in 1916. He became a FRS (Fellow of the Royal Society) in 1918 becoming the second and the youngest Indian, following Ardaseer Cursetjee in 1841. He became the first Indian to be elected as a Fellow of Trinity College, Cambridge in 1918. He could not tolerate the living conditions outside India being a vegetarian he fell ill and suffered with Tuberculosis and severe vitamin deficiency. He came back from there in 1919 and soon after he died of illness, malnutrition, and possibly liver infection in 1920 at the age of 32.

Ramanujan’s Notebooks

Ramanujan had written his work in 4 notebooks on loose leaves before going to England. Most of them were without derivation. Since paper was very scare in those days and the slate was very popular for solving problems, he must have used slate. The 1st notebook had 351 pages, with 16 somewhat organized chapters and rest unorganized. The 2nd notebook was of 256 pages, under 21 chapters and 100 unorganized pages. His 3rd notebook contained 33 unorganized pages. And the 4th notebook contained 87 unorganized pages – it was rediscovered in 1976.

He credited all his knowledge to his family Goddess, Namagiri of Namakkal. There are two interesting incidents in Ramanujan’s life that I wish to share here —

(1) Meeting with Mahalanobis

Once one Sunday morning PC Mahalanobis (who founded the Indian Statistical Institute in Calcutta and also become a Fellow of the Royal Society) went to meet Ramanuja in Whewell’s Court. Mahalanobis said to him – “Now, here is a problem for you.” He told he used to visit a Belgian friend whose house was on a long street. They were numbered on this side 1, 2, 3 and so on, and that all the numbers on one side of him added up exactly the same as all the numbers on the other side of him. He knew that there were more than fifty houses but not so many as five hundred. He wanted to know his friend’s house number.

Through trial and error, Mahalanobis had figured out the answer. Ramanujan figured out too but he gave a general solution to the problem using continued fraction. A continued fraction whose denominator consists of a number plus a fraction, ad infinitum. As stated, the problem had one solution – house number 204 in a street of 288 houses – i.e., 1 + 2 + 3 —- + 203 = 205 + 206 + —- + 288. But without the 50 to 500 house constraint, there were other solutions – for example, house number 6 in a street containing 8 houses – i.e., 1 + 2 + 3 + 4 + 5 = 7 + 8. Ramanujan’s continued fraction comprised within a single expression – all the correct answers. Mahalanobis was astounded. How interesting!

“The limitations of his knowledge were as startling as its profundity,” Hardy would write. Mahalanobis was also a great Mathematician. In fact, Pandit Nehru liked him a lot. And when Nehru asked him – “How to solve India’s poverty?”, Mahalanobis immediately wrote a very long equation with Partial Differentials that filled the whole black board. Then he explained, each and every variable in the equation again completely zapping Nehru.

(2) Meeting with Hardy

Once GH Hardy came to meet Ramanujan in a taxi from London. Hardy noticed the number – it was 1729. He thought little about the number and said it was “rather a dull number”. Ramanujan immediately retorted, “No, Hardy, it is a very interesting number. It is the SMALLEST NUMBER that can be expressed as the sum of TWO CUBES in TWO DIFFERENT WAYS”. That is, it is the smallest number that can be expressed as 12 x 12 x 12 + 1 x 1 x 1 but also as 10 x 10 x 10 + 9 x 9 x 9. How interesting! Coincidentally, 1729 is also a Carmichael Number.

His Work

During his short lifetime, Ramanujan independently compiled nearly 3900 results (mostly identities and equations). Most of his claims have now been proven correct, although a small number of these results were actually false and some were already known.

He has produced a square with numbers – it is really superb. Would you like to see it – Click here to see that square. Ramanujam has given the method as how to draw a magic square involving any important date, not only birth day, except a few. In Kharuraho Paarashwanaath Jain temple (10th century ) one can find a magic square engraved.

7 12 1 14

2 13 8 11

16 3 10 5

9 6 15 4

This is referred to as the Chaunteesaa Yantra, since each row, each column, each diagonal, 2 x 2 sub-squares the corners of each 3 x 3 and 4 x 4 square, the two sets of four symmetrical numbers (1 + 11 + 16 + 6 and 2 + 12 + 15 + 5), and the sum of the middle two entries of the two outer columns and rows (12 + 1 + 6 + 15 and 2 + 16 + 11 + 5), sums to 34.

Honors

In recognition of his contributions, the Government of India has declared Ramanujan’s birthday (22 December) should be celebrated every year as National Mathematics Day, and also declared 2012 as the National Mathematical Year.

Read: The Man who knew infinity : A life of the genius Ramanujan: Robert Kanigel