The genius who continues to inspire research even after his death

He was Indian who loved numbers so much that he spent all his life doing them. He lived and died young but his contribution is so enormous that mankind still remembers him as a genius.

Born in a orthodox Brahmin family, he quickly took to mathematics like a duck to water. His progress in mathematics was so rapid that his teachers did not know what to do with him. On his part, this genius lost interest in all other subjects except mathematics.

He is none other than Srinivasa Ramanujam. He  was born on December 22, 1887 in his maternal grandfather’s house in Erode. After his birth, he spent his childhood in a house in Sarangapani Street in Kumbokanam.

His father was  K. Srinivasa Iyengar, a clerk in a sari shop and his  mother, Komalatammal, was a housewife who occasionally sang at a local temple.    

Little Ramanujum went to Kangayan Primary School just across his house on Sarangapani Street in Kumbokanam. This house is now a museum.  Ramanujam was a brilliant child but the teachers then did not know how to deal with him. Both his teachers and his parents soon realised that Ramanujam was no ordinary child.

Ramanujam’s tryst with mathematics began when he was just ten years old. He read a book on Advanced Trignometry by Sidney Luxton Loney (1860-1939) and mastered it (S. L. Loney, a British mathematician, has written several books on trigonometry, including the one called Plane trigonometry).

His interest in trignomeotry led him to discover his own theorems. Moreover, he discovered Euler’s identity independently. At school he was brilliant in mathematics. By 17, he had conducted his own research on Bernoulli numbers (The Bernoulli numbers were discovered by a Swiss mathematician Jakob Bernoulli, after whom they are named. Bernolli is a sequence of  rational numbers in relation to the number theory) and Euler-Mascheroni Constant (The Bernoulli numbers appear in the Taylor series expansions of the tangent and hyperbolic tangent functions, in formulas for the sum of powers of the first positive integers, in the Euler- Mascheroni formula and in expressions for certain values of the Riemann Zeta functions-this is a function of complex variables that analytically continues the sum of  the infinite series).

When still in school, one of his teachers (others say it was a friend) in 1903 gave him a library book on maths by a British mathematician George  Shoobridge Carr with 5,000 equations, hoping that this would satiate the child prodigy’s appetite for numbers.  

The book, called Synopsis of Pure Mathematics (1880), opened a Pandora’s Box for the teenaged Ramanujam.

Ramanujam learnt about religion, Shastras and Puranas from his mother with whom he was close. He also learnt singing from her.

He passed his primary school examinations in 1897 with Arithematic, English, Tamil and geography as his subjects. He then joined the higher secondary school (Town High School) where he came across mathematics.

By 1904, Ramanajum began his own research. He began filling notebooks with his own equations and sums.

None understood them and he did not understand who others could not understand his equations. His interest in mathematics was so deep that he neglected other subjects. His college asked him to leave.

Meanwhile, his mother had arranged his marriage with nine-year-old Janaki. He married on July 14, 1909.  He then decided to go to Madras which then was the biggest city in south. He hoped that someone in this city would be able to understand and appreciate his equations. He took his notebooks with him.   

Even in Madras, none could understand the equations or make head and tail about them. Ramanujam then took to writing letters to mathematicians and one of them reached G. H. Hardy in Cambridge, England.

It was left to Hardy (1887-1947 ) to recognize this genius and call him to England. Now began one of mankind’s most famous collaborations-Ramanujam the genius and Hardy the reformer came up with a formula on integer partitions known as the Hardy–Ramanujan asymptotic formula.
This formula has been applied in physics to find quantum partition functions of atomic nuclei (first used by Niels Bohr) and to derive thermodynamic functions of non-interacting Bose-Einstein systems.
Unfortunately, this collaboration was cut short when Ramanujam contacted liver infection in 1919 and died in Kumbokanam in 1920, aged 32. By then, Ramanujam had left enough of his mathematical genius for generations to pore over. His wife, Janaki shifted to Bombay but returned to Madras in 1950 where she lived till she died in 1984.

He has to his credit over 3900 independently compiled equations and results and some of them are Ramanujan prime and the Ramanujan theta functions. These results have continued to inspire further research.

If Ramanujan's 1916 paper on modular forms created a sensation in the Western world, his last letter to Hardy, written literally from his deathbed in 1920,  on the functions of  “mock theta,” is now creating waves.

Mathematicians have ignored Ramanujan's theory of mock theta as he had not given any specific definition of it. However, he has  listed 17 protypical examples of these new functions. As there was no theory on these functions, the mock theta theory was largely ignored.

It was only in 2002 that the unifying conceptual framework of mock theta was discovered  by  Sanders Pieter Zwegers, a Dutch mathematician, in his doctoral theses. He made a connection between Maass forms (This is a function on the upper half plane that transforms like amodular form but need not be holomorphic. They were first studied by Hans Maass, a German mathematician, in 1949) and Ramanujam’s mock theta function in 2002.

Ramanujam was born on December 22 and the blog takes a bow to this great mathematician who has continued to inspire awe and admiration not only among the mathematical community but all of mankind.

When Bhaskara tried to cheat destiny

Bhaskaracharya or Bhaskara , the great mathematician and astronomer of the 12^th Century was also a well-known astrologer. When his daughter, Leelawathi, was born, he decided to cast her horoscope.

When he was calculating the planetary positions in the horoscope of his daughter, he found to his horror that Leelawathi would become a widow within an year of her marriage.

Bhaskara then decided to groom his daughter in the best possible way. He also wanted to educate her and her brother Lopamudra. Both seem to have taken after Bhaskara as they loved mathematics.

When Leelawathi reached marriageable age, the search for a suitable companion started. Bhaskara had not forgotten the horoscope he had cast of his daughter  and he decided to cheat fate by fixing an auspicious time for the marriage.

Bhaskara, by his precise mathematical calculations, concluded that if the mangalasutra was tied at a particular time, both his favourite child and his son-in-law would live a long and eventful life. He then determined to endure that nothing went wrong and himself determined the exact time when the mangalasutra has to be tied.

All arrangements for the marriage were being made and the marriage party had come to Bhaskara’s house in Ujjaian. Bhaskara was the head of the Ujjain Observatory and he had invited several people to the wedding.

Bhaskara had decided to rely on science to ensure that the mangalasutra was tied at the most auspicious time he had predicted. In this way, he thought he could change the destiny of his daughter.

Bhaskara had kept a simple device in the house to give him the exact mathematical time when the mangalasutra would have to be tied. He took two vessels and placed them one above the other. He filled the upper vessel with water and made a tiny hole in the centre. Water would trickle down to the lower vessel. The time was calculated by estimating the depth of the water that had collected in the lower vessel.

The unit of time was a Nadika. One Nadika was equal to 24 minutes. Thus, this was the water clock that was used in India. It was very accurate. 

Bhaskara had warned Leelawathi not to go near the clock. Curiosity, thy name is woman, says an adage. Leelawathi went near the clock as she was fascinated by the simple mechanism. She bent to look into the vessel. A small pearl set in her nose ring fell into the upper vessel. It sank down and it covered the tiny hole, thus blocking the free floor of water into the lower vessel.

Leelawathi was not aware of the nose ring having fallen into the vessel. Bhaskara too did not know what happened. When he saw that the water flow had stopped, he thought that the auspicious time had arrived.

He got the marriage ceremony performed. This was the very time he had wanted to avoid. Thus all the minute calculations  that Bhaskara had come up with went awry. His daughter was married at the most inappropriate time. Bhaskara realised that something had gone wrong when he went to check the time and found that the water flow had stopped.

However, the marriage had already taken place and nothing could be done to undo the prediction. Leelawathi’s husband died soon after marriage and she came back home.

Feeling cheated by destiny, Bhaskara consoled his daughter saying that he will make her immortal. “I will see you’re your name will live on forever”, he said.

He then began writing his magnum opus, on Mathematics and Astronomy and called it Sidhanta Shiromani.

The first portion of the book was called Leelawathi and it dealt with mathematics. He named it after his daughter.

Leelawathi was good at mathematics and in the book, Bhaskara addresses mathematical and algebraic problems to her. Leelawathi and other portions of Sidhanta Shiromani are in verse form.

Leelawathi has 278 verses and it deals with arithematic, algebra and geometry.

He poses the question to Leelawathi as follows:   

“Partha was determined to slay Karna. Posing his mathematical query, he asks, With half the arrows in his quiver Partha (Arjuna) destroyed those of Karna; with four times the square root of the total, he killed Karna’s horses. Six arrows were needed to kill Karna. With three arrows, Partha cut down Karna’s royal standard, his umbrella and bow, With the last arrow Arjuna killed Karna. Tell me O Leelawathi, how many arrows did Aruna have in his quiver?         
In another verse, he says a peacock which was sitting on a nine foot high pillar saw a snake going towards its anthill at the foot of the pillar. The snake was at a distance of three times the pillar’s height when the peacock swooped down. If both the peacock and snake travelled at the same speed, tell me O Leelawathi how far from the pillar did the peacock peck at the snake?

However, Bhaskara could never forget the incident of the pearl from the nose ring falling into the vessel and ruining Leelawathi’s marriage.

In perhaps the most poignant verse, he says

Whilst making love a necklace broke
A row of pearls mislaid
One sixth fell to the floor
One fifth upon the bed
The young woman saved one third of them
One tenth were caught by her lover
If six pearls remained upon the string
How many (total)  were there in the necklace.
   

Leelawathi was first translated into Persian by Abdul Faizi on orders of the Mughal Emperor Akbar. The other portions of Sidhanta Shiromani were translated into Persian during the reign of Akbar’s grandson, Emperor Shahajan.

The story of Leelawathi is narrated by Faizi in his Persian narration and other sources. Leelawathi remained the standard mathematical text for centuries till the advent of the British and the adoption of the English mathematical text.

If you have the time and interest, please go through Bhaskara’s Sidhantha. I can assure you that it is not boring as European mathematical texts. Nor is it uninteresting. It is in Sanskrit and it has been translated into almost all languages, including English.