A brief History on Ramanujan

 

                                                                            

“Dear Sir, I beg to introduce myself to you as a clerk in the Accounts Department of the Port Trust Office at Madras... I am now about 23 years of age… I have had no University education…”

Those were his initial sentence from his first letter to one of the professor of mathematics in Cambridge University, London.

Who might have thought that later their partnership will be one of the most fruitful in the history of Mathematics.

 Ramanujan, Srinivasa Iyengar Ramanujan.

Who was he? Why (majority Indians) haven’t heard or being taught about him?

Even I didn’t know until I was in University when one of my friend, Ajay, suggested me a movie named- "The man who knew Infinity". I have done my Higher Secondary from a college named after him – Ramanujan Junior College, Assam. But due to lack of Knowledge about him and his legacy I merely valued that name at that time.

If I were asked to introduce him in one sentence then I will frame as –

       The identity of India in the world of Mathematics in this modern era.

Why is he important and does he really deserve to get introduced as an ideal in our society?

I can’t answer from the prospective of 1.38 billion citizens but as my individual prospective - A big Yes. It is not that we have totally forgotten him or his legacy. I personally thank the western world and specially Matthew Brown (director of the movie – The man who knew Infinity) for his 10 years of dedication. The big brainy of our nation did recognized him, that is why we Celebrate National Mathematical day on 22^nd of December to mark his birth anniversary. When we talk about Ramanujan’s early life. He was born on 22^nd of December 1887 in Erode, Madras (now Tamil Nadu). He was born in a very poor Brahmin family and so most of his life was under poverty. His father, Kuppuswamy Srinivasa Iyengar was a clerk in a sari shop and mother was a housewife. I don’t know many story from his childhood but one story is mostly being told. Even our Prime Minister Narendra Modi too has mentioned in one of his speech and also in a documentary (https://www.youtube.com/watch?v=SYBOOjhAMsM) by Vigyan Prasar & IISER Pune a professor ( Prof. A Raghuram) did mentioned the same. It goes as-

        Mathematics class was going on and the teacher was explaining some mathematical concept (Ratio and Proportion) to the students. He was explaining that if there are 5 mangoes in a bucket and number of students are also 5 then each student will get one mango. Similarly, if there are 50 mangoes and number of students are also 50 then again each students will get 1 mango. Then a boy raised his hand and asked does it mean that if there are ‘0’ mangoes and number of students are also ‘0’ than still students will get one mango each?

For us, who rarely ever tried to understand the basics of mathematics, concept behind various mathematical properties like addition, subtraction, multiplication, division, concept of ‘0’ or 1 but take them as granted, this question seems meaningless because how can we divide zero mangoes among zero students and still end up with some finite mango?

Yes, this was a hypothetical scenario but with this contrive situation that kid could able to imagine a situation that has puzzled even the greatest Mathematicians of this world. He could imagine what will happen if we divide zero by zero (0/0).

In the world of mathematics, this are some of the most fundamental questions which needs to be answered with the proper knowledge of mathematics. For us it might seems not a great deal. Some of us might now have imagined answer as zero (most of us) or infinite. Let’s look this two statements:

1)      Anything to the power zero is one i.e. X⁰=1

2)      Zero to the power anything is zero i.e. 0^X=0

Then there is a junction point i.e. 0⁰, if I say 1 on the basis of statement-1, then I violate statement-2 and if I say 0 on the basis of statement-2 then statement-1 gets violated. So what could be the answer? Similar is the case of 0/0

Look at this statement again

1)      0/anything =0

2)      Anything/0=infinite (not defined)

3)      Anything/same thing=1 (i.e. x/x= 1)

So, how will you answer ‘0/0’? Which statement is justified? Logically 3^rd seems correct, Statement-1 and 2 are also not wrong. But only one statement can be used as it cannot have two solution.

Did we ever tried to imagine this? I never did.

Another interesting thing about his mathematical ability is that for him approximation in just a way of representing a never ending number because his results were mostly exact even in those math which involves many intermediate approximated results. His such ability even mesmerize present day mathematicians with super computer in their table.

To give a hint of his ability to calculate numbers with precision. Look at this calculation by him without any calculator:

There is something more interesting about this number and that is ‘e’ raised to some power involving ‘pi’ gives a number which is so close to an integer.

He spend most of his life under poverty. But that was not his excuse. He created his own opportunity. Although he introduced himself as a fellow without a degree in his first letter to G.H. Hardy but his academic records were not that bad. He was very good in study. He passed his primary examination scoring best in the district and by the age of 16 he was well known for his mathematical ability among his mates and faculties. Till than he had mastered many subjects in mathematics including advanced trigonometry, geometry, infinite series etc. this child prodigy also received many merit certificates and academic awards. Whatever books he received on mathematics, he mastered them all but it’s been often said that one particular book had greatly influenced him and was mostly credited to unlock his mathematical genius. The name of that book was- A Synopsis of Elementary Results in Pure and Applied Mathematics, by G.S. Carr.

This book contain a collection of 6165 theorems without proof.

Motivated by him I too took a print copy of the same but forgot that I was not 1% close to Ramanujan. For people like us mastering few initial pages of this book is a challenge. I wonder what short of brain might he possess to master this book.

Under the recommendation from his teacher and his academic records, he was provided a scholarship to study at Government Art’s college, Kumbakonam. But like his previous record he could not able to perform very well. He even failed in most of his academic subjects because by then he was so involved in mathematics that he gave no time for other subjects. He loss his scholarship. Although he continued in other college but could not get qualified and ultimately without an FA (Fellow of Arts) degree he left his college education and started his independent research in mathematics.

Even in extreme poverty, he never compromised with his passion. In one of his letter to G.H. Hardy he even wrote as- “I am already a half starving man. To preserve my brains I want food…” He got married at the age of 19 with a girl who was 9 years younger. The sacrifice made by Mrs. Janakammal (Ramanujan’s wife) to shape one of the greatest mind ever walk on this planet, deserves a salute. There is a video in YouTube where we could actually see and listen her voice. However, that video also gives a glimpse of Indian society. She was alive till 1994. She lived in poverty till her last breadth. Yes, that’s the reality of our society.

About his professional life it is very motivating. He was best in mathematics but could not improve in other subjects. To support his living he got and loss many jobs. With a letter of introduction he run among many individuals and finally to R. Ramachandra Rao. By than he had already left his education and was without a degree. This stage of his life gives us a beautiful lesson that- in this society with degrees and certificates, a designated failed fellow have to stand solo and if tried to stand among them he must be supported by a pillar of knowledge much taller than those with degrees and certificates.

Ramanujan stood, for R. Ramachandra Rao he proved that he is worthy. In the middle of 1912, he manage to get a job as a clerk in account office in madras with a salary of Rs.30/month. But this was not the only good thing that has happened in his life. The other good thing was that then he had someone who was willing to help him and his work to get recognized not only in India but in abroad. Ramachandra Rao and few other distinguished people helped him to write a letter to many British Mathematicians. The first letter he wrote was to a British Mathematicians M.J.M Hills of University College London. A reply came saying that although he had “a taste for mathematics and some ability but he lacked the necessary educational background and foundation to be accepted by mathematicians”.

In beginning of 1913, Ramanujan again wrote letter (9 page long) to three other British Mathematicians at Cambridge University, London. It was the third who took his work into account. He even replied saying that a single look was enough to show they could only have been written by a Mathematician of the highest caliber and even compare his work with Euler and Jacobi.

Now who are this two fellow and why did he compare him with them?

To answer this question in a sentence-

    Those are the individuals who are mainly qualified to be designated as “The greatest Mathematicians”.

I don’t know much about Jacobi, Gauss and Bernoulli but yes Leonhard Euler was (the greatest) mathematician. This are the people who laid the foundation of mathematics. Their every theorem is worthy to be awarded with Fields Medal. It has been often said that Euler published more paper (theorem) after his death then when he was alive.

The name of that professor who took his work into account was prof. G.H. Hardy (Godfrey Harold Hardy).Then here begins one of the most romantic story between an atheist and an orthodox (but not blind) Hindu. One of the most fruitful era in the history of Mathematics.

Now you may wonder what was in that nine page manuscript. And trust me if you see that manuscript once than possibly there will be two kind of reaction:

1)      1) Amazed and speechless seeing a letter with few words but mostly formulas.

2)      2) Amazed but laugh thinking this is how mathematician writes letter to each other.

Here you can have a look at his original manuscript to G.H. Hardy:

As you can see some of the results are very unusual like:

1.         Sum of positive integer is not only in fraction but a negative (i.e. -1/12)

2.      Sum of the cube of all positive integers is in fraction (i.e. 1/120)

3.       Integration of some of the continued infinite series etc.

As you can see most of his works are on analysis of infinite series. Some of the results are amazing. Actually I was not amazed because I hardly understand much of it. But yes, it was this letter which convince G.H. Hardy. It is said that the last page of that letter actually makes G.H. Hardy to get convinced as it contain results/solution to solve some of the most difficult continued fraction.

In context to last page prof. G.H. Hardy wrote as-

     “… the theorems defeated me completely; I had never seen anything in the least like them before and they must be true because if they were not true, no one would have the imagination to invent them”.

But being born in an orthodox Hindu family, it was forbidden for him to cross sea. By February he already got a response from G.H. Hardy showing interest on his paper and willing to work with him. Ramanujan refuse to go to Cambridge as forbidden by his religious belief and also restricted by his mother. But mathematical community, more specifically G. H. Hardy had already started the possible steps needed to bring him to Cambridge. He was then granted a research scholarship at the University of Madras for two years with Rs 75 per month. He worked there for few month and published many paper in Indian Journal. But later he accepted to go to Cambridge. It has also been said that on night his mother had a vivid dream in which the family goddess (the deity of Namagiri) commanded her not to stand between her son and the fulfilment of his life’s purpose.

Finally, on March 1914, he departed for England. Note the year (1914), this figure holds one of the darkest history of mankind. It’s the beginning of the World War-1 and later you will realize that it was also one of the cause why we lose him.

As soon as he reached Cambridge he begin his work in collaboration with G.H. Hardy and Littlewood (another professor of Mathematics in Cambridge). But soon there rises a problem for both Hardy and Ramanujan. Because he didn’t have any degree so he can’t published his paper and also most of his works were without proof as stated early that he was mostly influenced by a book named as - A Synopsis of Elementary Results in Pure and Applied Mathematics, by G.S. Carr. So, he too have the habit of writing formulas without proof. Some scholars also believe that because paper was a precious commodity then so he had to write only results which being derived in a slate. But in Cambridge he must give some logical explanation to his formulas.

As depicted in the movie, he was asked to attain some lectures as a formality which will designated him as a student and get a degree and then pursue his dream to publish his formulas. But for him it was a long process.

Those were good old days. Hardy and Ramanujan were passionately working on mathematical unsolved problems and coming with ground breaking results. Ramanujan worked day and night. Sometimes not sleeping for many days.

Yes, this is mathematics where two contrasting personalities whose collaboration was a clash of different cultures, beliefs and working styles yet work together and contribute some of the most fruitful results in the history of mathematics including partition theory which was considered as impossible. That is why I love mathematics.

What next than?

Published paper, got a bachelor degree, recognition etc.

In 1917, he was elected to the London Mathematical Society. On May, 1918, he was elected as a Fellow of the Royal Society. He was just 31 years old. He is one of the youngest fellow in the history of the Royal Society. Later on 13^th October 1918, he was the first Indian to be elected as a Fellow of the Trinity College, Cambridge.

I don’t know what more qualification is required to consider him as “the greatest Mathematician” in our society.

But those 5 years were not as much fruitful as it sounds.

Being an Indian, he regularly face racism. India was under British rule then, the western do have the feeling of superiority complex. Besides that, it was the era of World War-1 and England was not an exception. Also, he was a vegetarian and for English the word vegetarian was not new but they never had a live human example. He had to face many social challenges. It has been said that he used to cook his own food but as WWI begins soon there was a shortage of food mainly for a vegetarian. We don’t know how he manage to live. Soon he felt serious illness. He was diagnosed with Tuberculosis which at that time was an incurable diseases. Some other reference also state that it’s likely that what he had was a parasitic liver infection picked up in India. It was not that he felt such serious illness for the first time. When he used to be in India there too he frequently felt ill. But this was for the first time that he was diagnosed with an incurable diseases that too in a foreign land. When he was in hospital one famous event is said to happen- One day G.H. Hardy came to meet him in a taxi. He noticed the number of that taxi was 1729. Then while conversation with Ramanujan, Hardy mentioned that taxicab number was an unusual rather a dull number. But Ramanujan replied, no! it was not, it is a very interesting number. It is the smallest number expressible as the sum of two positive cubes in two different ways.

                               1³+12³=10³+9³=1729

In context to Ramanujan love for mathematics J.E. Littlewood also said that-“Every positive integers is one of ramanujan’s personal friends”.  

My mother once told me that everything comes at a price, but I never imagined that it could be this cruel.

Knowing that he has few days left he made his mind to return home and live rest of his remaining days with family more specifically with his wife who had sacrifice her entire life for him.

In 1919, he returned to kumbakonam and at the age of 32 on 26^th of April 1920, we lost one of the greatest mind and a human in the history of mankind.

We as a society had never recognized him. After his death it is said that as a Hindu tradition a dead body is always burned but during his funeral no priest was willing to conduct his ceremony. Whatever might be the reason – either it was due to the reason that he crossed the sea which was forbidden or he was diagnosed with tuberculosis (a communicable diseases), but this is how we have honored him. This is how it ends.

The legacy he left, even though he lived only 32 years, is something to be honored and kept as treasure. But in contrast, we rarely heard about him. No, folk’s song is ever sung in any media in India. There is no any reference.

With Sincerely 

Ajit Kumar Chetry