On the death anniversary of Srinivasa Ramanujan today, I take a look not at his life per se, but more on his mathematical works. Something that has always interested me as a maths lover, and this guy reached heights in mathematics sans any formal training.

Ramanujan had no formal training in mathematics, he was more or less self taught, yet he made significant contributions to number theory, infinite series, continued fractions. Cracked math problems that were considered unsolvable till then, and researched in maths on his own.

For some one who had no formal training in maths, he compiled nearly 3900 results on his own, gave us the Ramanujan Prime, Ramanujan Theta Function, partition formulae, and has a peer reviewed scientific journal named after him. His output is far more prodigious than any one.

Ramanujan was an enigma, in the pantheon of great modern mathematicians. He relied more on gut feeling, intitution, quite opposite to the standard Western methodology of proof, he credited his discoveries to divine intervention.

Again in school, Ramanujan was absolutely brilliant at maths, but quite often flunked in the other subjects. Consider this by the time he was 13, he mastered SL Loney's work on Advanced Trignometry, soon he developed his own methods to solve the quartic function.

The book that influenced Ramanujan the most though was Synopsis of Pure Mathematics by G.K.Carr that had around 5000 theorems, when he was 16. The very next year he investigated the Bernoulli numbers, calculated Euler-Mascheroni constant to 15 places.

His genius though would be a sort of curse for him, fact is Ramanujan was interested only in maths, he hardly cared about other subjects at school and college. So while he passed Maths with distinction, he was below par in the other subjects

And this meant that Ramanujan failed his Fellow of Arts exam in 1906, in spite of his brilliance at Maths, and with no formal degree, he had to pursue maths research on his own, at times living in dire poverty.

Chance meeting with V Ramaswamy Iyer in 1910, would be the turning point in Ramanujan's life. Iyer was the founder of the Indian Mathematical Society, and thanks to him, Ramanujan began to be recognized in the math circles of Madras.

Ramanujan however wanted a regular job to make ends meet, Iyer felt that a genius like him should not be wasted in clerical jobs. He sent letters to his friends in Madras, one of em was R.Ramachandra Rao, the district collector of Nellore.

Though Rao was initially sceptical about Ramanujan's work, a discussion on elliptic integrals, divergent series convinced him of his brilliance. Ramanujan however said he needed financial support, and Rao sent him to Madras, giving him the aid he needed too.

Ramanujan joined as a clerk at the Madras Port Trust, and used his spare time to pursue research into mathematics. While Rao, supported him financially, Ramaswamy Iyer ensured his work was published in the Indian Mathematical Society.

One of the first problems Ramanujan posted in the journel was to find the value of what is called the infinite nested radicals( pic1). After 6 months when none failed to answer it. he himself posted the solution, this equation was to solve infinitely nested radicals.

This was the nested radicals equation posted by Ramanujan, and this was the equation he formulated to solve the problem. Later on this equation would be used regularly in problems related to nested radicals.

Ramanujan's first formal paper for the Maths Journal was on Bernoulli Numbers and their properties. He came up with his own method of calculating Bn, based on previous Bernoulli numbers. The entire paper had 3 proofs, 2 corollaries and 3 conjectures.

Basically Ramanujan's method to calculate Bn was based on this
1) Bn is a fraction, and the numerator in it's lowest wud be a prime number
2) Denominator of Bn contains each of factors 2, 3 only once
3) 2^n(2^n-1) Bn/ is an integer.

I understand there is a clerkship vacant in your office, and I beg to apply for the same. I have passed the Matriculation Examination and studied up to the F.A. but was prevented from pursuing my studies further owing to several untoward circumstances. - Applying for a job.

Finally on the reccomendation of his Maths professor, E.W.Middlemast, Ramanujan managed to land a clerical post at the Madras Port Trust, and he spent his spare time in doing Maths research.