JOIN USHe was always game for numbers!
This year has been declared the national year of mathematics to mark the 125th anniversary of the birth of the genius, Srinivasa Ramanujan. It was 100 years ago, around 1912, that Srinivasa Ramanujan’s work came to notice, writes S Ananthanarayanan
It is a century since Srinivasa Ramanujan flashed for a short while through the world of mathematics. A century because it was around 1912 that Ramanujan’s talent came to relevant notice, to be followed by eight years of unprecedented, and since unequalled productivity – of work of originality and genius, a fascination for which has remained fresh and tantalising. This year is also the 125th anniversary of his birth and has been declared by the Government as the ‘National Mathematical Year’.
From Erode to Cambridge
Ramanujan was born in 1887 at Erode in Tamil Nadu and displayed remarkable talent even when in junior school. By the time he was 13, he had mastered a fairly advanced standard text book of mathematics, one which is normally used only in university. When he was 16, he chanced upon a collection of problems and outline of the theory that had been developed for aspirants to the exacting Tripos Examination of the University of Cambridge.
With little exposure to higher math either in his curriculum or from his teachers, this collection of advanced and difficult problems opened for Ramanujan the universe of number theory and higher algebra.
Ramanujan became fascinated by mathematics and worked at it to the exclusion of all else. While he never cleared his university exams, for want of credits in other subjects, and could not hold down a regular job, he rapidly discovered, in books or by himself, most of the mathematics that he would have learnt in a proper course of formal training.
And along the way, he developed insights and broke fresh ground in the form of new formulae or theorems of clarity and power that took professional mathematicians by surprise.
The work in these years were recorded, often with sketchy description of the method followed, in note books, which have now become celebrated as the first record of Ramanujan’s early work.
His work soon came to the attention of G W Hardy, a gifted professor at Cambridge, who was overwhelmed by its quality. The work was “certainly the most remarkable I have received,” he said and commented that Ramanujan was “a mathematician of the highest quality.”
Hardy lost no time in arranging for Ramanujan to come over to work in England and the young Indian was soon imbibing aspects of mathematics so far unknown to him, and learning that much of what he had worked out by himself, to his chagrin, had been discovered earlier.
But the rich, academic environment of Cambridge allowed his undoubted talent to flower and in the next five years, till he died at the young age of 32, he churned out a huge volume of work of the highest quality.
Nature of work
Ramanujan’s work had so much variety that it is not feasible to describe the range in the space of an article. But we can get a glimpse of one major area, which concerned formulae to evaluate numbers that are expressed as the converging sum of an infinite series of reducing components.
An example of an infinite series like: 1+1/2 +1/4 + 1/8 + 1/16 + 1/32 + ……. , where the terms rapidly get smaller, so that the later additions are small indeed, can be shown never to add up to more than the number 2 even if we consider infinite terms. But a series like 1+ ½ + 1/3 + ¼, +1/5 + 1/6 + ……….. , where successive terms do not diminish as rapidly as the previous case, if we go to infinite terms, adds up to infinity.
More complicated series even add up to values that can never be exactly stated, but need to be described only as an infinite series. One example of such a number is p, the ratio of the diameter of a circle to its radius. In this number, the terms after the decimal point continue for ever, and never repeating as in the case of recurring decimals.
The terms are therefore truly random, and series that add up to such numbers have been of interest since long.
Ramanujan, with his uncanny, nearly inspired insight came up with incredible formulas for such series, which had the property of getting very close to the final decimal numbers even on evaluating only a few of the terms of the series. These formulas are hence a powerful way of generating numbers of the nature of p, correct to many decimal places, faster than by using other series expansions.
Many of these formulas or procedures were described in the ‘note books’, or even written down later, without complete proofs or the systematic explanation of professional mathematicians.
Finding proofs for such propositions in the ‘note books’ has thus become a lifetime task for many, and for yet others, a task of following up on the remarkable results that he published, to discover efficiently creating long series of random numbers, which has also now found application in secret codes used in electronic commerce.