Ramanujan’s Puzzles

Source: Quanta Magazine, Jul 2016

The work that Ramanujan did in his brief professional life a century ago has spawned whole new areas of mathematical investigation, kept top mathematicians busy for their whole professional lives, and is finding applications in computer science, string theory, and the mathematical basis of black hole physics.

The mathematician Mark Kac divided all geniuses into two types: “ordinary” geniuses, who make you feel that you could have done what they did if you were say, a hundred times smarter, and “magical geniuses,” the working of whose minds is, for all intents and purposes, incomprehensible. There is no doubt that Srinivas Ramanujan was a magical genius, one of the greatest of all time. Just looking at any of his almost 4,000 original results can inspire a feeling of bewilderment and awe even in professional mathematicians: What kind of mind can dream up exotic gems like these?

Ramanujan indeed had preternatural insights into infinity: he was a consummate bridge builder between the finite and the infinite, finding ways to represent numbers in the form of infinite series, infinite sums and products, infinite integrals, and infinite continued fractions, an area in which, in the words of Hardy, his mastery was “beyond that of any mathematician in the world.”

What if Ramanujan had modern calculating tools?

Ramanujan, like many other great mathematical geniuses such as Gauss, loved to play with specific cases, which he then built into general results. These geniuses did prodigious calculations by hand — Ramanujan used chalk on slate in his early days, erasing intermediate results with his elbow.

Steven Wolfram has conjectured that if Ramanujan had modern calculating tools like Mathematica, “he would have been quite an adventurergoing out into the mathematical universe and finding all sorts of strange and wonderful things, then using his intuition and aesthetic sense to see what fits together and what to study further.” What do you think?

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