Three Puzzles Inspired by Ramanujan
Very interesting piece in Quanta by Pradeep Mutalik:
Srinivasa Ramanujan’s story is part of mathematical folklore, one of the most romantic in the history of mathematics. He started as a poor self-taught clerk in India. Working alone, he discovered highly original and unknown mathematical results that were far ahead of his time. In 1913, he sent a letter filled with strange-looking mathematical theorems to G. H. Hardy, a mathematician at the University of Cambridge. Hardy, one of the world’s leading experts on number theory, later said that “some of [Ramanujan’s theorems] defeated me completely; I had never seen anything in the least like them before.” He was referring to results like the one below.
Hardy arranged for Ramanujan to come to England, and the rest is history. 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.” While most of Ramanujan’s results are far beyond the scope of this column, it turns out that we can get a flavor for some simple infinite forms using nothing more than middle-school algebra. Let’s embark on a journey to the infinite.
1. Our first question is to prove the following equation involving an infinite nested radical. In 1911, Ramanujan sent the right-hand side of the following equation to a mathematical journal as a puzzle:
If you are intimidated by this at first sight, don’t be! As I mentioned above, you can prove this result with no more than middle-school algebra. All you need is the following elementary result: . . .