March 14th is celebrated every year as the International Day of Mathematics or alternately, the Pi Day (3/14 in the month/day format… as you know 3, 1, and 4 are the first three significant digits of π which goes 3.14159265358….. and so on).
Pi (π), the ratio of the circumference of a circle to its diameter, is a transcendental number whose decimal extent is infinite. It has been a fascination for mathematicians since time immemorial:
– in the 7th century, Brahmagupta declared it to be the square root of 10, giving it the value 3.16
– Archimedes took a geometric approach and arrived at a value between 3 10/70 and 3 10/71
– from the 16th century onward, western mathematicians including Issac Newton used calculus to develop several infinite series that converged to π.
Predictably, Ramanujan too wasn’t immune to the mysterious charm of π.
When Ramanujan started his romance with mathematics, the approximate value of π was already known. To him then, the joy was in discovering newer ways in which one could express π and developing formulae that would give more and more precise values of π.
He began with expressions that gave the approximate value to a modest number decimal places such as:
which gave the value of π up to 9 decimal places : 3.14159265380
which gave up to 14 decimal places : 3.14159265358979265 and
which gave up to 30 decimal places : 3.141592653589793238462643383279
One of the pages from his famous ‘Notebooks’ carries the following entries (highlighted in red):
Then, in 1914, the Quarterly Journal of Pure and Applied Mathematics carried Ramanujan’s publication, titled “Modular Equations and Approximations to π” that contained not one, but seventeen different series that converged rapidly to π. Two of these were:
which gave 8 correct digits for the decimal places for each term of ‘n’ and
which ‘spat out’ 14 correct digits for every ‘n’, allowing one to calculate to thousands of decimal places in a very short time.
While Ramanujan’s formulae were progressively more and more accurate, what is more important to us today is his approach to the calculations, which provided the foundation for the fastest- known algorithm that, in 1987, allowed mathematician and programmer Bill Gosper to use the computer to churn out the value of π to around 17 million decimal places. Later, mathematicians David and Gregory Chudnovsky used his formulae as the basis of their own variants that allowed them to calculate the value of π to an astounding 4 billion decimal places using their homemade parallel computer.
Of course, today, with the advent of super-computers, it has been possible to get the value of π with an ever-increasing precision. In January this year, Timothy Mullican successfully calculated the value to a record 50 trillion decimal places after 303 days of computing!!
When Ramanujan was a child, he liked to rattle off the numerical value of π and another transcendental number ‘e‘ to any number of decimal places. In my recently published picture book biography of Ramanujan (you can read more about it here), I’ve mentioned how he had been fascinated with calculating the length of the equator – an exercise for which he would certainly have had to use π.
In 1914, around the time he indulged in developing formulae for π, he calculated the length of the equator to be 40,078km. Today, with supercomputers at our disposal, we know the length to be around 40,075km.
Now, if that isn’t genius, what is?!!
Psst… want to read the picture book mentioned above? Here’s where you’ll find it: