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:
Three years back, in 2016, hubby dear bought the book -The Man Who Knew Infinity, by Robert Kanigel. It was supposedly a well researched biography of the mathematical genius Srinivasa Ramanujan. Not a fan of biographies then, I let the book rest on my shelf for a whole six months before I started reading it.
Until then, I had only heard passing mentions of Ramanujan, never read about him -surprisingly, not even an anecdote or snippet from his life. So, Kanigel’s book came as a surprise. It immediately drew me into its pages, taking me back to an era (Ramanujan was born on 22nd Dec, 1887) that was hitherto unknown to me except in the context of India’s freedom struggle.
For reasons unknown, the book had a profound effect on me. No sooner than I had finished reading it, I got an opportunity to go to Chennai. I immediately made up my mind to visit Kumbakonam, where Ramanujan had spent the better part of his short life. From Chennai, I took an overnight train to the small, dusty town on the banks of the Kaveri – the journey in itself was rather amusing, with my second class compartment being populated by men wearing white lungis and white shirts and women wearing gold jewelry that offset their lips and tongues that had turned red with chewing betel leaf.
Once in Kumbakonam, I spent half a day at Sarangapani Sannidhi Street that was, at one time, witness to the many eccentricities of my newfound muse. Ramanujan’s house -recently renovated after escaping demolition, thanks to our late President APJ Abdul Kalam’s intervention -sat sandwiched between two other houses that had been converted into shops. A set of steps invited me into this narrow house that stretched backwards in length. The blue columns weren’t blue during Ramanujan’s time; I’m sure they weren’t even painted then. The well laid Mangalore tiles weren’t there during his time either -the house had a thatch roof then.
But the columns and roof didn’t matter to me. What mattered was the high plinth on which Ramanujan sat as a child with a slate and chalk and worked on his mathematical ideas. What mattered was the window behind which was the tiny room with a single wooden bed under which Ramanujan hid to solve equations as a child because his father would get angry if he saw him do something so useless! Further inside was a small living room (that’s now a memorial of sorts), kitchen and the backyard with a well.
It was indeed a humbling experience to stand inside that house, on that street, in that dusty town from where Ramanujan had started a mathematical journey that took him all the way to England and back. But, as Kanigel’s book informed me, Ramanujan wasn’t just about mathematics. His short life (he was 32 when he passed away) was a tapestry woven from numerous strands, each as interesting as the other. He was a staunch Vaishnavite Brahmin, who not only knew his scriptures but dissected and discussed them even as he brooded on the concepts of ‘shunya’ and ‘infinity’. He was highly superstitious, with an interest in astrology as well as the occult. He was a quintessential ‘mama’s boy’ who fell back upon his mother for everything in his life. Be it narrating stories from the Ramayana, Mahabharatha and the Puranas, or imparting knowledge about their traditions or teaching him to pray and become a devotee of the Goddess Namagiri of Namakkal or playing with him, his favorite board game -the Aadu Puli Aatam (Goats and Tigers), or giving that vital push towards achieving his mathematical goal, Komalatammal was instrumental in nurturing her son’s love for Math and standing up for him when it mattered. Finally, there was also a strange dichotomy about Ramanujan – while he was confident about his mathematical prowess, he was extremely insecure about everything he did, yearned for recognition of his genius, and took offense at the tiniest of alleged faux pas by friends or peers.
Sitting on the steps in the backyard of his house, I recollected excerpts from Kanigel’s book and found myself drawn to this complex, intriguing character from the past. Soon, I was trying to imagine Ramanujan’s childhood, figuring out how his surroundings could have contributed to his love for mathematics. As I did so, quite unknown to me, a seed was sown into my thoughts -a seed of an idea for a book for children based on Ramanujan’s life story. Researching for the book, I ended up reading many more papers and books that talked about his life and works -most importantly, S R. Ranganathan’s Ramanujan: The Man and the Mathematician. Today, that idea is on its way to becoming a reality. My picture book biography on the life of Srinivasa Ramanujan has charted its own journey and is expected to be released soon.
(edit: the book was launched on 18/1/2020 and is now available in stores. Adding a pic of the cover page… you can read more about it here)But that is only half my story.
Being an interior architect, my first connect with people is through the spaces they inhabit. So, after visiting his Kumbakonam house, I wanted to now visit all other places he’d been to or stayed in. This wasn’t easy, given that my city of residence isn’t anywhere near these places and that I have a home, office and my kids to tend to. However, they don’t say -if you wish for something with all your heart it does become a reality- for nothing. So intense was my wish of understanding the enigma that was Ramanujan that serendipity offered me a few chances to catch up with my muse. Earlier this year, I got another opportunity to visit Chennai. I made the most of it by visiting the house he’d briefly stayed in, in Triplcane, thanks to a dear friend. I then hunted down the site of the house he’d breathed his last in -unfortunately, the house is no longer in existence- and another house that he’d briefly lived in after his return from England.
Then, as luck would have it, I got an opportunity for an academic visit to London. After my professional commitments, I stayed back for a few days to visit places in London and Cambridge that my muse had been to. And so, with the help of Richard Chapling, a young mathematician and Trinity alumnus, who started off as a total stranger but ended up being a dear friend, I traced Ramanujan and his mentor Godfrey Harold Hardy across London and Cambridge. Together, Richard and I walked the walk that Hardy once described as being the ‘most distinguished walk’* – from his home at St. George’s Square, along the Grosvenor Road and across the Vauxhall Bridge to the Oval; we visited the house in Putney that is famed for being the place where Ramanujan had the legendary conversation with Hardy about the number 1729; we traveled to Cambridge to visit the Trinity College where he had spent five precious years, the Wren Library where his original letter to Hardy has been preserved, the Centre for Mathematical Sciences where his bust sits in splendor; we paused at the houses he’d stayed in at different points of time, we walked down the streets he would have once walked… back in London, I hunted down the house on Cromwell Road that Ramanujan had stayed in when he first arrived in London -the building is now home to the French Embassy.
Visiting all these places was a strangely emotional journey for me – strange, because here I was, getting affected by places that I had no immediate connect to. I often sit and wonder why I was drawn towards Ramanujan’s story. A friend recently tried to impress upon me the idea of ‘past connections’ and ‘karma’. I’d be lying if I said I’m not tempted to agree with him in this context.
There are some more places with a Ramanujan connect in India that await me, and there is a more detailed story of his inside of me that needs to be told (perhaps, for older children). I guess I’ll get that sense of closure only when both these journeys are completed… and I hope it happens sometime soon enough.
Today, though, I’m going to celebrate Ramanujan’s birthday by sharing the Magic Square he’d worked out based on the date of his birth: 22-12-1887 (exploring possibilities with these squares was one of Ramanujan’s earliest mathematical preoccupations).
For those who do not know, a Magic Square is square grid in which a given set of unique positive integers are arranged such that each cell has a different integer and the sum of integers in every row, column and diagonal is equal. In the above case, the sum in each case is 139. It is a happy coincidence (I’m sure Ramanujan would’ve been pleased as a punch when he realized this) that 139 is a prime number and the sum of five consecutive prime numbers: 19 + 23 + 29 + 31 + 37!!
And on that note, here’s me wishing my mathematical muse a very happy birthday!!
*The original quote by G H Hardy (thanks to Richard) goes thus:
‘The half-mile from St. George’s Square to the Oval is my old brandy nomination for the most distinguished walk in the world.’
Richard tells me that Old Brandy came to mean a taste that was eccentric, esoteric, but just within the confines of reason.
**P.S: There’s a separate post begging to be written about my amazing visit to Cambridge with Richard. I hope I find time for it soon enough…