January 25, 2019

Srinivasa Ramanujan: The 1729 Man

A math column I wrote in December 2012 for Mint, reprised here.

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What I've been doing all morning is, I've been trying to come up with some numerically interesting factoid about the number 125. It's what you get when you cube 5 (5 x 5 x 5, or 5^3), but that's kind of ordinary, no? Well, you can write it as the sum of two squares in two different ways:

2 x 2 + 11 x 11 = 4 + 121 = 125
5 x 5 + 10 x 10 = 25 + 100 = 125

Now that's promising. But is 125 the smallest number that can be so expressed? No, because 65 is the sum of the squares of 1 and 8, as well as the squares of 4 and 7.

Hmm. Maybe 125 is the smallest cube that can be written as the sum of two squares in two different ways? It certainly is! Now we're talking!

Yet what exactly are we talking? And why am I pursuing this pointless pastime?

Well, the great mathematician Srinivasa Ramanujan would have been 125 tomorrow, December 22nd. Mathematicians all over India -- and indeed the world -- are celebrating his life. Delhi, for example, hosted a major international Ramanujan conference this week. And I suspect all who attended -- anyone, in fact, who knows something about Ramanujan -- will know the story about him that prompted my quest with 125.

This story: he was in hospital in Putney, England. In walked GH Hardy, his mentor and a fine mathematician in his own right. He told Ramanujan that he had come in a cab whose number, 1729, "seemed to me rather a dull one." (I get the feeling Hardy and Ramanujan rather liked playing around with numbers).

The sick Ramanujan disagreed. "Oh no, not at all!" he said. "It is the smallest number that can be written as the sum of two cubes in two different ways!"

Of course he was right. Here are the two ways:

1^3 + 12^3 = 1 + 1728 = 1729
9^3 + 10^3 = 729 + 1000 = 1729

And 1729 is indeed the smallest such number. And because of this incident, it is now known as the Ramanujan-Hardy number.

What touches me about this story is though Ramanujan was seriously ill that day, he was sharp enough to remember, and tell Hardy, this little nugget about a random number Hardy mentioned. And so, to mark his 125th birthday, I thought it only fitting to search for some nugget like that about 125. (Seeing as I've got other plans for his 1729th birthday).

Though I think I have to admit: 1729 is a more interesting number than 125.

Ramanujan's life-story is too well-known for me to spell it out here. Suffice it to say that when this clerk in Madras sent some of his homespun mathematical research to Hardy in 1913, Hardy saw genius in it and invited Ramanujan to work with him at Cambridge. What followed was an intensely productive five-year collaboration and friendship between these two remarkable men. For me, something of that relationship is captured in a famous tale about how Hardy rated his own and other mathematicians' raw talent for mathematics. On a scale of 0 to 100, Hardy awarded himself 25. And Ramanujan? 100.

But despite Hardy's respect, England took a toll on Ramanujan's health. In 1919, he returned to India. In April 1920, at just 32, he died.

I wouldn't presume to dissect Ramanujan's mathematical work here. The great majority of it ranges far beyond both my understanding and ability to explain. But what I do understand are his efforts to calculate the value of π (pi) that intangible number that tells us the ratio of a circle's circumference to its diameter. Much like Roger Federer used to rack up Grand Slam tennis titles, Ramanujan churned out formulae (the better word is series) to calculate π. One of them turns out to look like this:

1/π = 5/16 + 376/65536 + 19224/268435456 …

Take my word for it: those three terms alone give a value for π accurate enough for most purposes you might imagine. Other Ramanujan formulae have been used to calculate π to millions of digits after the decimal point.

All of which is fascinating and remarkable enough. Yet to a layabout like me, Ramanujan's real appeal lies in that story about 1729. Because in a quiet yet substantial way, it speaks for the way his mind worked. It speaks of his curiosity and passion for numbers and mathematics. It tells me that when you constantly search for and find wonder in the smallest things -- who would have thought, 1729? -- you prime yourself for greater things. And Ramanujan achieved some great things indeed.

That's the lesson I take from the life of this young genius from Kumbakonam.

As of last year, his birthday is observed as National Mathematics Day in India. So celebrate it tomorrow. Celebrate an exceptional mind. And if you stumble on something interesting about 125, please let me know.