A Letter from Madras

In January 1913 a twenty-five-year-old accounting clerk in Madras posted nine pages of mathematics to a professor at Cambridge he had never met.

He had no degree. He had failed college twice. He was earning about thirty rupees a month and doing mathematics on a slate because paper cost too much.

The pages contained about a hundred and twenty results, almost none of them proved. Some were already known. A few were wrong. And some were so strange that Hardy concluded nobody could have invented them.

They were true.

At about fifteen he got hold of a book by George Shoobridge Carr, a cramming manual for Cambridge examinations.

It is a list of roughly five thousand results in algebra, trigonometry, calculus and analysis, with the proofs compressed to almost nothing or omitted entirely.

For most students that would be useless. Ramanujan treated every line as a puzzle, worked out his own proof, and then kept going.

It also shaped him permanently. He learned mathematics as a set of results to be found, not arguments to be written down — which is exactly what made him so hard for professionals to deal with later.

He won a scholarship and lost it, twice, because he would not study anything except mathematics.

Without a degree there was no job in mathematics and no library. He spent years poor, often ill, doing original work in notebooks and showing them to anyone who would look.

In 1912 a sympathetic official got him a clerk’s job at the Madras Port Trust. His supervisor and a few local engineers recognised that something unusual was going on and encouraged him to write to England.

Two professors in England ignored the letters. The third did not.

The letter opens with an apology for being a clerk, and then simply starts listing theorems.

Hardy read it, decided it was probably from a crank, and put it aside. Then he could not stop thinking about a few of the formulas, and in the evening he fetched Littlewood.

They worked through it. Some results were known. Some were false. And some were, in Hardy’s words, scarcely possible to believe — and if they were not true, nobody would have had the imagination to invent them.

He wrote back within days. Getting Ramanujan to England took another fifteen months of letters, money and religious objection from his family.

Ramanujan reached London in April 1914. The war began in August.

He was a strict vegetarian in a country introducing rationing, cooking for himself in a college room, in a climate he had never experienced.

The work was extraordinary anyway. Hardy spent those years doing two things at once: collaborating with him, and teaching him to write proofs that other mathematicians could check — without, as Hardy put it, destroying his confidence or breaking the spell.

In 1918 he was elected a Fellow of the Royal Society at thirty-one, and then a Fellow of Trinity. By then he was seriously ill.

Here is a question a child can ask. In how many ways can you write a number as a sum of whole numbers?

Four can be written five ways: four; three plus one; two plus two; two plus one plus one; and one four times.

Ten has forty-two ways. One hundred has one hundred and ninety million, five hundred and sixty-nine thousand, two hundred and ninety-two.

There was no formula. You counted, and the counting became impossible almost immediately. Slide the number and watch it explode.

In 1918 Hardy and Ramanujan produced a formula. It is not exact — it is an approximation that gets better as the number grows.

The leading term alone is a few per cent high at a hundred. Add the correction terms they derived and the total comes out at one hundred and ninety million, five hundred and sixty-nine thousand, two hundred and ninety-one point nine nine six.

Getting there required inventing a technique, now called the circle method, which is still a standard tool in number theory.

Ramanujan also noticed something nobody else had: the values are not random. Every fifth one is divisible by five, every seventh of another family by seven, and so on. Press the formula button and compare.

In 1914 Ramanujan published a handful of formulas for pi. One of them is this.

Take one term and you get pi correct to six decimal places. Take two and you have about fourteen. Each additional term buys roughly eight more digits.

He gave no derivation. The numbers one thousand one hundred and three and twenty-six thousand three hundred and ninety are not arbitrary — they come from the theory of modular equations — but he simply stated the result.

It was not proved until 1987, seventy-three years later, and in the 1980s it was used to set world records for computing pi.

Add the terms and watch the digits lock in.

Asked where his results came from, Ramanujan said that the family goddess Namagiri revealed them to him, often in dreams.

Hardy, who was a famously combative atheist, did not argue with him about it and did not use it against him either.

That is the right response. Science does not care where an idea comes from. It cares whether it survives checking.

His notebooks contain around three thousand nine hundred results. Mathematicians spent most of the twentieth century going through them, and nearly all of them are correct.

In 1919 Hardy went to see Ramanujan, who was ill in a nursing home in Putney, and could not think of anything to say.

So he mentioned that he had arrived in taxi number one thousand seven hundred and twenty-nine, which seemed a rather dull number.

Ramanujan replied at once that it is not dull at all: it is the smallest number expressible as the sum of two cubes in two different ways.

One cubed plus twelve cubed. Nine cubed plus ten cubed. Both give one thousand seven hundred and twenty-nine. Search the cubes yourself and find it.

Ramanujan was seriously ill from 1917 onwards, in and out of English sanatoria, and sailed home in 1919.

In January 1920, three months before he died, he wrote to Hardy describing what he called mock theta functions: seventeen examples of a new kind of object, with properties but no definition.

Nobody could say what they actually were. They sat as a curiosity for eighty years, until in 2002 Sander Zwegers showed they are the visible part of a larger structure, now called mock modular forms.

Those objects now turn up in the mathematics of black hole entropy. He died on the twenty-sixth of April 1920, aged thirty-two, of an illness that would be curable today.

In 1976 the mathematician George Andrews was going through a box of papers at Trinity College, left by another mathematician who had died in 1965.

Inside were eighty-seven loose sheets in Ramanujan’s handwriting, containing around six hundred formulas from the last year of his life.

They had been sitting in the library for over half a century, catalogued under somebody else’s name.

It is now called the lost notebook, and mathematicians are still working through it.

Most mathematicians are read, used and absorbed. Ramanujan is still being decoded.

His partition congruences started a field. His tau function led to one of the deepest conjectures in number theory. His series for pi are in the software that computes it. His mock theta functions turned out to describe black hole entropy.

The results came without proofs, and finding the proofs has occupied very good mathematicians for a century.

He had five years of professional support, in the middle of a war, while ill. That is the whole of his career.

The part worth remembering is how close it came to not happening at all.

Ramanujan sent similar letters to two other British mathematicians. Both ignored them. If Hardy had done the same, the notebooks would have stayed in a drawer in Madras.

Talent of that kind is presumably not rare. Having someone open the envelope is.

The chapter stills in this film are AI-generated. The photograph of Ramanujan, the notebook page, the photograph of Hardy and Trinity Great Court are real. Sources are listed below.