Srinivasa Ramanujan, a self-taught Indian mathematician, compiled during his short life a vast number of deep results spanning analysis, number theory, and modular forms. His achievements rank among those of the greatest mathematicians. *The Man Who Knew Infinity* is a film about Ramanujan’s remarkable life.

The film is based on the biography of the same name by Robert Kanigel, published in 1991.^{1} Unlike the book, which provides a complete and accurate account of Ramanujan’s life, the film focuses primarily on his time at Cambridge University between 1914 and 1919. Ramanujan’s initial discoveries and early life in India are touched on only briefly.

The film explores Ramanujan’s work, his relationship with the mathematician G. H. Hardy, the resistance he encountered from some of his fellow mathematicians, and the difficulties he experienced living in England, where he fell seriously ill. Ramanujan returned to India in 1919 and died the following year, aged thirty-two.

Although *The Man Who Knew Infinity* recasts a number of events for dramatic effect, Ramanujan’s life is sufficiently fascinating and intriguing that dramatization is largely unnecessary. Ramanujan’s life story is enough to move an audience to tears.

## His Life

Before examining the film in more detail, a brief biographical sketch may be helpful for readers unfamiliar with the subject. Ramanujan was born on December 22, 1887 in his maternal grandfather’s hometown, Erode, in the southern Indian state of Tamil Nadu. He grew up in the nearby town of Kumbakonam. Ramanujan’s father was a lowly-paid accountant to a cloth merchant. His mother, Komalatammal, was an exceptionally strong willed woman.

Ramanujan and his family were orthodox Iyengars, a subsect of the Brahmin caste of the Hindu religion. The goddess Namagiri, worshipped at the Temple of Namakkal, was held in veneration by the family. Prior to the arrival of Ramanujan, Komalatammal had been childless. The family prayed to Namagiri in the hope that she would bless Komalatammal with children. Ramanujan was born shortly thereafter.

Ramanujan’s mathematical talents were evident from an early age. As a boy, he often woke in the middle of the night and wrote mathematical formulas on a piece of slate he kept beside his bed. These were subsequently recorded in his notebooks.^{2} Legend has it that it was Namagiri, appearing in Ramanujan’s dreams, who gave him the formulas. No proofs appear in his notebooks.

Unable to find anyone in India who could evaluate his work, Ramanujan wrote two letters to Hardy in 1913. Ramanujan included examples of the formulas he had discovered, but did not provide any proofs or reasoning. On the basis of these letters, Hardy, in consultation with his colleague J. E. Littlewood, came to the conclusion that Ramanujan was a genius.^{3} Several of the formulas Ramanujan provided were profoundly original, but some were already well known, and others incorrect as stated.

Hardy felt that Ramanujan’s time should not be wasted rediscovering past work. He invited Ramanujan to join him at Cambridge to provide him with formal training and a proper sense of direction.

Orthodox Brahmins believed that it was a sin to cross the oceans. For this reason, Komalatammal would not allow Ramanujan to travel to England, even though he was willing to do so. Namagiri seems to have played a role in resolving this impasse. Komalatammal had a dream, in which Ramanujan was honored at an assembly of white men. In the dream, Namagiri ordered Komalatammal not to stand in the way of her son’s recognition. The next morning, Komalatammal gave her permission for Ramanujan to sail to England.

Ramanujan lived and worked in England for just five years. A number of the theorems he had discovered in India were published in prestigious journals. He made several new and fundamental discoveries and collaborated with Hardy on two major projects, one of which was the asymptotic series for the partition function by the powerful and innovative circle method.^{4}

Daily life in wartime England was tough, and especially so for Ramanujan. A strict vegetarian, he found it hard to obtain suitable food. He was also unaware of how to protect himself from the winter cold. As a result, Ramanujan often fell ill and spent time in hospitals and nursing homes. Despite these difficulties, his mathematical work was unaffected.

On the basis of his work and achievements, Hardy firmly believed that Ramanujan deserved to be elected a fellow of Trinity College and the Royal Society. Ramanujan’s health, now declining rapidly, created a sense of urgency in this respect. Hardy worked tirelessly to convince his colleagues. His advocacy on Ramanujan’s behalf was ultimately successful and both honors were conferred.

In March 1919, Ramanujan returned to India in very poor health.^{5} His condition deteriorated further in the months that followed. Ramanujan, undeterred by his plight, continued working. Three months before his death, he wrote a final letter to Hardy that outlined his groundbreaking discovery of the mock theta functions of orders 3, 5, and 7. These are now considered to be among Ramanujan’s deepest contributions.

Ramanujan died in Madras on April 26, 1920.

In the last months of his life, Ramanujan frequently asked his wife Janaki Ammal for loose sheets of paper to record new results. Following Ramanujan’s death, Janaki, who had no formal education, delivered the papers to the University of Madras. They were subsequently forwarded to Hardy, who passed them on to G. N. Watson, the world's premier authority in the field of special functions. After Watson’s death in 1965, the papers, comprising eighty-seven pages of handwritten results and more than six hundred formulas, were placed in storage at the Trinity College Wren Library. Forgotten by the mathematical world, the papers became known as “The Lost Notebook.” They were rediscovered by George Andrews in 1976 and have been the focus of intensive research ever since.^{6} *The Lost Notebook and Other Unpublished Papers *was finally published in 1987, in honor of the Ramanujan Centennial.^{7}

Ramanujan’s work was profoundly original, revealing new and unexpected connections between disparate fields. He typically recorded only the most striking or significant case of a general result. Investigating his identities reveals vast underlying theories. Ramanujan’s work has had a deep impact on number theory, analysis, combinatorics, and the theory of modular forms, to name just a few fields, as well as on domains outside of mathematics, such as computer science and physics. More specifically, the influence of his work can be seen in hypergeometric and *q*-hypergeometric series, partitions and combinatory analysis, additive number theory via the circle method, probabilistic number theory, elliptic and theta functions, modular forms and automorphic functions, special functions and definite integrals, continued fractions, Diophantine equations, irrationality and transcendence, Fourier analysis, Lie algebras, statistical mechanics and conformal field theory in physics, computer science, and computer algebra. This list is not exhaustive and, in fact, continues to grow.^{8}

According to Hardy, the real tragedy of Ramanujan’s career was that he wasted so much time on rediscovery during his formative years. Hardy believed that a mathematician’s best work is done at a young age. Had Ramanujan lived longer, in Hardy’s view, he would have discovered more theorems, but likely never surpassed the originality of his early work.^{9} And yet Ramanujan’s work on mock theta functions in the months prior to his death revealed, if anything, *increasing* originality.

According to Paul Erdös, Hardy rated mathematical talent on a scale of 1 to 100. Hardy assigned himself a score of 25, his colleague Littlewood a score of 30, David Hilbert a score of 80, and Ramanujan a perfect score of 100.^{10}

## The Dramatization

Producing a film for a general audience that accurately portrays complex mathematical content yet still retains broad appeal is extremely challenging. Antipathy towards mathematics is not uncommon and many view mathematical research as pointless abstraction. In the case of Ramanujan, fortunately, his life story has all the drama, excitement, and intrigue one might want for a film script.

Filmmakers often include sex and violence to ensure the success of a production. *Good Will Hunting* is a case in point. The story of a mathematical genius lacking a sense of purpose, the film portrays geniuses as freaks of nature. Although extremely successful, the film failed to fully explore an unusually interesting story as a means to convey the importance and excitement of mathematics when done by brilliant minds.

*The Man Who Knew Infinity* depicts Ramanujan’s life and contributions in a tasteful manner. The involvement of mathematicians such as Ken Ono and Manjul Bhargava as associate producers ensured that the film’s mathematical content would be presented accurately. On occasions when events have been recast for dramatic effect, these variations do not detract from the overall quality of the film.

Dev Patel, of *Slumdog Millionaire *fame, plays the role of Ramanujan. A good actor, Patel is nonetheless much taller, slimmer and, in particular, fitter than Ramanujan ever was. The choice of Patel for the lead role is understandable. A good looking and internationally-known actor helps sell the character to the public. Hardy is portrayed brilliantly by Jeremy Irons. Although Irons bears a striking resemblance to Hardy, he looks much older than Hardy would have been at the time.

Ramanujan’s wife Janaki is played by Devika Bhise. Among the few scenes portraying Ramanujan’s early life in India, the newlyweds are shown arriving at their new home in Madras.^{11} Depicted as a mature girl in the film, Janaki was, in fact, barely nine years old at the time of their marriage in 1909. She remained with her parents until she reached puberty, joining Ramanujan in 1912 when he was working as a clerk at the Madras Port Trust. Child marriage was common in India at that time, but including it in the film would have likely proved a distraction.

Ramanujan’s dominating mother, Komalatammal, is aptly portrayed by Arundhathi Nag. The film depicts Janaki writing to Ramanujan many times while he was in England and, as a dutiful daughter-in-law, passing the letters on to Komalatammal to mail them. The film shows Komalatammal hid all these letters, as well as those that Ramanujan wrote to Janaki. The aggravation and anxiety that this caused for the couple is emphasized in the film. In one such letter, Ramanujan informed Janaki that he was returning to India in ill health and asked her to meet him upon his arrival in Bombay. Janaki never received the message.^{12}

Ramanujan’s supervisor at the Madras Port Trust, Narayana Iyer, with whom Ramanujan regularly discussed his work, recognized Ramanujan’s unique mathematical talents. Iyer’s supervisor, Sir Francis Spring, is depicted in the film initially rejecting Ramanujan’s mathematics as worthless. But, thanks to Iyer’s influence, Spring eventually comes to appreciate Ramanujan’s brilliance. Iyer did indeed play a key role in convincing Spring how special Ramanujan was, but there are no records of Ramanujan being initially spurned by Spring. There are several instances in the film where Ramanujan is shown being treated poorly by the British. Some are exaggerated and others are entirely fictional.

The scenes depicting discussions between Iyer and Ramanujan carrying on late into the night effectively convey Ramanujan’s total preoccupation with mathematics and Iyer’s almost parental interest in him. Consequently, Ramanujan failed to spend much time with his young wife. In a touching scene, Janaki interrupts a late night discussion between Ramanujan and Iyer, reminding her husband not to forget her.

Mathematics is as much art as it is science. Hardy, for one, never cared about the possible application of his results. This is not to say that mathematics is not useful, but that its beauty alone can be the primary attraction. This was certainly true for Ramanujan. In the film he is shown telling Janaki that his mathematical equations are as lovely as the flowers in a garden, and that he seeks mathematical patterns as one would marvel at the patterns in nature. The idea that mathematics can be beautiful is quite alien to much of the public. Thus it is all the more important that *The Man Who Knew Infinity* emphasizes aesthetic appeal as a driving force behind the pursuit of mathematics.

A romantic scene in the film depicts Ramanujan and Janaki frolicking on the beach in Madras. Although Ramanujan was a loving and caring husband, he hailed from an orthodox Hindu family and thus would never have behaved this way in public. This scene is pure fantasy, albeit thoroughly enjoyable for the audience.

Ramanujan’s devotion to his religion and the inspiration he derived from the goddess Namagiri is depicted in several scenes, such as when Ramanujan is shown writing formulas on the stone slabs of a temple floor. The film also depicts Ramanujan’s religious practices during his time in England. At one point, Ramanujan remarks to Hardy that “an equation to me has no meaning unless it expresses a thought of God.”^{13}

Ramanujan’s final few months, during which he suffered a great deal, are not shown in the film. Most of Ramanujan’s relatives boycotted his cremation, believing he had sinned by crossing the oceans. As would have been the case with addressing the issue of child marriage, depicting how unfairly Ramanujan was treated by his orthodox relatives would likely have distracted from the film’s focus on his mathematical achievements.

## Ramanujan and Hardy’s Collaboration

The two letters Ramanujan wrote to Hardy in 1913 are perhaps the most important mathematical letters ever written. They contain dozens of incredible formulas. His first letter begins as follows:

I beg to introduce myself to you as a clerk in the Accounts Department of the Port Trust Office at Madras … After leaving school I have been employing the spare time at my disposal to work at Mathematics. I have not trodden through the conventional regular course which is followed in a University course, but I am striking out a new path for myself.^{14}

The letter concludes:

[I]f you are convinced that there is anything of value I would like to have my theorems published. I have not given the actual investigations nor the expressions that I get but I have indicated the lines on which I proceed. Being inexperienced I would very highly value any advice you give me.^{15}

The letters contained formulas for the number of primes up to a given magnitude with some finer statements as to their distribution, some definite integral evaluations, continued fractions evaluations, modular identities for elliptic and theta functions, and so on.

Two examples from his letters are

- $\frac{1}{1+}\frac{{e}^{-2\pi \sqrt{5}}}{1+}\frac{{e}^{-4\pi \sqrt{5}}}{1+}\dots =\left\{\frac{\sqrt{5}}{1+{\left\{{5}^{3/4}{\left(\left(\sqrt{5}-1\right)/2\right)}^{5/2}-1\right\}}^{1/5}}-\frac{\sqrt{5}+1}{2}\right\}{e}^{2\pi /\sqrt{5}}$

and

- $\text{thecoefficientof}{x}^{n}\text{in}\frac{1}{1-2x+2{x}^{4}-2{x}^{9}+2{x}^{16}\dots}$

$\text{isthenearestintegerto}\frac{1}{4\pi}\left\{cosh\pi \sqrt{n}-\frac{sinh\pi \sqrt{n}}{\pi \sqrt{n}}\right\}.$

The first formula is an incredible evaluation of the celebrated Ramanujan continued fraction at ${e}^{-2\pi \sqrt{5}}$, and the second is related to the famous asymptotic series for the partition function that is central to the film’s plot.

The scene depicting Hardy receiving the letter from Ramanujan and reading it in curiosity and amazement is beautiful. Hardy’s initial reaction was that Ramanujan must have been a fraud. He quickly realized after a conversation with his distinguished colleague Littlewood that Ramanujan was a genius ranking with the greatest mathematicians in history. The subsequent scenes depicting Hardy informing his colleagues about the letter and his decision to invite a hitherto unknown Hindu clerk to Cambridge are really impressive.

Upon Ramanujan’s arrival in England, Hardy was understandably keen to see proofs for Ramanujan’s remarkable claims, such as the first formula. In their discussions, Hardy points out that Ramanujan’s claims for formulas relating to prime numbers were actually false as stated. But the vast majority of Ramanujan’s results on infinite series, products, and integrals were, as Hardy realized, correct.

It remains unclear clear how Ramanujan arrived at these results. Hardy, a sworn atheist, dismissed the story of Namagiri giving the formulas to Ramanujan as a fable. Hardy is seen in the film asking Ramanujan how he discovered his results. In reality, this conversation never took place. Hardy noted that he was in the best position to resolve this mystery, but admitted not even once asking Ramanujan about the source of his inspiration.^{16}

The Hardy–Ramanujan asymptotic series for the partition function is the main focus of the mathematical discussions in *The Man Who Knew Infinity*.^{17} There are several reasons for this. Partitions are relatively easy to explain, unlike other aspects of Ramanujan’s work. The Hardy–Ramanujan formula for partitions is also one of the greatest achievements in number theory and the crowning glory of their collaboration. Additionally, this discovery illustrates beautifully how the intuition of Ramanujan and the sophistication of Hardy combined to establish a truly remarkable result.

A partition of a positive integer *n* is a representation of *n* as a sum of positive integers, two such representations being considered the same if they differ only in the order of the summands, or parts. For example, 2 + 2 + 1 is the same partition of 5 as 2 + 1 + 2. The number of partitions of *n *is denoted by by *p*(*n*). There are five partitions of 4: 4, 3 + 1, 2 + 2, 2 + 1 + 1 and 1 + 1 + 1 + 1 Thus *p*(4) *= *5. For a small number like 4, it is easy enough to work out all its partitions. But *p*(*n*) grows very rapidly for larger values of *n*. Consider *n = *200*,* for example:

- $p\left(200\right)=\mathrm{3,972,999,029,388}.$

How then do we know the value of *p*(200)? In the mid-eighteenth century, Leonhard Euler, founder of the theory of partitions, discovered the following remarkable recurrence relation:

- $p\left(n\right)=p\left(n-1\right)+p\left(n-2\right)-p\left(n-5\right)-p\left(n-7\right)+$

$p\left(n-12\right)+p\left(n-15\right)-p\left(n-22\right)-p\left(n-26\right)+\dots .$

The numbers 1, 2, 5, 7, 12, 15, 22, 26, … are the pentagonal numbers, given by the formula $\left(3{k}^{2}\pm k\right)/2$. One can use (4) to calculate *p*(*n*) from the values of the partition function at smaller integers. For example, *p*(11) = *p*(10) + *p*(9) - *p*(6) - *p*(4). Although Euler’s recurrence relation is remarkable, it is not a closed form evaluation of *p*(*n*).^{18} Hardy and Ramanujan were seeking a representation of *p*(*n*) in terms of familiar continuous functions. This seemed impossible because partitions represent a discrete process.

In the film, Hardy says simply, “Partitions can’t be done.” In real life, Hardy would never have spoken like this. But his remark conveys the near impossibility of such a representation using continuous functions without resorting to technical jargon that would be indecipherable for the public.

Euler was not only interested in *p*(*n*), but in other partition functions as well. Many of these have generating functions with elegant infinite product or series representations. These generating function evaluations yield beautiful relations among various partition functions. Euler’s famous formula for the generating function of *p*(*n*) is

- $F\left(z\right)\u2254\sum _{n=0}^{\infty}p\left(n\right){z}^{n}=\prod _{m=1}^{\infty}\frac{1}{\left(1-{z}^{m}\right)},\text{for}\left|z\right|1.$

From the fundamental Cauchy residue theorem in complex variable theory—formulated a few decades after the time of Euler—it follows that for each positive integer *n*,

- $p\left(n\right)=\frac{1}{2\pi i}{\int}_{C}^{}\frac{F\left(z\right)}{{z}^{n+1}}dz,$

where *i* is the imaginary square root of –1, and the integral is to be taken counterclockwise over a simple closed contour *C* encircling the origin and within the unit circle. The difficulty here is in determining the contour that would lead to the evaluation of *p*(*n*). Notice that the product part in (5) indicates that *F*(*z*) would become large when *z* is near a root of unity. Hardy and Ramanujan’s brilliant idea was to start with a simple circular contour and then to deform it, taking it close to the roots of unity, to evaluate *p*(*n*). This is very deep, difficult, and sophisticated, and is at the heart of the powerful circle method that they introduced in their famous paper of 1918. The final result they proved is that

- $\left|p\left(n\right)-\frac{1}{2\sqrt{2}}\sum _{q=1}^{v}\sqrt{q}{A}_{q}\left(n\right){\psi}_{q}\left(n\right)\right|=O\left(\frac{1}{{n}^{1/4}}\right),$

where *v* is of the order of magnitude $\sqrt{n}$, ${A}_{q}\left(n\right)$ is summed over certain *q ^{th}* roots of unity,

*O*(.) is less than a constant multiplied by (.), and

- ${\psi}_{q}\left(n\right)=\frac{d}{dn}\left(exp\left\{\pi \sqrt{\frac{2}{3}\left(n-\frac{1}{24}\right)}/q\right\}\right).$

What (7) means is that since the error term $O\left({n}^{-\frac{1}{4}}\right)$ tends to zero as *n* tends to infinity, the value of *p*(*n*) is the nearest integer to the remarkable sum of continuous functions up to *v*.

Hardy then informs his Cambridge colleagues that, “He has done it!” This creates the impression that Ramanujan was solely responsible for (7). While it could not have been accomplished without his insight, Hardy’s technical prowess with complex variables was just as important.

There is a wonderful convention in mathematics that for a joint paper, all authors are equal. Hardy was a firm believer in this practice and did not elaborate on the respective contributions of the pair. Littlewood reviewed Ramanujan’s collected papers and, having secured Hardy’s consent, elaborated on this point.^{19} Ramanujan had insisted all along that it should be possible to obtain an accurate formula involving continuous functions that would yield the value of *p*(*n*); Hardy was initially unconvinced. With *n* in the place of $n-\frac{1}{24}$, they could determine the asymptotic size of *p*(*n*). It was Ramanujan’s insight to use $n-\frac{1}{24}$; in this way they could show that, by summing the series up to a fixed number of terms, the remaining error would at most be of the order of magnitude of the next term.

Thus (7) yields an asymptotic series for *p*(*n*).

Ramanujan also came up with an intuitive understanding of the functions ${A}_{q}\left(n\right)$ and ${\psi}_{q}\left(n\right)$, and insisted that there ought to be a representation of *p*(*n*) with a bounded error. At this point Hardy asked P. A. MacMahon, a noted combinatorialist, to check the formula for numerical accuracy.^{20} MacMahon calculated the value *p*(200) in (3) using (4) and demonstrated the astonishing accuracy of (7). Using complex variables, Hardy could thus demonstrate, as surmised by Ramanujan, that if the sum in (7) is taken up to an order of magnitude of $\sqrt{n}$ terms, then one does indeed get *p*(*n*) as the nearest integer to the sum.

The determination of the precise form of ${\psi}_{q}\left(n\right)$ was crucial, and for this Littlewood credits Ramanujan’s intuition:

The form of the function ${\psi}_{q}\left(n\right)$ is a kind of indivisible unit; among the many asymptotically equivalent forms, it is essential to select exactly the right one. Unless this is done at the outset, and the $n-\frac{1}{24}$ (to say nothing of thed/dn) is an extraordinary stroke of formal genius, the complete result can never come into the picture at all. There is, indeed, a touch of real mystery. … [t]here seems no escape … from the conclusion that the discovery of the correct form was a single stroke of [Ramanujan’s] insight. We owe the theorem to a singularly happy collaboration of two men, of quite unlike gifts, in which each contributed the best, most characteristic, and most fortunate work that was in him. Ramanujan’s genius did have this one opportunity worthy of it.^{21}

One of the peculiar properties of (7) is that if *n* is replaced by ∞, the sum would diverge. This was established by Derrick Henry Lehmer in 1937.^{22} That same year, Hans Rademacher proved that if the exponential function used in the definition of ${\psi}_{q}\left(n\right)$ is replaced by a suitable hyperbolic function this would convert the series in (7) to an infinite convergent series, whose value would yield *p*(*n*).^{23} Although Ramanujan did not write down the Rademacher series, he intuited that something like it ought to exist. Hardy felt that a convergent infinite series for *p*(*n*) was too good to be true and settled for the asymptotic series for *p*(*n*). In (2), the expression on the right is equal to

- $\prod _{m=1}^{\infty}\frac{\left(1+{q}^{m}\right)}{\left(1-{q}^{m}\right)},$

which is the generating function for partitions, in which each partition is counted with the weight 2 raised to the power of the number of different parts in the partition. Thus (2) and (9) are generating functions of a weighted version of *p*(*n*). In (2) Ramanujan was using hyperbolic functions, just as Rademacher did for *p*(*n*).

While (2) is not correct as stated by Ramanujan in his first letter to Hardy, it does yield a genuine approximation. “Ramanujan’s false statement,” Hardy remarked, “was one of the most fruitful he ever made, since it ended by leading us to all our joint work on partitions.”^{24}

In a thought-provoking lecture on the centenary of Ramanujan’s birth, Atle Selberg noted that the Rademacher convergent series involving hyperbolic functions was somewhat more natural than the asymptotic series of Hardy–Ramanujan.^{25} Selberg pointed out that although Ramanujan felt such a convergent series ought to exist, he agreed to stop at the asymptotic series for *p*(*n*) out of respect for his mentor. Selberg emphasized that if Hardy had fully trusted Ramanujan’s insight, they would have arrived at Rademacher’s convergent series for *p*(*n*).^{26}

Hardy credits MacMahon’s calculations as being crucial for the final version of his result (7) with Ramanujan.^{27} A charming scene in the film depicts Hardy bringing Ramanujan to MacMahon’s office so that he can meet the Indian genius. During their meeting, MacMahon challenges Ramanujan to compute the square root of a particular number. Ramanujan responds instantly. MacMahon then invites Ramanujan to pose a computational problem for him to solve. MacMahon provides the answer, just as quickly. From time to time, MacMahon and Ramanujan engaged in friendly computational contests.^{28} Of the pair, Hardy felt that MacMahon was the faster and more accurate.^{29}

The remarkable formula for partitions formed the basis for Hardy’s campaign to get Ramanujan elected as a fellow of the Royal Society and a fellow of Trinity College. Despite Ramanujan having found several spectacular results, Hardy needed something totally unexpected and staggering, such as the asymptotic series for *p*(*n*), to obtain these honors. The scenes depicting the tremendous efforts made by Hardy attempting to convince the academic aristocracy that Ramanujan should be honored are deeply moving. Hardy’s first attempt to get Ramanujan elected a fellow of Trinity was unsuccessful. He was, however, successful in getting Ramanujan elected a fellow of the Royal Society.

Thereafter the fellowship of Trinity was awarded with no further difficulties.

## Life in England

Shortly after Ramanujan arrives in Cambridge, he is shown walking across a lawn at Trinity College only to be stopped by a guard. Ramanujan is ordered to stay off the grass and follow the path running around the edge of the lawn. Unbeknownst to Ramanujan, only a privileged few, such as fellows of the college, are entitled to traverse the sacrosanct and immaculate lawns. After Ramanujan is elected a fellow of Trinity, Hardy informs him that he may now walk across the lawn with confidence.

While Ramanujan did not pursue the convergent series for *p*(*n*) in deference to Hardy, he was, generally speaking, actually quite confident about the correctness of his results and therefore was not as timid in the presence of Hardy as he is depicted. Similarly, while Hardy insisted on proofs and conveyed their importance to Ramanujan, he did not chastise Ramanujan to the extent shown in the film. In one scene Hardy tells Ramanujan that he will not see him again unless he provides proofs for certain results. Hardy was never this harsh with Ramanujan. In the preface to Ramanujan’s *Collected Papers*, he wrote:

It was impossible to ask such a man to submit to systematic instruction … I was afraid too that, if I insisted unduly on matters which Ramanujan found irksome, I might destroy his confidence or break the spell of his inspiration … He was never a mathematician of the modern school, and it was hardly desirable he should become one; but he knew when he had proved a theorem and when he had not.^{30}

In reality, Hardy helped Ramanujan write up his results and filled in missing steps so that his work could be published. One scene depicts Ramanujan jumping with joy when Hardy hands him a reprint of his paper on highly composite numbers.^{31} The film dramatizes the differences between the pair regarding proofs to create interest for the audience, which is understandable.

By contrast, the depictions of discrimination and racial prejudice against Ramanujan in England are overdone. At Hardy’s suggestion, Ramanujan attended some lectures to build a level of basic knowledge about important areas in mathematics. Ramanujan is shown taking part in a class in which the lecturer is discussing formulas involving special functions. The lecturer notices that Ramanujan is not taking notes and asks if he is following the discussion. Ramanujan replies that he is and that he knows the answer. Somewhat irritated by this response, the lecturer asks Ramanujan to come to the board and demonstrate his solution. Ramanujan does so with ease and astonishing speed. The infuriated lecturer throws Ramanujan out of the class and asks him not to attend any more of his lectures. While this scene conveys Ramanujan’s brilliance and the ways in which he often startled professors with his talent, it misrepresents their admiration for his genius. P. C. Mahalanobis, a college mate of Ramanujan in Cambridge recalled:

I used to do my tutorial work with Mr. Arthur Berry, Tutor in Mathematics of King's College. One day I was waiting in his room for my tutorial when he came in after having taken a class on elliptic integrals. He asked me: “Have you met your wonderful countryman, Ramanujan?” I told him I had heard that he had arrived but that I had not met him so far. Mr. Berry said: “He came to my elliptic integrals class this morning” … I asked “What happened? Did he follow your lecture?” Mr. Berry said, “I was working out some formulae on the black board. I was looking at Ramanujan from time to time to see if he was following what I was doing. At one stage, Ramanujan’s face was beaming and he appeared to be excited. I asked him whether he was following the lecture and Ramanujan nodded his head. I then enquired whether he would like to say anything. He got up from this seat, went to the black board and wrote some of the results which I had not yet proved.” I remember that Mr. Berry was greatly impressed.^{32}

In another scene, Ramanujan goes to the post office to see if there are any letters from his wife. When a dejected Ramanujan leaves the post office empty-handed, he is approached by a group of young men in military uniform. Ramanujan is mocked, kicked, pushed to the ground, and his face left bloodied. This event never occurred. There was not a single instance when Ramanujan was physically abused because of racial prejudice.

## 1729

No account of Ramanujan’s life is complete without mention of the famous taxi cab episode. Hardy recalled visiting Ramanujan in a nursing home in Putney: “I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen.”^{33} To Hardy’s surprise, Ramanujan replied, “No, it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways.”^{34}

- $1729=1728+1={12}^{3}+{1}^{3}=1000+729={10}^{3}+{9}^{3}.$

Hardy was stunned.^{35} But, as it turned out, Ramanujan had worked out the parametrization of all the integer solutions to the equation

- ${x}^{3}+{y}^{3}={z}^{3}+{w}^{3}$

in India. He was aware that 1729 was the smallest solution for *x*, *y*, *z*, and *w*. The elegance of Ramanujan’s parameterization is undeniable, but Euler had obtained the parametrization of all the solutions to this equation long beforehand, and thus it is known as Euler’s equation.^{36}

Ramanujan and Hardy are depicted in several scenes discussing partitions while Ramanujan lies ill in hospital. The number 1729 is never mentioned. Instead, the 1729 episode appears near the end of the film, when Ramanujan takes a taxi to the port to sail back to India. The lovely property (10) of 1729, mentioned by Ramanujan to Hardy, eases the emotion of the scene when they part, which is touching and beautifully done. The 1729 story is very well-known in India and a local audience would spot this discrepancy immediately.

## Legacy

Nearly a century after his death, Ramanujan’s influence continues to grow. Books have been published about Ramanujan’s notebooks, explaining their significance for contemporary research.^{37} *The Ramanujan Journal*, devoted to the areas of mathematics influenced by Ramanujan’s work, was launched in 1997. The journal publishes three volumes of three issues each year, a threefold increase since its debut.^{38} Prizes for outstanding research in areas influenced by Ramanujan are awarded to young mathematicians.^{39} Bhargava, an associate producer on the film, was the inaugural winner of the Shanmugha Arts, Science, Technology & Research Academy (SASTRA) Ramanujan Prize in 2005. Popular books, plays, documentaries, films, and even an opera about Ramanujan’s life and work have been produced.^{40}

Although there are some discrepancies in the film’s plot, *The Man Who Knew Infinity* is unquestionably one of the finest films ever made about mathematics. In the preface to the debut issue of *The Ramanujan Journal*, I wrote, “The very mention of Ramanujan’s name reminds us of the thrill of mathematical discovery.” *The Man Who Knew Infinity* not only conveys the thrill of such discoveries, but also movingly depicts the remarkable life and exceptional talents of Srinivasa Ramanujan.

- Robert Kanigel,
*The Man Who Knew Infinity: A Life of the Genius Ramanujan*(New York: Charles Scribners, 1991). Kanigel’s biography was far from the first published work on Ramanujan’s life. Biographies were published previously by Ragami, written in Tamil, and by S. R. Ranganathan, to mention just two examples. See Ragami [pseud], “Ramanujan, the Mathematical Genius” (in Tamil), Series,*Dinamani Kadir*(Madras weekly newspaper), 1,985–86; S. R. Ranganathan,*Ramanujan**– the man and the mathematician*, (Bombay: Asia Publishing House, 1967).

For my review of Kanigel’s book, see: Krishnaswami Alladi, “Discovery and Rediscovery of Mathematical Genius – A Review of The Man Who Knew Infinity: The Life of the Genius Ramanujan,”,

*The American Scientist*80 (1992): 388–89. My only criticism of Kanigel’s book is that he spends too much time discussing Hardy’s homosexual inclinations without any concrete evidence. Hardy was a confirmed bachelor, as were many Cambridge dons totally consumed by their work, and the comment Hardy made, namely “my association with Ramanujan was the one romantic incident in my life,” should be taken in the spirit of mathematical romance, and not in sexual terms. ↩ - Digitized versions of these were later published. See Srinivasa Ramanujan,
*Notebooks*(Bombay: Tata Institute of Fundamental Research, 1957). ↩ - For sheer manipulative ability, Hardy believed Ramanjuan’s talent was rivaled only by that of Leonhard Euler and Carl Jacobi. ↩
- G. H. Hardy and Srinivasa Ramanujan, “Asymptotic Formulae in Combinatory Analysis,”
*Proceedings of the London Mathematical Society*2 (1918): 75–115. ↩ - Hardy had written to Francis Dewsbury, Registrar of the University of Madras, beforehand, remarking that Ramanujan “will return with a scientific standing and reputation such as no Indian has enjoyed before.” Bruce Berndt and Robert Rankin,
*Ramanujan, Letters and Commentary*, American Mathematical Society and London Mathematical Society History of Mathematics Series, Vol. 9 (Providence, RI: American Mathematical Society, 1995), 200. ↩ - See, for example: George Andrews, “An Introduction to Ramanujan’s Lost Notebook,”
*American Mathematical Monthly*86 (1979): 89–108. ↩ - Srinivasa Ramanujan,
*The Lost Notebook and Other Unpublished Papers*(New Delhi: Narosa, 1987). ↩ - Launched in 1997 by Kluwer and now published by Springer,
*The Ramanujan Journal*is devoted to the areas of mathematics influenced by Ramanujan. For an article on the conception and scope of the journal see Krishnaswami Alladi,*Ramanujan’s Place in the World of Mathematics*(New York: Springer, 2012), 147–51. ↩ - In
*A Mathematician’s Apology*, Hardy claims:[M]athematics, more than any other art or science, is a young man’s game. … Galois died at twenty-one, Abel at twenty-seven, Ramanujan at thirty-three, Riemann at forty. There have been men who have done great work later; … [but] I do not know of a single instance of a major mathematical advance initiated by a man past fifty. … A mathematician may still be competent enough at sixty, but it is useless to expect him to have original ideas.

G. H. Hardy,*A Mathematician’s Apology*(Cambridge: Cambridge University Press, 1992), 70–72. ↩ - Bruce Berndt,
*Ramanujan’s Notebooks, Part 1*(New York: Springer, 1985), 14. ↩ - In traditional Hindu society, the woman moves into the man’s home following a marriage. This often means living with the in-laws. Thus Ramanujan is shown accompanied by his mother when he and Jananki arrive on a bullock cart at their new residence. ↩
- Kanigel notes that when the pair were reunited in Madras, Ramanujan found out from Janaki that his mother had intercepted their letters. Robert Kanigel,
*The Man Who Knew Infinity: A Life of the Genius Ramanujan*(New York: Charles Scribners, 1991), 318. ↩ - In his biography on Ramanujan, S. R. Ranganathan says Ramanujan made this statement to a friend, but this friend was not Hardy. S. R. Ranganathan,
*Ramanujan**– The Man and the Mathematician*(Bombay: Asia Publishing House, 1967). ↩ - G. H. Hardy, P. V. Seshu Aiyar, and B. M. Wilson, ed.,
*Collected Papers of Srinivasa Ramanujan*, (Cambridge: Cambridge University Press, 1927), xxiii. ↩ - G. H. Hardy, P. V. Seshu Aiyar, and B. M. Wilson, ed.,
*Collected Papers of Srinivasa Ramanujan*, (Cambridge: Cambridge University Press, 1927), xxiii. ↩ - Hardy pointed out that Ramanujan often showed him a dozen new theorems each day and that he was preoccupied with understanding them. See G. H. Hardy,
*Ramanujan**– Twelve Lectures on Subjects Suggested by His Life and Work*(Cambridge: Cambridge University Press, 1940), 11. ↩ - In 2005, I saw a play about Ramanujan entitled
*Partition*, which was also centered around the amazing Hardy–Ramanujan formula for the partition function. My review appeared in*The Hindu*, India's national newspaper in May 2005, and is reprinted in Krishnaswami Alladi,*Ramanujan’s Place in the World of Mathematics*(New York: Springer, 2012), 143–144. ↩ - François Arago remarked that “Euler calculated without apparent effort as men breathe or as eagles sustain themselves in the wind.” See E. T. Bell,
*Men of Mathematics*(New York: Simon and Schuster, 1937), 139. ↩ - See J. E. Littlewood, “Review of Ramanujan’s Collected Papers,”
*Math. Gazette*XIV (1929);*Littlewood's Miscellany*- Edited by Bela Bollobas, (Cambridge: Cambridge University Press, 1986), 94–99. ↩ - MacMahon, a former Major in the British Army, had been stationed in Madras at one time. He was a fellow of St. John’s College, Cambridge, but never a Cambridge faculty member. ↩
- J. E. Littlewood,
*A Mathematician’s Miscellany*(London: Methuen & Co. Ltd., 1953), 89–90. ↩ - Derrick Henry Lehmer, “On the Hardy–Ramanujan Series for the Partition Function,”
*Journal of the London Mathematical Society*12 (1937): 171–76. ↩ - Hans Rademacher, “On the Partition Function
*p*(*n*),”*Proceedings of the London Mathematical Society*2, no. 43 (1937): 241–54. ↩ - G. H. Hardy,
*Ramanujan**– Twelve Lectures on Subjects Suggested by His Life and Work*(Cambridge: Cambridge University Press, 1940), 9. ↩ - Selberg remarked that he had independently obtained such a convergent series, but did not publish it because he realized that it had already been done by Rademacher. ↩
- Selberg’s comments can be found in Atle Selberg, “Reflections around the Ramanujan Centenary,” in
*Collected Papers*, Vol. 1 (Berlin: Springer, 1989), 695–701. This is a write-up of another talk he gave a few weeks later in Bombay. ↩ - G. H. Hardy,
*Ramanujan**– Twelve Lectures on Subjects Suggested by His Life and Work*(Cambridge: Cambridge University Press, 1940), 119. ↩ - See Srinivasa Ramanujan,
*Collected Papers*(Cambridge: Cambridge University Press, 1927), xxxv; G. H. Hardy,*Ramanujan**– Twelve Lectures on Subjects Suggested by His Life and Work*(Cambridge: Cambridge University Press, 1940), 119. ↩ - Srinivasa Ramanujan,
*Collected Papers*(Cambridge U.K.: Cambridge Univ. Press, 1927), xxxv. ↩ - Srinivasa Ramanujan,
*Collected Papers*(Cambridge U.K.: Cambridge Univ. Press, 1927), xxx–xxxi. ↩ - Srinivasa Ramanujan, “Highly Composite Numbers,”
*Proceedings of the London Mathematical Society,*2, no. 14 (1915): 347–400. ↩ - This quote of Mahalanobis is taken from P. K. Srinivasan’s wonderful book
*Ramanujan**– Letters and Reminiscences*, Ramanujan Memorial Number, Volume I (Madras: Muthialpet High School, 1968), 145–48. Kanigel describes this story without referring to Mahalanobis (see Robert Kanigel,*The Man Who Knew Infinity: A Life of the Genius Ramanujan*New York: Charles Scribners, 1991), 201–202), but he does acknowledge Srinivasan’s book in describing this incident. ↩ - Avery Carr, “Ramanujan’s Taxi Cab Number,”
*AMS Blogs*, August 15, 2013. ↩ - Avery Carr, “Ramanujan’s Taxi Cab Number,”
*AMS Blogs*, August 15, 2013. ↩ - It was such incidents that prompted Littlewood to remark that every number was a personal friend of Ramanujan. It was a sheer coincidence that Hardy arrived in the taxi numbered 1729, although I prefer to think of it as the handiwork of the goddess Namagiri. ↩
- See Bruce Berndt and S. Bhargava, “Ramanujan—For Lowbrows,”
*The American Mathematical Monthly*100 (1993): 644–56. ↩ - George Andrews and Bruce Berndt,
*Ramanujan’s Lost Notebook, Parts I**–V*(New York: Springer, 2005–2013); Bruce Berndt,*Ramanujan's Notebooks, Parts I**–V*(New York: Springer, 1985–1998). ↩ - Krishnaswami Alladi,
*Ramanujan’s Place in the World of Mathematics*(New York: Springer, 2012), 147–51. ↩ - See Krishnaswami Alladi,
*Ramanujan’s Place in the World of Mathematics*(New York: Springer, 2012), 161–66. ↩ - See Krishnaswami Alladi, “Srinivasa Ramanujan Going Strong at 125, Part I,”
*Notices of the American Mathematical Society*59 (2012): 1,522–37. ↩