Welcome back to the deep dive. So our mission
today, it's a big one. We're going to take this
the stack of sources that detail an almost impossible
life. It really is. We're talking about a man
who. Well, who arrived at the front lines of
mathematics having never really studied the rulebook.
Exactly. And our job is to extract the insights
that explain not just what he did, but how. We
are diving into the extraordinary life and, frankly,
the shocking mathematical legacy of Srinivasa
Ramanujan. Yeah, we're looking at a true mathematical
anomaly. Srinivasa Ramanujan, born in India,
1887. He was a figure in the early 20th century
whose whole career, his entire life, It just
defies every standard academic story we know.
It does. The journey he took from, you know,
extreme poverty in Madras all the way to the
very rigid formal halls of Cambridge. It's just
this powerful testament to raw, undiluted genius.
OK, let's unpack that right away. Let's define
the central paradox here because the sources
are so clear on this. Remigian had almost no
formal training in pure math, no access to modern
European textbooks, no degree, no rigorous instruction
in things like complex analysis. And yet. He
made these huge contributions to some of the
most difficult fields out there. Number theory,
infinite series, continued fractions. Right.
And he was solving problems that establish academics,
you know, the best in the world. They considered
them completely unsolvable. It's just it's an
unprecedented achievement. And when you look
at the sheer output, the numbers themselves are
almost unbelievable. Let's get into that. Ramanujan
independently compiled nearly 3 ,900 results.
We're talking mostly. identities and equations,
all recorded in his famous notebooks. And the
really astonishing part. The astonishing part
is that the vast majority of them over the decades
have been proven correct. And so many of them
were completely new, opening up research areas
that are still defining modern math today. Can
we just stop there for a second? 3 ,900. It's
mind -boggling. I mean, you have to put that
in perspective for someone listening. A typical
PhD thesis in pure mathematics, what does that
have? Five, maybe six major provable results.
If you're lucky, yeah. And Ramanujan generated
3 ,900 of them in isolation with no formal proofs
attached. Exactly. And that context is precisely
why his early attempts to get noticed failed.
As Hans Heisling pointed out, he couldn't get
the attention of leading mathematicians at first
because his work was, and I'm quoting here, too
novel, too unfamiliar, and additionally presented
in unusual ways. They didn't know what they were
looking at. Not at all. They'd look at his equations
and couldn't even figure out what branch of math
they belonged to, let alone start to verify them.
His genius was so far outside the established
framework that to them it probably just looked
like nonsense. It's this classic clash, isn't
it? Intuition versus rigor, East versus West.
So our mission today is to explore two key ideas.
First, how did the solitary genius find his voice
through that extraordinary partnership with G
.H. Hardy? A crucial relationship. And second,
how do his insights continue to inspire such
profound research? I mean, we're not talking
about dusty historical facts here. We're talking
about things, mere comments in his writings.
that researchers were still uncovering and proving
his profound number theory as late as 2012. His
work is a perpetual mathematical goldmine. It
really is. So let's go back to the beginning
of this incredible story, the isolated prodigy.
Srinivasa Ramanujan was born December 22, 1887,
in Erode, India. He was from a traditional Tamil
Brahmin Iyengar family in what was then the Madras
presidency of British -ruled India. And his roots
were very modest. His father was a clerk in a
small shop. His mother was a housewife who also
sang at a local temple. And she'd often take
him with her, right? Which gave him this really
early, strong connection to spirituality that
we'll see come up again and again. Absolutely.
And his childhood, I mean, it immediately signaled
that this was no ordinary mind. The sources say
that by age 11, he had completely exhausted the
mathematical knowledge of two college students
who were boarding at his family's home. Just
to think about that. These are guys studying
for higher exams, and this 11 -year -old kid
is basically running them out of things to teach
him. Until they had nothing left to give. It's
an incredible developmental leap. Most kids that
age are, what, struggling with fractions. And
he's already pushing into university -level concepts.
And it didn't stop. By age 13, he gets his hands
on a book by S .L. Looney on advanced trigonometry.
And this wasn't some simple textbook. Louie's
book is a real challenge, even for university
students at the time. But Ramanujan doesn't just
master it. He starts discovering his own sophisticated
theorems, completely independent of any curriculum.
He was just a natural problem solver. He had
this high aptitude for logistics, too, even as
a teenager. Oh, the school scheduling problem.
Exactly. The story goes that when he was 14,
he helped his school solve this really complex
scheduling challenge. How to assign 1 ,200 students
to about 35 teachers. That's not just arithmetic.
That's a complex optimization problem. It shows
he had this innate, practical, applied intelligence
long before those fields were even properly formalized.
It does. But the really crucial text, the one
that shaped his whole unique methodology, was
a book by G .S. Carr. A Synopsis of Elementary
Results in Pure and Applied Mathematics. That's
the one. He got a copy in 1903 when he was 16.
And it just... It fundamentally changed how he
approached mathematics. So what was this book?
It wasn't a normal textbook, was it? No, not
at all. It's arguably the single most important
document in his entire development. It was this
massive two -volume collection of about 5 ,000
theorems and results. Wow. But... And this is
the key part. They were stated largely without
any detailed proofs or derivations. It's just
the answers. It's just the final answers. Carr
designed it to be a memory aid for students cramming
for exams. It gave you the destination without
showing you the route. So imagine you're a 16
-year -old prodigy, and you get a book with 5
,000 mathematical final answers, but no explanation
of why they work. And because he studied it so
intensely, line by line, he internalized this
method where intuition and the final result were
everything. The formal proof was, well, secondary.
Or maybe just something you didn't need to write
down. And that's the period of isolated discovery,
right, where he has to invent the steps and the
proofs himself. This is why he later struggled
so much with the Western demand for rigor. Absolutely.
And yet, even completely on his own, his mind
was just exploding with discoveries. He independently
developed algorithms for Bernoulli numbers. He
calculated the Euler -Mascheroni constant to
15 decimal places. That's just world -class analysis.
Performed by a young man with no formal guidance
whatsoever. Which brings us to the... The tragedy
of his academic career, because despite this
obvious genius, his formal performance was a
complete disaster. A total disaster. Yeah. He
got a scholarship to government arts college,
but his obsession with math was so all consuming
that he just he neglected everything else. He
failed every other required subject. Especially
English and physiology, the sources say. He literally
couldn't force his brain to care about anything
that wasn't mathematics. Yeah. So the scholarship
was revoked instantly. Yeah. And that was followed
by this long, dark. He failed his fellow of arts
exam multiple times, left college without a degree.
And this isn't just an academic setback. In that
rigid colonial system, not having credentials
meant extreme poverty. Real poverty. The sources
describe him as often being on the brink of starvation.
This wasn't a struggling student with a support
system. This was genuine economic desperation.
And it was made worse by his refusal to work
in any field that wasn't math. it took until
about 1910 he's in his early 20s newly married
to his wife janaki That he begins this systematic
search, not just for a job, but for mathematicians
who could actually understand what he was doing.
And that search leads him to V. Ramaswamy Ayer,
the founder of the Indian Mathematical Society.
And Ayer was just instantly struck by the power
and originality in Ramanujan's notebooks. And
Ayer made a crucial decision. A really humane
and critical decision. He later said he had,
quote, no mind to smother his genius by an appointment
in the lowest rungs of the revenue department.
In other words, he refused. refused to let this
talent get buried in bureaucratic paperwork.
Right. He became his crucial early advocate.
He helped him get letters of introduction to
European scholars and, importantly, helped him
secure a small research scholarship, about 75
rupees a month, at the University of Madras.
That small bit of recognition and Ayer's refusal
to just give him a menial job, it really saved
him from being lost to clerical obscurity. It
saved his genius. But even with that small grant,
the financial pressure was still immense. The
sources detail it clearly, especially after he
married Janaki in 1909 and then had to recover
from surgery for hydrosil testis. He needed a
stable income. So he eventually secures a post
as an accounting clerk at the Madras Port Trust.
He's earning a very meager 30 rupees a month.
The irony is just, it's incredible. The man who
would redefine number theory is working as an
accountant. But what's so telling is that he
would finish his clerical work easily. very quickly,
which left him with all this spare time for his
actual research. And his colleagues and his boss,
they supported him. They recognized his unique
focus. That support was absolutely vital. It
let him continue his true work, which brings
us to the moment that changed absolutely everything.
The letter to Cambridge. Before G .H. Hardy,
though. Ramanujan's work went through what you
could call a rejection filter. Right. He'd sent
it out before. Yes. His manuscript pages were
sent to several established British mathematicians,
and they just failed to recognize what they were
seeing. The sources cite a couple of examples.
M .J .M. Hill, a prominent mathematician, sent
the papers back with this incredibly condescending
comment that they were riddled with holes and
that Ramanujan lacked the necessary educational
background. Wow. And even worse. H .F. Baker
and E .W. Hobson, both very respected names.
They just returned the work. No comment at all.
It was so unconventional. They didn't even think
it was worth a rejection letter. So three strikes.
It wasn't until January 1913 that he wrote to
G .H. Hardy at Cambridge. And Hardy was, well,
arguably the most brilliant and rigorous mathematician
in Britain at the time. And he receives these
nine pages of complex handwritten mathematics
from a completely unknown clerk in India. Hardy's
initial suspicion must have been off the charts.
Oh, absolutely. It was warranted. He got unsolicited
manuscripts all the time, usually from cranks
claiming they, you know, trisected the angle
or something. His first reaction was that it
was probably a fraud or the work of a lunatic.
But that suspicion quickly changed into something
else. He and his colleague, J .E. Littlewood,
started looking at the work more closely. And
Hardy later confessed that, yes, some of the
theorems were recognizably advanced, which confirmed
Ramanujan had genuine talent. But the others,
the truly novel results, he said they defeated
me completely. I had never seen anything in the
least like them before. It was that sheer novelty,
the fact that they had no traceable lineage in
Western mathematics, that forced Hardy to reconsider.
And this brings us to that famous quote. the
one that basically validated Ramanujan's genius
for all time. Hardy concluded that the theorems
must be true because if they were not true, no
one would have the imagination to invent them.
That quote is just. It's pivotal. It's the ultimate
recognition of pure intuition. The idea is so
beautiful and strange that it can't be a mistake.
It has to be a revelation. It took a genius like
Hardy to recognize a genius that looked nothing
like himself. And let's get into the specifics
of what convinced him, because it really shows
the technical gulf Ramanujan had bridged on his
own. The sources mention a complex infinite integral
formula. But it was the infinite series for pi
that really blew Hardy away, wasn't it? It was.
For the listener, an infinite series is just
a way to express a number by adding up an infinite
sequence of smaller and smaller terms. Reminiscence
series for, specifically, two -piler was derived
from something called a hypergeometric series.
Yeah. It was completely new to Hardy. And he
said it was much more intriguing than even the
work of giants like Gauss and Legendre. Yes.
And it's a masterpiece of technical complexity.
But what makes it so groundbreaking is its convergence
rate, how fast it gets you to the answer. Right.
Let's slow down on that. What does that actually
mean, extraordinarily rapid convergence? Well,
if you use the older standard methods, you need
thousands of terms to get, say, a dozen decimal
places of pi. But with Ramanujan's formula, if
you just take the first two terms of the sum.
Just two. Just the first two. You get a value
that's already correct to 14 decimal places.
Wow. That phenomenal efficiency is the basis
of some of the fastest algorithms we use today
to calculate pi. It shows he wasn't just generating
true ideas. He was generating ideas that were
computationally superior. So after this, Hardy
immediately starts trying to get Ramanujan to
Cambridge. But then he hits this huge cultural
and religious barrier. Right. Initially, Ramanujan
refused to go. Why? He cited his Orthodox Brahmin
upbringing. There was intense parental and societal
opposition to him traveling overseas to a foreign
land. It was a massive religiously fraught decision.
You could lose caste. You'd be forced to break
strict dietary laws. So what was the turning
point? The turning point is one of the most remarkable
anecdotes in the whole history of science. It
took E .H. Neville, a colleague of Hardy's who
was lecturing in Madras, to help convince him.
but the final decision came from his family the
dream the dream The sources say Ramanujan's mother
had this vivid, decisive dream where the family
goddess Namagiri Thayer commanded her to let
her son go to England to fulfill his life's purpose.
I love that. A divine mandate that just overrides
these deeply ingrained social rules. So this
wasn't just a decision. It was a spiritual turning
point for the whole family. It was. So in March
1914, Ramanujan boards the SS Navassa, leaving
his wife behind in India. And he sets sail for
a life he couldn't possibly have imagined. A
life that would define his genius, but also tragically
break his body. So Ramanujan arrives in Cambridge
and this incredible collaboration begins. And
this is where the tension, but also the real
genius, really comes to the fore. It's a study
in contrasts. You have G .H. Hardy, who is the
personification of the Western academic establishment.
He's an atheist, an apostle of mathematical rigor,
classically educated. For him, it was all about
proof, proof, and more proof. And then you have
Ramanujan. Deeply religious, relying almost entirely
on intuition and completely lacking any formal
grounding in the systematic methods of the day.
Hardy himself was just astonished to find out
that Ramanujan had what the vaguest idea of what
a function of a complex variable was. And that
he'd never even heard of fundamental cornerstones
of modern analysis like Cauchy's theorem. Okay,
but how is that even possible? How can a mathematician
be world -class in number theory, a field that
leans so heavily on complex analysis, but have
never heard of one of its most basic theorems?
It just tells you that his mind worked in a fundamentally
different way. He was discovering these complex
analytical results through a process that Hardy
described as a mingled argument, intuition, and
induction. A process that, frankly, Ramanujan
himself couldn't really explain coherently. So
Hardy is faced with this unbelievable intellectual
challenge. He has to mentor this guy, teach him
the systematic rules of formal proof, fill these
enormous gaps in his education. But he has to
do it without crushing the very source of that
extraordinary intuition. It's an incredibly delicate
balancing act. He had to treat him less like
a student who needed to be fixed and more like
an oracle who sometimes needed translation. Exactly.
It required tremendous patience and respect from
Hardy. He was trying to bring Ramanujan up to
speed on like 150 years of European mathematics,
while Ramanujan was simultaneously leaping 50
years into the future with entirely new theorems.
And the key struggle was just translating that
instantaneous, intuitive mathematical truth into
the language of rigorous, verifiable proof that
the academic world demanded. And all this is
happening during the worst possible time. World
War I. From 1914 to 1918, which just compounded
his physical struggle, the silent tragedy of
his time in England. Ramanujan was a strict Brahmin
vegetarian. And this is a point that often gets
overlooked, but it's absolutely critical. Maintaining
that diet, it wasn't just about not eating meat.
No, it was about abstaining from eggs, fish,
often certain root vegetables, and sometimes
even requiring self -preparation to ensure ritual
purity. This was nearly impossible in Edwardian,
and especially wartime England. And wartime rationing
made it exponentially harder. Access to specialized
Indian ingredients was non -existent. He was
forced to cook all his own food, which took up
hours of his time. He became deeply isolated,
and that was compounded by the cold, the climate,
the relentless intellectual pressure. His health
just... deteriorates he's eventually diagnosed
with tuberculosis coupled with severe vitamin
deficiency which was a direct result of his diet
and his isolation and yet despite this overwhelming
physical and mental distress the academic establishment
was finally forced to acknowledge his genius
We see that through this rapid succession of
honors. In March 1916, he gets a Bachelor of
Arts by research degree for his massive paper
on highly composite numbers. A paper Hardy later
said displayed Ramanujan's extraordinary mastery
over the algebra of inequalities. And then come
the historic elections in 1918. In May, he's
elected a fellow of the Royal Society, the FRS,
a monumental achievement. He was only the second
Indian ever admitted. And then in October of
that same year. He becomes the first Indian to
be elected a fellow of Trinity College, Cambridge.
That swift series of high honors all within about
seven months was absolutely crucial. It cemented
his legacy. And maybe more importantly, it completely
validated Hardy's initial, highly controversial
belief in this isolated clerk from Madras. It
proved that even if his methods were unconventional,
the depth of his work was simply undeniable.
So let's turn to the mathematics itself, the
architecture of his genius and that distinction
between his insight and the formal proof. That
remains the core theme. He was just generating
this abundance of formulae that were completely
novel. I mean, he was opening up entire new fields
of research before anyone else even realized
those fields existed. Let's go back to some of
the specific examples to really understand the
depth here. We talked about his famous infinite
series for Tarkas. Which is a highly technical
formula based on something called the negative
fundamental discriminant dollar dollars equals
known as just 232. It's a complex concept from
number theory. And we noted its speed, its convergence.
But let's just pause on what that means for us
today for modern computation. Because that series
is so efficient, it's basically the ancestor
of algorithms that are being used right now by
researchers and engineers who need extremely
precise values for pi. His intuitive algebraic
structure provided the most practical path forward,
even a century later. Okay, now let's switch
gears completely. Let's go from that cold computational
rigor to pure instantaneous insight. Which brings
us to the famous taxi cab number anecdote. 1729.
1729. This story, it takes place when Hardy is
visiting Ramanujan in the hospital at Putney,
and it just perfectly captures his unique relationship
with numbers. It does. Hardy, you know, trying
to make small talk, mentions he arrived in a
cab with the number 1729, and he remarks that
it seemed rather a dull one. It's such a beautifully
reserved British understatement. But Ramanujan,
despite being severely ill, he instantly corrects
him. He says, no, it is a very interesting number.
It is the smallest number expressible as the
sum of two cubes in two different ways. That's
just that immediate recall and computation. The
ability to see the deep properties of a four
digit number instantly is astounding. And the
solution is $1729 equals 13 plus $129 equals
93 plus 103 to 3. It shows that numbers weren't
just abstract concepts to him. They were personal.
As Hardy later recalled his collaborator Littlewood
saying, every positive integer was one of Ramanujan's
personal friends. He wasn't calculating. He was
just recognizing a friend's hidden talent. And
this instantaneous insight wasn't limited to
small numbers. It extended to solving entire
classes of difficult problems, like the one posed
by his Indian colleague, P .C. Mahalanobis, the
houses problem. Right. The problem is a clever
one. Imagine a street with houses numbered 1,
2, 3. up to some number nullar. And there's a
specific house, let's call it $6, where the sum
of the house numbers to the left of it equals
the sum of the house numbers to the right of
it. Mahalanobis had worked out a specific case
and asked Ramanujan for the solution. Most mathematicians
would spend hours setting up the equations. But
Ramanujan's response was just... astounding he
didn't just give the specific solution he immediately
gave the answer as a continued fraction which
solved the entire class of problems the infinite
set of possibilities mahalanobis asked him how
he did it and ramojin just said it is simple
the minute i heard the problem i knew that the
answer was a continued fraction which continued
fraction i asked myself then the answer came
to my mind He saw the structure, not the calculation.
That's the essence of his genius. And this focus
on integers led to what is probably his greatest
collaboration with Hardy in 1918, the study of
the partition function, P -S -A -9. Okay, so
the partition function, P -A -N -A, it asks a
simple question. In how many different ways can
you break up an integer, noller, into a sum of
positive integers? So for the number four, for
example. The number four can be partitioned in
five ways. You can have 4, 3 plus 1, 2 plus 2,
plus 1 plus 1, or 1 plus 1 plus 1 plus 1. It
seems simple, but as the number gets bigger...
The numbers just explode. The number of partitions
for 200 is a number that has 14 digits. It's
computationally impossible to list them all.
So how did they solve it? Their work together
yielded a non -convergent asymptotic series,
which is a highly complex formula that lets you
calculate to act exactly. And crucially... This
collaboration pioneered what became known as
the powerful circle method. Okay, so what is
the circle method in simple terms? It sounds
very abstract. It is abstract, but we can make
it accessible. Think of a complex number not
as a point on a line, but as a point on a two
-dimensional plane. Right. The circle method
is a technique that uses complex integration,
basically calculating an area, but on this complex
plane, by integrating over a circle. By carefully
dividing that circle into what they called major
and minor arcs, they could isolate the contribution
of specific mathematical properties. It was a
revolutionary way of thinking about how to count
extremely difficult things. Using advanced geometry
to solve what seems like a simple counting problem.
Exactly. And even in the last year of his life,
when he was frail and suffering, he made one
final, crucial discovery that remained a mystery
for decades. Mach theta functions. Yes, this
is maybe the ultimate example of his insights
being ahead of their time. He generated these
functions, and they were related to modular forms,
these mathematical structures with incredible
symmetry, but they didn't quite fit the definitions
of the time. For years, mathematicians didn't
know what to do with them. They were like puzzle
pieces from a future game. So what's their significance
today? Where did they fit in? Well, it wasn't
until modern mathematics caught up that we fully
understood them. They are now known as the holomorphic
parts of harmonic weak mass forms. Which means?
It means they are specific parts of a much larger,
highly symmetric structure that wasn't even fully
defined until the 21st century. It just proves
that even on his deathbed, Ramanujan was generating
insights that were literally 70 or 80 years ahead
of the field's ability to classify or even understand
them. When we talk about Ramanujan's immense
output, those almost 3 ,900 results, we have
to go back to the physical record of his work,
the famous notebooks. And they were filled almost
entirely with final results, virtually no derivations
or proofs recorded anywhere. And this is a huge
point of fascination for mathematicians. This
lack of proof led to the early and incorrect
assumption that he just received the results
through divine insight and couldn't actually
prove them. But that's not the case. No. Brucie
Berndt, the mathematician who has spent decades
meticulously studying these notebooks, he asserts
that Ramanujan most certainly was able to prove
the vast majority of his results. So why the
radical omission? Why only record the destination
and not the route? Well, the practical reasons
are really compelling, and they speak volumes
about his impoverished circumstances. First...
Caper was extremely expensive in India at the
time. He couldn't afford to waste it. He couldn't
afford it. So he adopted a common practice for
math students in the Madras presidency. He worked
out the full proofs and all the complex algebra
on a reusable slate. Which you can just... wipe
clean for the next problem. Right. So imagine
the physical labor involved. He's not just working
out 3 ,900 results. He's calculating, writing,
erasing, and refining these monumental themes
on a slate, and then committing only the final
distilled identity to the expensive paper notebook.
That contrast between the physical struggle for
resources and the intellectual sophistication
is just, it's staggering. And second, the format
was also heavily influenced by Carr's synopsis,
the book that started it all for him, which just
stated results without proofs. He was likely
using the notebooks as a personal inventory,
needing only the final equations as reference
points. And speaking of records, we have to mention
the rediscovery in 1976 of a fourth notebook.
A lost notebook. 87 unorganized pages containing
discoveries from his very last year of life.
And its discovery caused a firestorm of excitement
among mathematicians. It's fueled decades of
work that's still going on today. It just emphasizes
that his output was so vast, so rich, that even
the unorganized messy notes he wrote in the final
months of his life contained foundational new
mathematics. And now we have to address the spiritual
dimension because he himself linked it inextricably
to his work. Ramanujan was a rigorously orthodox
Hindu. And he credited his mathematical acumen
directly to his family goddess, Namagiri Thayar.
For him, mathematics wasn't just abstract logic.
It was a way of accessing profound eternal truths.
He often said he looked to the goddess for inspiration.
He even described this vivid dream similar to
the one that convinced his mother to let him
travel. He saw a red screen formed by flowing
blood. And on this screen, a hand began to write
complex elliptic integrals that just stuck instantly
in his mind. And as soon as he woke up, he wrote
them down. It all leads to one of his most powerful
and famous quotes. An equation for me has no
meaning unless it expresses a thought of God.
And while Hardy, the staunch atheist, really
struggled to reconcile this mystical view with
academic rigor, there's no question that for
Ramanujan, the deep truths of mathematics and
divine insight were one in the same. It gave
him this unwavering confidence in his results.
It gives you a balanced understanding of his
work. His intuition wasn't just a psychological
phenomenon. For him, it was a deeply personal
spiritual connection that allowed him to bypass
the standard process and arrive directly at these
fundamental realities. But despite this staggering
flow of insights he was providing right up to
the end, his failing health really dictated his
final days. The sources note this immense strain
of his life in England, which led to a devastating
physical and mental collapse. Including a suicide
attempt in late 1917 or early 1918, he tried
to jump onto the tracks of a London underground
station. An attempt that Hardy intervened to
prevent? Which led to his release from arrest.
His illness, combined with the dietary strain
and the miserable climate, it all compelled him
to return to India in 1919. He was severely weakened,
but even back home, he kept working. He died
the following year, April 26, 1920, at the tragic
age of 32. And it's just heart -wrenching to
read that even in severe pain in his final days,
his wife recounts that he continued doing his
mathematics, filling sheet after sheet with numbers.
And what makes his early death even more tragic
is the modern analysis of his medical condition.
A 1994 review of his medical records suggested
his illness was actually more consistent with
hepatic amoebiasis. Which is a serious complication
from untreated dysentery, something that was
rampant in Madras at the time. Not the tuberculosis
he was originally diagnosed with. And if we connect
that to the bigger picture, the tragedy just
deepens because amoebiasis was treatable. It
was a treatable and often curable disease, especially
by the time he left England. If he had received
an accurate diagnosis and the right medical care,
it is highly probable he would have lived. The
fact that a treatable illness, compounded by
the strain of living abroad in wartime, might
have been the cause. It's arguably the single
greatest tragedy in modern mathematical history.
It is. Yet his work transcended his short life.
It laid the groundwork for the next century of
math. A major focus of that posthumous work is
the Ramanujan conjecture. This focuses on the
size of something called the tau function, which
comes from his work on modular forms. And the
Ramanujan conjecture became absolutely central
to 20th century mathematics. It was finally proven
in 1973 as a consequence of Pierre de Lyon's
proof of the Weyl conjectures. And that work
earned de Lyon the Fields Medal, the highest
honor in math, in 1978. But the connection goes
even deeper. This entire theoretical framework,
which sprang directly from Ramanujan's insight,
was later essential to the work needed to prove
Fermat's last theorem. Maybe the most famous
and difficult problem in mathematical history.
So think about that intellectual flow. A conjecture
made by a self -taught clerk, based on intuition
in the early 1900s, provided the structural blueprint
needed almost a century later to solve a problem
that had stumped mathematicians since the 17th
century. That's a legacy that defines the entire
field. And this overwhelming contribution led
directly to Hardy's ultimate, iconic assessment
of him. Hardy famously devised this scale of
pure mathematical talent from 0 to 100. He rated
himself at a modest 25, his great collaborator
Littlewood at 30, and David Hilbert, one of the
foundational titans of 20th century math, at
80. And Ramanujan. Hardy gave him a score of
100. A perfect score, placing him above his own
greatest contemporaries. You'd have to pause
and reflect on the sheer magnitude of that, placing
a self -taught figure above a giant like Hilbert.
It speaks volumes about the raw, explosive power
of his genius. But Hardy's eulogy, which was
published in 1940, it was ultimately balanced.
He noted both the staggering depth and the glaring
limitations imposed by his lack of formal education.
He said Ramanujan's insight was quite amazing
and altogether beyond anything I've met with
in any European mathematician. He called him
the greatest formalist of his time, meaning the
best at manipulating complex expressions. But
he also noted that his lack of knowledge was
as startling as its profanity. For instance,
Ramanujan had to rediscover many known results
because he just hadn't read the modern texts.
His achievements were greatest in algebra, specifically
hypergeometric series and continued fractions.
But despite those limitations, Hardy's final
words on his collaborator were so prophetic.
He believed that what Ramanujan had accomplished
was wonderful enough. When the researches which
his work has suggested have been completed, it
will probably seem a good deal more wonderful
than it does today. And that prediction has absolutely
proven true. Today, Ramanujan's influence extends
far beyond academic papers. He's rightly recognized
as a global cultural icon. India celebrates his
birthday, December 22nd, as National Mathematics
Day. And the institutional honors are plentiful.
There's the Sastra Ramanujan Prize, which gives
$10 ,000 annually to mathematicians under 32
working in areas he influenced. There's even
the Ramanujan IT City, a huge special economic
zone in Chennai. Cementing his image as a national
figure of intellectual prowess and innovation
and his life story is just, it's tailor -made
for film. Oh, absolutely. The 2015 film The Man
Who Knew Infinity brought his struggles and his
partnership with Hardy to a massive international
audience. And even in broader pop culture, his
name has become shorthand for intuitive, effortless
genius. He was famously mentioned in the 1997
film Good Will Hunting. Where Professor Lambeau
compares the main character's raw mathematical
brilliance to Ramanujan's. And he inspired the
character Amita Ramanujan in the TV series Numbers,
who was also steeped in number theory. So what
does this all mean for us? For the learner trying
to grasp the magnitude of his life? If there's
one summary takeaway from this deep dive, for
me, is that Ramanujan proved that mathematical
rigor, while absolutely essential for validation,
can sometimes follow intuition and inspiration.
It's a powerful reminder that pure talent can
emerge, unstructured and profound, from the most
isolated circumstances. Indeed. And the study
of his work continues to yield surprises even
now. Bruce Burnt noted that nearly all of the
thousands of conjectures he made have been proven
correct. often leading to brand new insights.
Which brings us to a final provocative thought,
building on the profound depth of his unfinished
legacy. Even the casual passing comments in his
notebooks, where he'd remark about certain formulas
having simple properties or similar outputs to
others, have turned out to be profound and subtle
number theory results that remained completely
unsuspected until almost a century after his
death. So you have to ask... What other profound
mathematical secrets, entire new fields of inquiry,
might still be hidden within those unexamined
notes, just waiting for us to catch up to the
mind of a genius who lived a century too soon?
The exploration is... far far from over. Thank
you for joining us for this deep dive into the
extraordinary life and shocking mathematical
legacy of Srinivasa Ramanujan.