This is the story of Srinivasa Ramanujan, a man whose intellect was so singular that it frightened the brilliant minds of Cambridge, and whose "archaic" mathematics, written a century ago, has curiously resurrected itself to solve the deepest mysteries of String Theory and Black Holes.

Part I: The improbable Genius

Most students would have used it to memorize formulas for exams. Ramanujan used it as a launching pad. He began to see the gaps between the theorems, filling them with his own intuition. He worked on a slate because paper was too expensive, wiping his calculations away with his elbow, recording only the final "gems" in his now-famous notebooks.

The Letter That Changed History

In 1913, Ramanujan took a gamble. He sent a letter to G.H. Hardy, the most famous mathematician in England. The letter was a mess of wild-looking formulas—infinite series for Pi, strange integrals, and assertions about prime numbers that looked impossible.

Hardy’s first instinct was to throw it in the trash, assuming it was the work of a crank. But something stopped him. He looked at a formula for continued fractions that seemed to defy logic. Later, he would famously remark:

This invitation to Cambridge marked the beginning of one of the most romantic and tragic partnerships in the history of science. It was a clash of cultures: Hardy, the rigorous atheist who demanded proof for everything; and Ramanujan, the devout Brahmin who claimed his theorems were placed on his tongue by a goddess.

Part II: The Magic of Pi and the Infinite

One of Ramanujan’s earliest obsessions was the number Pi ($\pi$). For thousands of years, mathematicians had struggled to calculate digits of Pi efficiently. The formulas they used were slow and plodding.

Ramanujan saw Pi differently. He saw it not just as a geometric ratio, but as a deep spectral value arising from complex modular functions. In 1914, he published a series of formulas for $1/\pi$ that were bizarrely fast. The most famous one looks like this:

$$ \frac{1}{\pi} = \frac{2\sqrt{2}}{9801} \sum_{n=0}^{\infty} \frac{(4n)!(1103+26390n)}{(n!)^4 396^{4n}} $$

To a normal mathematician, this looks like a random collection of numbers. Why 9801? Why 1103? Why 26390?

But this formula is a "modular machine."

Ramanujan didn't arrive at this by crunching numbers. He arrived at it by understanding Modular Equations—functions that stay symmetric when you fold and stretch the complex plane in specific ways. He didn't know it, but he was building the toolkit for String Theory.

Part III: The "Magic" Number 24 and String Theory

Here is where the story shifts from pure math to the bedrock of modern physics.

In the 1960s and 70s, physicists began developing String Theory—the idea that the fundamental particles of the universe (electrons, quarks, photons) are not tiny dots, but tiny vibrating "strings."

But there was a problem. The math of String Theory kept breaking. When physicists tried to calculate the probabilities of these strings moving through space, the equations produced "anomalies"—mathematical nonsense like probabilities greater than 100% or infinite energies.

To fix the math, they found that the strings needed "room" to vibrate. They couldn't just vibrate in our 3 dimensions of space + 1 of time. They needed more dimensions. But exactly how many?

The Calculation

When physicists sat down to calculate the "critical dimension" for the original Bosonic String Theory, they encountered a strange sum in their equations related to the zero-point energy of the string modes. The sum looked like this:

$$ 1 + 2 + 3 + 4 + ... $$

This is the sum of all natural numbers. Intuitively, it should be infinity. But in the weird world of "zeta function regularization" (a technique Ramanujan was a master of), this sum behaves as if it equals -1/12.

To make the physics consistent (specifically, to preserve Lorentz invariance), the dimension $D$ of the universe had to satisfy a specific equation involving this sum:

$$ \frac{D - 2}{24} = 1 $$

Solving for $D$:

$$ D - 2 = 24 $$

$$ D = 26 $$

Where does the number 24 come from? It comes from the Ramanujan Modular Discriminant (often denoted as $\Delta$). This function, which Ramanujan studied extensively, has a "weight" of 24.

Decades before physicists even dreamed of strings, Ramanujan was obsessed with the number 24. It appears in the Dedekind eta function, a modular form that Ramanujan manipulated with ease. The "24 modes" in the Ramanujan function correspond exactly to the 24 physical vibrations of a string in the transverse directions.

(Note: In Superstring theory, which includes fermions, the dimension count drops from 26 to 10, utilizing a generalization of Ramanujan’s work from the number 24 to the number 8).

Part IV: The Lost Notebook and Black Holes

The most chilling chapter of Ramanujan’s legacy comes from his deathbed.

In 1920, Ramanujan was back in India, dying of what was likely hepatic amoebiasis. He was in pain, wasting away, yet his mind was more active than ever. In his final letter to Hardy, dated January 1920, he wrote about a new class of functions he had discovered. He called them "Mock Theta Functions."

He listed 17 examples, gave no proofs, and then died.

For decades, these functions were a mystery. They didn't fit into the standard definitions of modular forms. They were "almost" modular, but they had a "shadow"—a chaotic term that ruined their symmetry. The mathematical community largely ignored them, viewing them as a dying genius's final, confused scribbles.

The Discovery of 1976

In 1976, American mathematician George Andrews was digging through a box of old papers in the Trinity College library in Cambridge. He found a sheaf of papers with Ramanujan’s distinctive handwriting. It was the "Lost Notebook."

The discovery was electric—Andrews described it as the mathematical equivalent of finding Beethoven’s 10th Symphony. It contained over 600 formulas, mostly focused on these mysterious Mock Theta Functions.

The Black Hole Connection

Fast forward to the 2000s. Physicists were trying to solve one of the biggest paradoxes in science: Black Hole Entropy.

Stephen Hawking had famously predicted that black holes have entropy (a measure of disorder). But in quantum mechanics, entropy is a count of "microstates"—the different ways the insides of the black hole can be arranged. To prove Hawking right, String Theorists needed to count these microstates for a specific type of black hole.

When physicists Atish Dabholkar, Sameer Murthy, and Don Zagier looked at the math required to count these quantum states, they were stunned. The equations governing the entropy of black holes were exactly the Mock Theta Functions Ramanujan had scribbled on his deathbed.

Ramanujan’s "mock" symmetry was precisely the tool needed to describe the "imperfect" modularity of a black hole’s quantum states. He had discovered the mathematics of black holes 50 years before the concept of a black hole was even fully accepted by physicists.

Part V: The Taxicab and the "Personal Friends"

No article on Ramanujan is complete without the story that best illustrates his relationship with numbers: The Taxicab Number.

While Ramanujan was ill in Putney, Hardy visited him. Hardy, awkward as always, arrived and said, "I rode here in taxicab number 1729. It seemed to be a rather dull number."

Ramanujan, lying sick in bed, immediately replied:

$$ 1729 = 1^3 + 12^3 $$

$$ 1729 = 9^3 + 10^3 $$

This wasn't a calculation he did in his head at that moment. Ramanujan "knew" 1729. To him, numbers weren't just quantities; they were personalities. 1729 is now known as the Hardy-Ramanujan Number, and it has spawned an entire class of "Taxicab Numbers" in number theory.

Curiously, the number 1729 also appears in String Theory and the physics of "K3 surfaces," which are shapes used to compact extra dimensions. It seems that even his "little jokes" with Hardy had deep roots in the structure of reality.

Part VI: A Legacy of "Divine" Algorithms

Today, Ramanujan’s influence extends far beyond theoretical physics.

1. Signal Processing: We now use "Ramanujan Graphs"—graphs that are highly connected but sparse—to build efficient communication networks. These graphs are "good expanders," meaning they allow information to flow rapidly without clogging the network. They are essential for modern internet architecture.
2. Computer Algebra: The algorithms used by Mathematica and Maple to calculate fundamental constants often rely on Ramanujan’s 1914 breakthroughs.
3. Crystallography: His work on "quasi-crystals" and partition theory helps chemists understand the structure of matter in ways that were previously impossible.

The Goddess Namagiri

Ramanujan always attributed his success to his family goddess, Namagiri Thayar. He famously said:

Whether you believe in the divine or in the subconscious processing power of a singular genius, the result is the same. Ramanujan tapped into a vein of mathematical truth that runs deeper than human logic usually allows.

Conclusion: The Man Who Knew Infinity

Srinivasa Ramanujan died at age 32. He left behind three notebooks and a "lost" pile of papers. He never saw a supercomputer. He never heard of a black hole. He never knew what a "string" was in the context of physics.

And yet, when we push the boundaries of human knowledge today—when we try to count the states of a black hole or map the 10th dimension—we find Ramanujan there, waiting for us. He had already sketched the map.