Ramanujan: The Maths Behind the Man

Ramanujan: Life between Madras and Maths. Art & article by Samskrith Raghav.

Did you know that the first Indian to become a Fellow of the prestigious Royal Society (UK) AND Trinity College, Cambridge didn't even have a college degree? He loved math. He did math. Math was all he wanted to do. He forgot everything else. This was Srinivasa Ramanujan, arguably the greatest mathematician of all time. Today, April 26th, 2020, is the hundredth anniversary of his passing. Even today there are still mathematicians working to prove his theorems. There may still be notebooks upon notebooks of his work, filled with formulas waiting to be proven.

Srinivasa Ramanujan is one of the greatest mathematicians who ever lived. He filled notebooks upon notebooks with formulas, theorems, and functions, always noting down the answer without showing any steps or work whatsoever. His work can be found in the Wren Library at Trinity.. But what makes this Indian mathematician who died a hundred years ago so relevant today? Ramanujan the epitome of mathematical prodigy. When he was 11, he quickly exhausted the mathematical knowledge of not one, but TWO college students who were living with him. He assisted in the logistics of assigning his school’s 1,200 students to 35 teachers and was independently discovering things discovered by the likes of Euler and Jacobi. He was invited to Trinity College in England by Professor G. H. Hardy, who was intrigued by the work Ramanujan had sent in his letter. Together, he and Ramanujan (who had no formal Mathematics training) tackled partitions, figuring out a formula (with proofs) that could predict the number of partitions of 200 to within 0.004.

Even at the end of his life, Ramanujan did not stop writing formulas. After he left England, he spent the last year of his life in India writing down formula after formula (all without proof). His famous mock theta function as well as his mock modular forms were discovered during this time. He wrote over 600 formulas in this short period of time on 100 pieces of paper on 138 sides. He died on April 26, 1920, exactly 100 years ago today.

Ramanujan also has some other recognition. He has a number named after him, called the Hardy-Ramanujan number-aka 1729. In a widely known story, Hardy remarked to Ramanujan that his cab number was a rather boring one - 1729. Ramanujan expressed his opinion to the contrary, stating that 1729 was actually rather interesting; it was the smallest number able to be expressed as the sum of two cubes in two ways: 13 + 123 and 93 + 103. He was also named a Fellow of the Royal Society, the first Indian to be honored in such a way.

But had Ramanujan lived a much longer life, or even lived to the average human lifespan, what would all his accomplishments be? Ramanujan’s true genius is probably still yet to be discovered. He discovered an infinite series for calculating pi which converges rapidly and is the basis for all modern calculations of pi to date. He was the first person to discover K3 surfaces, an object more complex than elliptical curves (though he didn’t share his discoveries until someone found his notebooks years after his death). His theta function is at the heart of String Theory and his mock modular forms may be the key to unlocking the secrets of black holes.

One of the defining features of Ramanujan’s work is his infinite series. Many a mathematician has spent a great portion of his career attempting to work out the missing steps in his infinite series. One of the most puzzling of these is:

1 + 2 + 3 + 4 + … = -1/12

How Ramanujan managed to get this result from that infinite series is intriguing. Let me attempt to explain it:

1. The first thing Ramanujan does is equate the infinite series to a constant c (not a sum)

1 + 2 + 3 + … = c

or

c = 1 + 2 + 3 + 4 + ...

2. Next, Ramanujan does something totally natural - he multiplies the entire equation by 4. Reasonable enough? Here’s the twist: He skips every other number.

4c =  4 +  8

3. He then proceed to subtract this new equation from the original:

  c = 1 + 2 + 3 + 4 + … 

4c =       4     +    8 + … 

After subtraction, it leaves you with:

-3c = 1 - 2 + 3 - 4 + … 

4. Somehow, Ramanujan manages to convert this:

-3c = 1 - 2 + 3 - 4 + … 

Into this:

-3c = 1/(1 + 1)2

So how does he do this? Well, this actually consists of two steps. Firstly, he knows that this:

1 - 2 + 3 - 4 … 

Is the same as this:

(1 - 1 + 1 - 1 + … )2

Here’s the reasoning behind this:

Try multiplying:

(1 - 1 + 1 - 1 + … )

(1 - 1 + 1 - 1 + … )

Let’s use a more visual representation for this. Here’s how it looks:

	+1	-1	+1	-1
+1	+1	-1	+1	-1	…
-1	-1	+1	-1	+1
+1	+1	-1	+1	-1
-1	-1	+1	-1	+1
	…

As you can see, I have multiplied everything already. It’s pretty simple, all you're really doing is multiplying 1’s and -1’s. After that, just add everything diagonally(Use the color codes to help you) and you get:

1 - 2 + 3 - 4 + … 

Which proves this:

1 - 2 + 3 - 4 + … = (1 - 1 + 1 - 1 + … )2

5. The second step is that Ramanujan knew that this:

1 - 1 + 1 - 1 + … 

Is the same as this:

1/1 + 1

This comes from the sum of the geometric series:

1 + r + r2 + r3 + … = 1/1 - r

This is only valid if:

-1 < r < 1

For example, if r = 1/2:

1 + 1/2 + 1/4 + 1/8 + … = 2

However, many people, including Ramanujan substitute r as -1 anyways. So you get:

1 - 1 + 1 - 1 + … = 1/1 + 1

6. What you get eventually is:

-3c = 1/(1 + 1)2

1/(1 + 1)2 is obviously the same as 14. So:

-3c = 1/4

So you divide and get:

c = -1/12

In the end, you get this:

1 + 2 + 3 + 4 … = -1/12

But many of you readers will protest, saying that this is incorrect. Obviously, this is absolute mathematical insanity, but it is also genius. And luckily for Ramanujan, most of his work was correct mathematically. For example, his work on partitions. Ramanujan was the co-creator (with G. H. Hardy) of the influential circle method aka the Hardy-Littlewood method of partitions. Partitions are all the ways to sum up to a number. For example, the partition number for 4, aka P(4), is 5: 1 + 1 + 1 + 1, 2 + 2, 1 + 1 + 2, 3 + 1, and 4. Ramanujan discovered a formula for estimating the number of partitions for numbers beyond 100 which came within two percent. And when the numbers rose higher and higher, the percent of difference became smaller and smaller. In fact, at infinity, the percentage of difference was at 0. Eventually, Ramanujan refined his formula to come within 1 percent, and then eventually 0.004 of the real thing.

But is that all that Ramanujan did? No. In fact, Ramanujan also independently discovered many theorems or formulas which made mathematicians like Euler famous. He independently discovered the results of Gauss, Kummer, and others on hypergeometric series. He investigated at a young age and calculated Euler’s constant to 15 decimal places. He then began to study the Bernoulli numbers, although his discovery of them was independent of any other source. Had these mathematicians not discovered them earlier, Ramanujan would have been the first - making him all the more amazing.

But then again, how is this hundred years old mathematics from an Indian genius relevant to the world of today? Even today, Ramanujan’s genius still applies. His theta function is at the heart of the String Theory in Particle physics, a theory in which all particles are described as different vibrations or modes of a string. It requires ten dimensions, but is still a possible Theory of everything, because unlike the theory of the point particle, it can accommodate for All 4 Fundamental forces: the Strong Nuclear Force, the Weak Nuclear Force, ElectroMagnetism, AND Gravity. The fact that Ramanujan’s theta function is at the center of this possible theory of everything is amazing.

And Ramanujan’s work may have some applications which will help unravel the most mysterious secrets of the cosmos. In Ramanujan’s letter to G. H. Hardy, he described several functions which behave differently than theta functions, or modular forms, but closely mimic  them - so called mock modular forms. Ramanujan predicted that his mock modular forms corresponded to ordinary modular forms which produced similar outputs for roots of 1. His predictions proved correct, and an expansion of mock modular forms is used to predict entropy levels in black holes. In fact, these mock modular forms may be the key to unraveling the mysteries black holes pose.

In the end, this untrained mathematician often on the brink of starvation and desperate for work did some of the finest work in mathematics. He is so renowned that one of his notebooks is on display at the Wren Library in Trinity College. His work is still being proved today and it will take many lifetimes just to prove all his work. But with absolute and true genius, like Ramanujan, the time is worth spending.

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