A century later, why the world still needs Ramanujan

The opening lines of the letter were startling. Dated February 27, 1913, it was addressed to Godfrey Harold Hardy, the most prominent mathematician in England at that time. It had come from afar India and was written by a humble Indian clerk. There were about 10 sheets of papers filled with 120 incredible and what appeared to be improbable mathematical formulas.

The letter read, “Dear Sir,…the sum of an infinite number of terms of the series: 1+2+3+4..+ ∞ = -1/12  under my theory. If I tell you this you will at once point out to me the lunatic asylum as my goal…you will not be able to follow my methods of proof.”

What? Really? The sum of all the positive integers is a fraction; that too a negative one? What’s more, the Indian known as Srinivasa Ramanujan had the audacity to tell the world’s leading mathematician that, “you will not be able to follow my methods of proof!”

GH Hardy was astounded. Every prominent mathematician gets letters from deluded cranks who promise solutions to unsolved puzzles in mathematics. It was tiresome and annoying. Hardy thought, perhaps he should trash the letter and forget about it. But, something stopped him from doing that.

Later recalling that day, Hardy said, “They (the formulas) defeated me completely; I had never seen anything in the least like them before. A single look at them is enough to show that they could only be written down by a mathematician of the highest class. They must be true because, if they were not true, no one would have had the imagination to invent them.”

Also read: How cryptic Ramanujan still inspires, boggles minds of young researchers

If the letter had made its way to the dustbin that day, the world today would have been poorer in knowledge. Ramanujan would have died unknown, perhaps Hardy would not have been able to solve the many intricate problems that he cracked later and the mobile and communication revolution we enjoy today would not have been possible.

Counter intuitive

What is 1+2+3? It’s 6. Easy.
What about 1+2+3+4? 10.
Hmm. Fine then 1+2+3+4+5? 15.

What about 1+2+3=4+5+ …..+200=? Well if you recall your high school mathematics and the formula “the sum of integers up to n = nx(n+1)/2,” then it’s easy.

Plugging in the formula n=200, we get,
200 x (200+1)/2 = (200 x 201)/2 = 20,100.

One thing you can notice is that, as the value of “n” increases, the value of the sum escalates. In mathematicians’ lingo, the sum ‘diverges’.

Now let us ask a question.
1+2+3+4+…+∞ =?

The three dots “…” means ‘repeat endlessly’. That keeps adding next number endlessly.
So what is 1+2+3+4+…+∞ =?
We saw that 1+2+3=4+5+ …..+200=20,100. Therefore this value must at least more than 20,100.

However Ramanujan claimed 1+2+3+4..+ ∞ = -1/12

Method in madness

Although Ramanujan said Hardy would not follow his ‘method’, it is not that difficult. What we need is simple arithmetic and a bit of patience.

What is 1-1; obviously zero.
What about 1-1+1-1=? It is zero + zero, so zero.
What about 1-1+1-1+1-1+1-1+…=?
Recall the three dots “…”? It means you repeat the last step endlessly.
It does not matter how many zeros you add. The sum is zero.

Let us call 1–1+1–1+1–1⋯ = A
that is A = 1–1+1–1+1–1⋯

Let us now do neat little trick mathematicians do. Let us take A from 1. That is 1-A. Now, what is 1-A=?

1-A=1-(1–1+1–1+1–1⋯)
But then 1-(1–1+1–1+1–1⋯) is = 1-1+1-1+1-1+… = A
That is 1-A= A
that is 1= 2 A
That is A = ½
Or 1–1+1–1+1–1 ⋯ = ½. Voila!

Bewildering. Is it not? This is not Ramanujan’s original discovery. Italian mathematician, philosopher, and priest Guido Grandi had discovered it 200 years before Ramanujan.

Now let us go to the next step
Let us say B =1–2+3–4+5–6⋯
Of course, this series is bizarre, but wait it will turn into a beauty.
Now we are going to subtract B from A.
A-B = (1–1+1–1+1–1⋯) — (1–2+3–4+5–6⋯)
or if we rearrange,
A-B = (1–1+1–1+1–1⋯) — 1+2–3+4–5+6⋯
and if we shuffle
A-B = (1–1) + (–1+2) +(1–3) + (–1+4) + (1–5) + (–1+6)⋯
A-B = 0+1–2+3–4+5⋯

Now look at the RHS, 0+1–2+3–4+5⋯ ; it is nothing but B
So we have A-B=B

A= 2B
We know that  A = ½, So B is =1/4
or 1–2+3–4+5–6⋯ = ¼

This was also not Ramanujan discovery. Many mathematicians have got these results many years before.

Ramanujan took the next step.

C = 1+2+3+4+5+6⋯ That is the sum of all-natural integers.
B-C = (1–2+3–4+5–6⋯)-(1+2+3+4+5+6⋯)
If we shuffle and rearrange,

B-C = (1-2+3-4+5-6⋯)-1-2-3-4-5-6⋯
B-C = (1-1) + (-2-2) + (3-3) + (-4-4) + (5-5) + (-6-6) ⋯
B-C = 0-4+0-8+0-12⋯
B-C = -4-8-12 -16⋯

Now look at 4, 8, 12… They are multiples of 4. Therefore I can take 4 out and write
-4-8-12-16⋯ as -4(1+2+3+4…)
B-C = -4(1+2+3+4…)
But (1+2+3+4…) is C
so, B-C = -4C
So B= -3C
But from earlier we know B = ¼
1/4 = -3C
1/-12 = C or C = -1/12
or  1+2+3+4+…=  -1/12

This now has a fancy name: Ramanujan Summation.

Sleight of hand

But, before you go crazy and conclude mathematics doesn’t make sense, here’s a word of caution. In the above, the “=” sign does not have the conventional meaning “is equal to”. When you actually set to add infinite terms, it is not an action that you can actually perform, and hardly can we talk about “is equal to”. In Ramanujan Summation of a divergent series, the meaning of symbol “=” changes from “equal to” to “is associated with”.

What appears to be sleight of hand is, in fact, useful in understanding how the world works. Discovered a hundred years ago, Ramanujan Summation helps understand the bizarre world of quantum mechanics. It sheds light on the enigmatic workings of the black holes. The digital signal processing, especially in identifying periodic components of signals buried in noise is hugely benefited by Ramanujan’s Summation. The decoding of genomic signal processing too requires the application of Ramanujan Summation.

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Take, for example, the idea of the quantum vacuum. Often we understand void as ‘nothingness’. However, our modern understanding of the quantum fluctuations tells us that a steady stream of pairs of particle and anti-particle pairs is continuously created and annihilated in a vacuum. Thus vacuum is filled with virtual particles. Two uncharged but conducting plates kept in the vacuum will have no attractive forces between them in classical physics. However,  Hendrik Casimir, a Dutch physicist used the Ramanujan’s sums to study the interaction of the uncharged plates, called Casimir effect, and computed the attractive force between these plates in the presence of virtual particles.

In the recent years, Prof PP Vaidyanathan, California Institute of Technology and his colleagues, have used the Ramanujan Summation to identify the structured signals such as periodic signals. Using it, they have built a novel theory for filter band for signal processing that helps power the internet and cell phones.

(The author is a science communicator with Vigyan Prasar, New Delhi)