Ramanujan's contributions

What Did Ramanujan Contribute To Mathematics & How’s It Being Used To Understand The Universe?

Some people get to ride the roller coaster bike of their life for only a few years but can significantly impact humankind even in their unfortunately short lifespan. And it wouldn’t be exaggerating if I say that Srinivasa Ramanujan truly swore by these words. Born on December 22, 1887, exactly 133 years from today, Srinivasa is a genius who carved an unparalleled niche for himself in the field of mathematics and that too in a visibly brief lifetime of 32 years.

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Srinivasa Ramanujan (Image: Britannica)

So today, on Ramanujan’s 133rd birth anniversary, let’s have a look at the genius’s mathematical prowess and also acknowledge him for the recent applications that his work has found in the field of physics.

Background and Love for Mathematics:

Ramanujan was born in Madras, British India, in the late nineteenth century to a very humble family. Ramanujan’s father worked as a clerk in a shop, while his mother was a housewife who used to sing at a local temple. Srinivasa showed signs of his mathematical genius at an early age, but he struggled a lot at other subjects, often failing in them. 

When he turned 11, he got a book on trigonometry as his birthday present. This was when Srinivasa started solving mathematics religiously. Ramanujan mastered trigonometry in just 2 years. The book that really changed his life was the one containing 5000 theorems that he studied in detail. To everyone’s surprise, Ramanujan had started developing his own theorems by the age of 16.

Unfortunately, back in the 1900s, Mathematics was not an option to make a living in India. But Srinivasa wasn’t ready to let go of his passion due to this hurdle. Rather, he started showing his work to foreign mathematicians. However, all of them said that he was unconventional and lacked formal education.

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But one fine day, Ramanujan’s work came across G.H Hardy’s desk in England, which was when Ramanujan’s mathematical career took off. When Hardy took a look at Ramanujan’s efforts,  he could not believe his eyes. So Hardy decided to call him to England. Hardy and his colleague Littlewood further examined Ramanujan’s work, after which Littlewood said, “I think we can compare him with the great German mathematician, Jacobi.” 

Ramanujan (centre) and his colleague G. H. Hardy (extreme right), with other scientists, outside the Senate House, Cambridge (Image: Wikipedia)
Ramanujan (centre) and his colleague G. H. Hardy (extreme right), with other scientists, outside the Senate House, Cambridge (Image: Wikipedia)

Some of Ramanujan’s Great Mathematical Contributions:

Infinite Series:

In England, Srinivasa made great progress working with Hardy and Littlewood. He was exceptionally good at infinite series. Before him, there was an infinite series for pi. However, the problem was that it was prolonged. It took about 600 terms to come to a value of 3.14. However,  Ramanujan, on the other hand, developed a series that would converge to 3.141592 just after 1 term. 

Ramanujan Summation:

Srinivasa did interesting mathematics in the field of infinite summation. He invented Ramanujan summation, which is a technique for assigning a value to divergent infinite series. It is essentially a property of the partial sums rather than a property of the entire sum. This method for summation of numbers points to the fact that ‘S’= -1/12, where S = 1+2+3+4+5+6+7+……..Though this result is shocking, string theory, quantum field theory, and some complex analytics make extensive use of this to derive equations.

Highly Composite Numbers:

Ramanujan worked intensively on highly composite numbers. A highly composite number is basically a positive integer that has more divisors than any smaller positive integer. He coined this term in 1915. There is an infinite number of highly composite numbers, the first few being 1, 2, 4, 6, 12, 24, 36, 48, 60…and so on. The corresponding numbers of divisors are 1, 2, 3, 4, 6, 8, 9, 10, 12… and so on. In 1915, he listed 102 highly composite numbers up to 6746328388800. Later in 1983 and 1988, Robin & Nicholas modified this list.

Hardy-Ramanujan Number:

Another famous incident that shows Ramanujan’s love for numbers was when Hardy once met him in the hospital. When Hardy got there, he told Srinivasa that his cab’s number, 1729, was “rather a dull number” and hoped it doesn’t turn out to be an unfavorable omen. To this, Ramanujan said, “No, it is a very interesting number. It is the smallest number expressible as a sum of two cubes in two different ways.” 1 cube + 12 cube and 9 cube + 10 cube. This number is now known as the Hardy-Ramanujan number. 

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Hardy – Ramanujan number

Apart from this, Srinivasa also made substantial contributions to the analytical theory of numbers and worked on elliptic functions, continued fractions, hypergeometric series, infinite series, and many other fields. He also provided solutions to some mathematical problems, earlier considered unsolvable.

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It is said that Hardy once came up with a scale of mathematical ability that went from 0 to 100. While grading everyone, he put himself at 25. According to Hardy, David Hilbert, the great German mathematician, was at 80 while Ramanujan stood at 100.

Applications of Ramanujan’s Work in Fields Beyond Mathematics:

All the abovementioned contributions made by Srinivasa and the related facts stand as verifiable proof that he reached unimaginable heights in his short life of little over 32 years and defined his own standard of genius. There is no doubt that the mathematical genius of Srinivasa Ramanujan made him leave behind a rich legacy of problems for future mathematicians to solve.

However, Ramanujan’s contributions are not confined to mathematics only. This might have come as a surprise for many. Still, it’s true that due to the remarkable originality and power of Ramanujan’s intellect, the ideas that he created a century ago are now finding applications in different areas of physical sciences.

Signal Processing:

One of the several areas where Ramanujan’s work has found its consumption is that of signal processing. All the signals that are processed digitally comprise certain patterns that repeat over and over again. These signals can be anything like that of speech, music, and even more, research-oriented ones such as DNA and protein sequences. While processing such signals, being able to extract and identify periodic information is of fundamental concern.

To achieve this, some of the best-known methods to extract periodic components in signals involve Fourier analysis. Now, this is where Ramanujan’s work comes into the picture. Motivated by Ramanujan’s sum, which is basically a sequence that repeats periodically, Prof. Vaidyanathan and his student Srikanth Tenneti have developed the concept of “Ramanujan subspaces” that works better than Fourier analysis when the region of periodicity is short.

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Ramanujan’s manuscripts (Image: thehindu.com)

In Understanding the Black holes:

Srinivasa is also well known for his work on the possible ways of partitioning an integer. For example, the integer 3 can be written as 1+1+1 or 2+1. As the integer gets larger and larger, it becomes quite intricate to calculate the number of ways to partition it. However,  he related this counting problem to some special functions called “modular forms.” A modular form is a symmetric one. In other words, it does not change under a set of mathematical operations and thus follows a modular symmetry. In his famous letter to Hardy in 1919, he introduced the “mock theta functions” and observed that they were “almost modular.” 

Nowadays, Ramanujan and Hardy’s results on partitions and the mock theta functions have found an important role in understanding the very quantum structure of spacetime, particularly in the comprehension of quantum entropy of a type of Black Hole in string theory. Thus Ramanujan’s otherwise simple mathematical calculation is now being utilized in revealing the complicated properties of black holes.

In a nutshell, Srinivasa is truly an intellectual who has served humankind with his genius and will continue to inspire millions with the extraordinary journey he went through and the marvelous work he did!

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