#### The Genius of Ramanujan

##### September 15, 2015

### A glimpse of the great Indian mathe-magician’s work…

History has seldom seen a person who was so passionate, unorthodox, as well as gifted in a field as was Srinivasa Ramanujan, the self-taught Indian genius, who made several startling discoveries in the realm of Mathematics.

Despite abject poverty and lack of formal training and encouragement, Ramanujan’s love for numbers never waned. And thanks to a chance encounter and ensuing collaboration with G.H.Hardy of Cambridge, one of the most eminent mathematicians of the world, his hidden genius came to light.

Ramanujan went on to make thousands of discoveries with the apparent ease of experiencing and recording a series of religious epiphanies by a mystic in a trance. The methods he followed are still shrouded in a veil of mystery, since he usually skipped the formal rigour (and hence made mistakes too sometimes) and relied more on leaps of intuition to arrive at sudden, surprising results.

The several ‘Notebooks’ left behind by Ramanujan are strewn with cryptic formulae and equations, and are still being *‘mined’* by mathematicians all over the world for beautiful gems and nuggets.

His life, tragically brief though it was, goes to prove the adage:

TALENT does what it CAN, but

GENIUSdoes what it MUST.

Here, we consider a few samples of his work that are accessible *(and hence inspiring too?)* to people who are familiar with basic high-school mathematics.

### The innocuous pair of equations

While in early high-school, the math teacher called up little Ramanujan and gave him the following pair of equations to solve:

**Fig. 1.** The simple-looking problem posed to young Ramanujan in school.

The solution for this pair of equations is quite obvious \( \Rightarrow (x = 9, y = 4) \) but is tricky to arrive at (**If you disagree, please go ahead; give it a try!**). And it is said that Ramanujan came up with the solution in a jiffy, showing early signs of his prodigious gift.

### The mysterious equation

\[ 2^{n} - 7 = x^{2} : n, x \in Z \]

**Fig. 2.** The little equation with a surprisingly small set of solutions ( `{ 3, 4, 5, 7, 15 }`

)

There was this simple equation(see Fig. 2. above) which Ramanujan solved, providing only a small set of solutions. Extensive experiments by mathematicians later verified that there exist no other solutions other than the ones suggested by Ramanujan.

### The strange nested radical

**Fig. 3.** The strange, infinitely nested expression/radical whose value Ramanujan computed to be equal to 3 (Image by @raghu_ugare using Mathematica)

That is, \[ \lim_ {n \rightarrow \infty } \sqrt{1 + 2 \sqrt{1 + 3 \sqrt{1 + 4 \sqrt{1 + \ldots (n-1)\sqrt{n}}}}} = 3 \]

Then, there was a question as to what the infinitely nested expression shown above would evaluate to, which Ramanujan showed to be equal to 3. (**Yes! Simply 3**)

### The astonishing Prime-product ratio

**Fig. 4.** The amazing Prime-product ratio of Ramanujan of Ramanujan (Visualisation by @raghu_ugare using Mathematica)

Yet another result connects an infinite product of a specific ratio involving Prime numbers to an incredibly simple finite value(2.5, to be precise).

That is,

\[ \lim_ {n \rightarrow \infty} \prod_{i=1}^{n} \frac{p_i^{2}+1}{p_i^{2}-1} = 2.5 \]

This result is all the more startling, given the fact that Prime numbers are still considered to be among the most mysterious entities in the number-universe. And yet here was Ramanujan, conjuring a result like this almost out of thin air! (which was verified to be true too.)

### Closing remarks

The above examples are just a few among many such instances which exhibit Ramanujan’s amazing faculty with numbers.

His passion and brilliance in Mathematics place him permanently in the ‘hall of fame’ along with other eccentric and prolific geniuses such as Paul Erdős, who dedicated their lives for the sole purpose of ruminating and working tirelessly on esoteric things that laymen dare not imagine.

Indeed, it would be fitting to end this tiny *‘peek’* into the beautiful mind of Ramanujan with a quote that expresses his deep conviction and almost fanatic faith in the ‘Divine beauty’ of numbers and equations:

❝ An equation means nothing to me unless it expresses a thought of God. ❞

:~

Srinivasa Ramanujan

**Fig. 5.** Srinivasa Ramanujan(1887-1920)

#### Further reading

- “The Man Who Knew Infinity: A Life of the Genius Ramanujan”, a narrative biographical sketch by Robert Kanigel
- “Ramanujan’s Notebooks”—a series of volumes collecting and analysing Ramanujan’s body of work, by Bruce Berndt & George E. Andrews
- “The collected works of Ramanujan”, published online.
- “The manuscripts of Ramanujan”, courtesy the University of Madras.