Thursday, January 22, 2009

Srinivasa Ramanujan and His Number Theory

Unjustly ignored during his tragically brief lifetime, will the mathematical genius of this minor bureaucrat from Madras, India be finally recognized?

By: Vanessa Uy

At present, his body of work on number theory is somewhat unknown beyond the confines of academia. But Srinivasa Ramanujan, whose body of work on his musings on number theory eventually gained the interest of the eminent Cambridge don Godfrey H. Handy. Which Ramanujan eventually gained “immortality” even though that fame seldom reaches beyond the cadre of the world’s leading mathematicians still deciphering the full worth of his forays into number theory. In order to appreciate the paramount importance of Srinivasa Ramanujan’s work on number theory and it’s indispensability in our current information-based society of the Internet, here is a primer on the mysterious yet wondrous world of number theory.

Number theory forms one of the facets of pure mathematics – i.e. mathematics for its own sake, without any practical goal in mind. It probably began when men (it was probably men who had the luxury of time to think of such things at the dawn of civilization) first thought of numbers as numbers – apart from the counting of his head of sheep and / or cattle.

As time went on, the “number theorists” have returned to the deceptively simple steps we count by. And repeatedly they have concluded that integers, or ordinary whole numbers – one, two, three, and so on – are the most baffling, stimulating and entertaining of all mathematical subjects. In ancient Greece at around 300 BC, the famed mathematician Euclid was famed for his exploration and discussion on prime numbers. Prime numbers are numbers - which cannot be evenly divided except by them or by the number 1. Euclid’s knowledge on prime numbers now forms the backbone of college-level number theory.

Later still, the analysis of numbers that has grown from Diophantine equations – after Diophantus’ famous “guess my age riddle” solvable by an algebraic equation since named in his honor – is called the “theory of numbers”. Which thus became the purest of pure branches of present mathematics. Its development back then by Diophantus, probably between 100 to 400 AD, helped algebraists to take on an equation as a way of categorizing all numbers of a given type rather than as a relationship solely between the numbers of a specific problem. Because of this, algebra slowly began to unfold as a separate discipline.

When the 19th Century “Mathematical Giant” Carl Friedric Gauss was 19, he began to flood the pages of his notebooks with new mathematics of his own – new theorems in the abstruse realm of number theory – which profoundly contributed to the further development of number theory. Add to that the discovery of Carl Friedrich Gauss’ “star pupil” named Bernhard Riemann. Which later continued Gauss’ work in the abstruse realm of number theory that without it, our contemporary use of very large prime numbers as a basis of providing secure and foolproof Internet transaction and security would not have been possible.

At the dawn of the 20th Century, Srinivasa Ramanujan with a day-job as a minor bureaucrat in Madras, India had the good fortune of finding time to dabble into the “still abstruse realm” of number theory. Ramanujan tried several times to gain the interest of professional and tenured mathematicians in his spare-time dabbling with numbers. But the mathematicians he contacted were somewhat jaded when it comes to the antics of “numerological crackpots” so they tend to ignore recent submissions, including Ramanujan’s.

Fortunately, his work gained the notice of the eminent Cambridge don Godfrey H. Hardy who took the time to decipher the young Ramanujan’s somewhat idiosyncratic scrawls which Hardy had realized that he was witnessing the work of an undiscovered mathematical genius. Unlike university tenured mathematicians of the time, Ramanujan knew his speculations about numbers were true, so he didn’t bother to prove them. Which unfortunately Hardy finds his fresh approaches on number theory somewhat wanting despite of their brilliance.

Eventually Hardy brought Ramanujan to England in 1914, in which Hardy had spent four years in teaching the largely self-taught Ramanujan in proving his intuitively brilliant conjectures on number theory. Unfortunately, both prevailing climatic and cultural conditions of World War I era England was not to Ramanujan’s overall liking and the great mathematician-in-ascendant died of a virulent strain of tuberculosis in 1920 at the age of 32.

With so much of Srinivasa Ramanujan’s mathematical opus still left unproved, brilliant mathematicians worth their salt from all over the world are up to this very day still hard at work in examining and proving Ramanujan’s number theories. Even the late, great Pakistan-born theoretical physicist Abdus Salam – the first Muslim Nobel Laureate – first gained fame as a brilliant mathematician during his early school days by solving a math problem posed by the Indian-born Ramanujan using an approach far more elegant than that previously provided by Ramanujan himself. Thus also proving that India and Pakistan were not always at each other’s throats.

You may now wonder of what use does number theory has in our modern day-to-day Internet-based society? Well, starting during the dark days of World War II, the concept behind number theory was used to great effect by the British computer software pioneer Alan Turing in cracking NAZI-era Enigma codes. The same principle is still in use today when law enforcement agencies decipher coded communiqu├ęs being sent by criminal organizations like the Aryan Brotherhood. And also in curbing illicit on-line activity like corporate fraud and the peer-to-peer sharing of child pornography.

Even though Srinivasa Ramanujan’s work in number theory still remains largely unknown beyond the confines of university tenured mathematicians and computer “hackers” working for Internet security companies. There had been attempts to make the rest of the population aware of his brilliant and indispensable contributions to mathematics. In the TV series NUMB3RS / Numbers, one of the characters, a women mathematician of ethnic Tamil descent was named Anita Ramanujan in honor of Srinivasa Ramanujan.