By I was recently on a cleaning spree and came across one of my textbooks on the subject, ‘Real Analysis’. For those of the readers who are from a non-mathematical background, let me first introduce the subject:
Real Analysis is a branch of mathematics that deals with the study of sequences and series, mathematical functions and is also the foundation for calculus. (It is a different matter that calculus is often taught before real-analysis in the regular school/college curriculum)
I would, however, define real-analysis as anything but ‘real’; it does not deal with any real-world problem as the subject title misleads. All the concepts in the subject seem abstract and is a shock to the regular student of mathematics, who progressed through SAT-levels, under the impression that he/she knew math!
Anyway, if you happen to find yourself pursuing a course in mathematics or statistics or any other allied subjects- your fate is tied with such subjects.
The subject starts with theorems and will go on-and-on to define concepts to prove these theorems. To top it all; when you find yourself at the juncture of feeling to have understood these concepts and re-construct the proof of the theorem; you will find your vanity shattered into a thousand pieces when you read the proof of the theorem that – ‘It’s trivial’.
Yes, trivial! – of all the words that mathematicians could find to define a concept that’s intuitive or relatively simple- the word they choose to mock you – is trivial as if murdering your new-found ego.
There are certain other subjects similar to Real Analysis that also follow a similar pattern of mockery such as Linear Algebra, Abstract Algebra, Complex Analysis, Numerical Analysis, Topology etc. etc.
Anyway, while studying these courses, I recollect an incident that has stayed with me over the years.
Any lecture of theoretical mathematics or statistics involves extensive use of the white-board or black-board. I remember one professor at my graduate-school who wrote down a theorem and reconstructed the proof of the same on the black-board and was waiting for the students to understand it, perhaps write-it-down if they so wished and ask questions.
Strangely, whenever a particular student asked a query about a certain step; the professor would point at that step, take a few-steps away back from the blackboard and say,
“Let us stare at it for some time.”
What?! Stare at it?!
No explanations, no ifs, no buts, simply stare at it!
Not even, the usual mockery that we were accustomed to – ‘it’s trivial’! As if textbooks printing, ‘it’s trivial’ was not insulting enough, we had someone say, ‘stare at it’!
There were a few chuckles and giggles; but at the end we did that – we stared at it!
After a while, one of the students raised his/her hand and explained the step – sometimes it would be the same student who asked the query in the first place.
Thus, complex theoretical mathematical problems were solved not by referring to any fat books written by authors of repute or even by googling; they were solved by the simple technique of learning how to stare.
I laugh at my younger self who giggled at this phrase and thought, “ Stare karne se solve ho jayega kya? (will it be solved by staring?)”
I was too naïve to understand the depth of this phrase.
The truth is that staring is a universal mantra for problem solving. Please note the word is ‘stare’ and not ‘watch’ or ‘observe’. Staring means prolonged gazing or fixed look.
And just like that, when you find yourself staring; either one of two things can happen – either the answer becomes ‘trivial’ or you grow out of the problem.
After all these years, I have finally learnt the art of staring.
