Category Archives: memory power concentration retention

The personality of Leonhard Euler

The portrait of Euler that emerges from his publications and letters is that of a genial man of simple tastes and conventional religious faith. He was even wealthy, at least in the second half of his life, but ostentation was not part of his lifestyle. His memory was prodigious, and contemporary accounts have emphasized this. He would delight relatives, friends, and acquaintances with a literal recitation of any song from Virgil’s Aenesis, and he would remember minutes of Academy meetings years after they were held. He was not given to envy, and when someone made an advance on his work his happiness was genuine. For example, when he learnt of Lagrange’s improvements on his work on elliptic integrals, he wrote to him that his admiration knew no bounds and then proceeded to improve upon Lagrange!

But, what is most characteristic of his work is its clarity and openness. He never tries to hide the difficulties from the reader. This is in stark contrast to Newton, who was prone to hide his methods in obscure anagrams, and even from his successor, Gauss, who very often erased his steps to present a monolithic proof that was seldom illuminating. In Euler’s writings there are no comments on how profound his results are, and in his papers one can follow his ideas step by step with the greatest of ease. Nor was he chary of giving credit to others; his willingness to share his summation formula with Maclaurin, his proper citations to Fuguano when he started his work on algebraic integrals, his open admiration for Lagrange when the latter improved on his work in calculus of variations are all instances of his serene outlook. One can only contrast this with Gauss’s reaction to Bolyai’s discovery of non-Euclidean postulates. Euler was secure in his knowledge of what he had achieved but never insisted that he should be the only one on top of the mountain.

Perhaps, the most astounding aspect of his scientific opus is its universality. He worked on everything that had any bearing on mathematics. For instance, his early training under Johann Bernoulli did not include number theory; nevertheless, within a couple of years after reaching St. Petersburg he was deeply immersed in it, recreating the entire corpus of Fermat’s work in that area and then moving well beyond him. His founding of graph theory as a separate discipline, his excursions in what we call combinatorial topology, his intuition that suggested to him the idea of exploring multizeta values are all examples of a mind that did not have any artificial boundaries. He had no preferences about which branch of mathematics was dear to him. To him, they were all filled with splendour, or Herrlichkeit, to use his own favourite word.

Hilbert and Poincare were perhaps last of the universalists of modern era. Already von Neumann had remarked that it would be difficult even to have a general understanding of more than a third of the mathematicians of his time. With the explosive growth of mathematics in the twentieth century we may never see again the great universalists. But who is to say what is and is not possible for the human mind?

It is impossible to read Euler and not fall under his spell. He is to mathematics what Shakespeare is to literature and Mozart to music: universal and sui generis.


Euler Through Time: A New Look at Old Themes by V S Varadarajan:

Hindustan Book Agency;;

Amazon India link:



Some fun – Math Late Show with David Letterman and Daniel Tammet

You and your research ( You and your studies) : By Richard Hamming, AT and T, Bell Labs mathematician;

Although the title is grand (and quite aptly so)…the reality is that it can be applied to serious studies for IITJEE entrance, CMI entrance, highly competitive math olympiads, and also competitive coding contests…in fact, to various aspects of student life and various professional lifes…

Please read the whole article…apply it wholly or partially…modified or unmodified to your studies/research/profession…these are broad principles of success…


What motivated Einstein?

The most beautiful thing that we can experience is the mysterious. It is the source of all true art and sciences.

— Albert Einstein, in What I believe, 1930.

E. T. Bell’s Men of Mathematics, John Nash, Jr., genius mathematician, Nobel Laureate and Abel Laureate; and Albert Einstein

(From A Beautiful Mind by Sylvia Nasar)

The first bite of mathematical apple probably occurred when Nash at around age thirteen or fourteen read E. T. Bell’s extra ordinary book Men of Mathematics — an experience he alludes to in his autobiographical essay (of Nobel Prize, Economics) Bell’s book, which was published in 1937, would have given Nash the first glimpse of real mathematics, a heady realm of symbols and mysteries entirely unconnected to the seemingly arbitrary and dull rules of arithmetic and geometry taught in school or even in the entertaining but ultimately trivial calculations that Nash carried out in the course of chemistry and electrical experiments.

Men of Mathematics consists of lively — and, as it turns out, not entirely accurate — biographical sketches. Its flamboyant author, a professor of mathematics at California Institute of Technology, declared himself disgusted with “the ludicrous untruth of the traditional portrait of the mathematician” as a “slovenly dreamer totally devoid of common sense.” He assured his readers that the great mathematicians of history were an exceptionally virile and even adventuresome breed. He sought to prove his point with vivid accounts of infant precocity, monstrously insensitive educational authorities, crushing poverty, jealous rivals, love affairs, royal patronage, and many varieties of early death, including some resulting from duels. He even went so far in defending mathematicians as to answer the question : “How many of the great mathematicians have been perverts?” None, was his answer. ‘Some lived celibate lives, usually on account of economic disabilities, but the majority were happily married…The only mathematician discussed here whose life might offer something of interest to a Freudian is Pascal.’ The book became a bestseller as soon as it appeared.

What makes Bell’s account not merely charming, but intellectually seductive, are his lively descriptions of mathematical problems that inspired his subjects when they were young, and his breezy assurance that there were still deep and beautiful problems that could be solved by amateurs, boys of fourteen, to be specific. It was Bell’s essay on Fermat, one of the greatest mathematicians of all time, but a perfectly conventional seventeenth century French magistrate, whose life was “quiet, laborious and uneventful,” that caught Nash’s eye. The main interest of Fermat, who shares the credit for inventing calculus with Newton and analytic geometry with Descartes, was number theory — “the higher arithmetic.” Number theory, investigates the natural relationships of those common whole numbers 1, 2, 3, 4, 5…which we utter almost as soon as we learn to talk.

For Nash, proving a theorem known as Fermat’s (Little) Theorem about prime numbers, those mysterious integers that have no divisors besides themselves and one produced an epiphany of sorts. Often mathematical geniuses, Albert Einstein and Bertrand Russell among them recount similar revelatory experiences in early adolescence. Einstein recalled the “wonder” of his first encounter with Euclid at age twelve:

“Here were assertions, as for example the intersection of three altitudes of a triangle at one point which — though by no means evident — could nevertheless be proved with such certainty that any doubt appeared to be out of the question. This lucidity and certainty made an indescribable impression on me.”

Nash does not describe his feelings when he succeeded in devising a proof for Fermat’s assertion that if n is any whole number and p any prime number, then n multiplied by itself p times minus p is divisible by p. But, he notes the fact in his autobiographical essay, and his emphasis on this concrete result of his initial encounter with Fermat suggests that the thrill of discovering and exercising his own intellectual powers — as much as any sense of wonder inspired by hitherto unsuspected patterns and meanings — was what made this moment such a memorable one. That thrill has been decisive for many a future mathematician. Bell describes how success in solving a problem posed by Fermat led Carl Friedrich Gauss, the renowned German mathematician, to choose between two careers for which he was similarly talented. ‘It was this discovery …which induced the young man to choose mathematics instead of philology as his life work.”…

For those readers who are interested:

  1. Who wants to be a mathematician:

2. Resonance Journal (India):

3. Ramanujan School of Mathematics; Super30 of Prof Anand Kumar:


Nalin Pithwa




From Passive to Active Learning: India Today: Jamshed Bharucha: Aug 19 2019

(By Jamshed Bharucha; Vice Chancellor, SRM Amravati University)


An Irish lady mathematician :-) :-) :-)

Hats off to Synges!

from Nalin Pithwa.

Motivation: Zakir Hussainji on tabla; horse running

This comes from real “sadhana”. I think I had read somewhere that Zakir Hussain used to get up at 2:00 am in the morning to do “riyaaz” with his father Ustad Allah Rakha.

The same kind of passion, spirit and “sadhana”/tapasya are required to do real mathematics or to acquire “siddhi”.

I was wondering if Prof. Manjul Bhargava too could play such stuff on his tabla. 🙂 🙂 🙂

Hats off to “the” tabla maestro and another a “Fields” medalist-cim-tabla player.

-Nalin Pithwa.

The Association of Mathematics Teachers of India

(I found this v nice organization and the list of its v cheap, high quality publications in Math for kids in a blog of Mr. Gaurish Korpal.)

Quite frankly, these mathematics teachers are doing/have done profound service to India’s budding, aspiring generations of child mathematicians!! 🙂 And, also to many parents in India, who mostly (in my personal opinion) think of only law, engineering, and medicine as the only respectable professions…:-( like Professor “Virus” of the famous movie, Three Idiots ! 🙂

Hats off to AMTI !!!

Nalin Pithwa.

Make reading a part of your daily routine

Reference: The DNA newspaper, Mumbai print edition, June 25 2017, Sunday. (Rya Jetha: (The writer is a 17 year old Indian-American, currently studying at Bombay International School)

Note: Although this article may not be directly to the topics of this blog, namely, math or the kinds of levels of concentration, memory power and retention required for intensely competitive Math exams like IITJEE, RMO/INMO (of Homibhabha/TIFR), it certainly suggests the benefits of good reading habits, which directly impact the development of the intellect and overall personality of a person/student of any age. I only mean to share this “feeling” or “opinion” of mine…) – Nalin Pithwa.


What’s In It for you:

  • Reading facilitates interaction between our own experiences and the world beyond our own.
  • Reading allows us to create an image for ourselves that is intimate and personal.
  • Reading gives us the opportunity to rekindle the bond with ourselves that is neglected and wilting.


Our brains are perpetually fed with images. Images of skyscrappers billowing flames, images of celebrities strutting down the red carpet, images of scrumptious Mediterranean food platters. Images, images, images. Everywhere. On billboards as we navigate through the intersections and flyovers of our city, on our social media feeds, on television, on newspapers covers. With the tentacles of the media being ever more agile and developing finer capillaries by the second, nothing is left to imagination.

What about creating the image? What about allowing our brains to somersault, cartwheel, and back flip once in a while? How do we periodically escape the reality impressed upon us, where our brains are spoon fed and mollycoddled by our circumstances? How do we give ourselves the space to be active creators, and not dormant receivers?

The answer is reading. Fellow teenagers, I bet you just rolled your eyes, thinking “easier said than done!” I couldn’t agree more. Who has the time or inclination to read when there are standardized tests to rigorously practise for, laps to swim, tracks to run around, places to be and people to meet? It doesn’t strike me as particularly surprising when teenagers regards reading as a pastime belonging to a previous era, a time when minutes elapsed slower and free time was available in enviable abundance.

Apart from the obvious advantage of reading —- expanding our vocabularies and knowledge, improving focus and enhancing writing and comprehension skills —- I see another benefit to reading critical for teenagers in the 21st century. Reading facilitates interaction between our own experiences and the world beyond our own. Reading allows us to create an image for ourselves that is intimate and personal —- an ability we have lost as thousands of visuals assault us on a daily basis. Sidetracked by the hyperactivity of the world we live in, we lack a connection with our own experiences  and creativity, and reading gives us the opportunity to rekindle the bond with ourselves that is neglected and wilting.

Teenagers are both victims and beneficiaries of the overwhelming bombardment of the 21st century. Instead of suffering from the onslaught of sensational news and enduring the blitzkrieg of our social media feeds, how about we acclimatize ourselves to a different kind of bombardment…of suspense, of plots, of twists, of complex characteers and vibrant settings! The possibilities are endless, and I assure you, the vibrancy and exhilaration is unparalleled by any Snapchat filter, Buzzfeed article or Trump meme. How about exploring the Taliban agenda in A Thousand Splendid Suns, or viewing the world through the lens of a migrant in The Sympathizer, and Americanal, or experience the trauma of World War II through the perspective of a blind French girl in All The Light We Cannot See, or delving into the history of The Gene, or exploring British occupied Burma in The Glass Palace, or living the psychological tension in The Girl on the Train. Take your pick!


On my part, if you are interested in Math and Physics, you can try reading some of the following books:

  1. The Man Who Knew Infinity: A Life of the Genius Ramanujan by Robert Kanigel,
  2. Men of Mathematics by E. T. Bell, this is the book which had inspired (to some extent) John Nash, Jr., as a boy, and he later on became a Nobel Laureate, an Abel Laureate, whose life inspired the Hollywood movie, A Beautiful Mind. The movie in turn is based on the biography ” A Beautiful Mind” by Sylvia Nasar; you can buy these books also from Amazon India or some other place.
  3. Taming the Infinite: The Story of Mathematics by Ian Stewart.
  4. A Brief History of Time: From Big Bang to Black Holes by Stephen Hawking.
  5. The Physics of Superheroes by James Kakalios.
  6. Physics for Entertainment by Ya. Perelman;
  7. Surely, you are joking, Mr. Feynman!
  8. Sherlock Holmes, (Unabridged complete works) by Sir Arthur Conan Doyle,
  9. Super Memory: it can be yours by Shakuntala Devi,
  10. P. G. Wodehouse ! (This is, of course, for light reading and besides, who can forget The Inimitable Jeeves, the Empress of Blandings, and Bertie Wooster? 🙂 🙂 🙂 )
  11. S. Chandrasekhar, Man of Science, by Radhika Ramnath,

(I will try to suggest more such literature later, although, you can find many many like these on your own…)

– shared by Nalin Pithwa.