## 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…

https://www.cs.virginia.edu/~robins/YouAndYourResearch.html

## Tricky Trigonometry questions for IITJEE mains maths practice

Prove the following:

1. $\frac{\sin{A}}{1-\cos{A}} = \frac{1+\cos{A}-\sin{A}}{\sin{A}-1+\cos{A}}$
2. $\frac{1+\sin{A}}{\cos{A}}=\frac{1+\sin{A}+\cos{A}}{\cos{A}+1-\sin{A}}$
3. $\frac{\tan{A}}{\sec{A}-1}=\frac{\tan{A}+\sec{A}+1}{\sec{A}-1+\tan{A}}$
4. $\frac{1+\csc{A}+ \cot{A}}{1+\csc{A}-\cot{A}} = \frac{\csc{A}+\cot{A}-1}{\cot{A}-\csc{A}+1}$

Hint: Directly trying to prove LHS is RHS is difficult in all the above; or even trying to transform RHS to LHS is equally difficult; it is quite easier to prove the equivalent statement by taking cross-multiplication of the appropriate expressions. 🙂

Regards,

Nalin Pithwa.

## 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:

http://www.ams.org/publicoutreach/students/wwtbam/wwtbam

2. Resonance Journal (India):

https://www.ias.ac.in/Journals/Resonance_%E2%80%93_Journal_of_Science_Education/

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

http://www.super30.org/rsm.html

Cheers,

Nalin Pithwa

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

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

https://www.indiatoday.in/magazine/education/story/20190819-from-passive-to-active-learning-1578655-2019-08-09

## Some practical uses of maths

1. MPEG 4, audio/video/speech recognition/speaker identification/face recognition/HDTV and mp3 —- all these use logarithms and trigonometry. The special jargon is — Fourier Series, and Fourier Transforms.
2. In finance, compound interest is calculated by using a power function; the inverse problem of finding the duration of deposit is calculated using logarithms.
3. All (digital) phones are touch phones and they use DTMF (Dual Tone Multi Frequency) standard — implemented using sines and cosines.
4. Quadratic equations are used to design/model/develop certain kind of electronic amplifiers.
5. Probability theory is used in computer networks, routing of telephone calls, and also in Wall Street — stock market !!
6. There are ways to compute the numerical value of the irrational number $\pi$ up to a million digits and these ways are used to test the efficiency and efficacy of supercomputers.
7. Quadratic equations are used to study projectile motion (or to put it playfully, suppose we throw a pebble at a certain angle from horizontal ground, (angle less than 90 degrees (which would mean vertically up)) — the projectile is subject only to the force of gravity of the earth — the path or curve or trajectory of the projectile is a parabole, which is characterized by a quadratic equation. This can be easily proved using laws of straight line motion in two dimensional using resolution of vectors.

More later,

Nalin Pithwa.

## Elementary algebra: fractions: IITJEE foundation maths

Expertise in dealing with algebraic fractions is necessary especially for integral calculus, which is of course, a hardcore area of IITJEE mains or advanced maths.

Below is a problem set dealing with fractions; the motivation is to develop super-speed and super-fine accuracy:

A) Find the value of the following: (the answer should be in as simple terms as possible, which means, complete factorization will be required):

1) $\frac{1}{6a^{2}+54} + \frac{1}{3a-9} - \frac{a}{3a^{2}-27}$

2) $\frac{1}{6a-18} - \frac{1}{6a+18} -\frac{1}{a^{2}+9} + \frac{18}{a^{4}+81}$

3) $\frac{1}{8-8x} - \frac{1}{8+8x} + \frac{x}{4+4x^{2}} - \frac{x}{2+2x^{4}}$

4) $\frac{x+1}{2x^{3}-4x^{2}} + \frac{x-1}{2x^{3}+4x^{2}} - \frac{1}{x^{2}-4}$1

5) $\frac{1}{3x^{2}-4xy+y^{2}} + \frac{1}{x^{2}-4xy+3y^{2}} -\frac{3}{3x^{2}-10xy+3y^{2}}$

6) $\frac{1}{x-1} + \frac{2}{x+1} - \frac{3x-2}{x^{2}-1} - \frac{1}{(x+1)^{2}}$

7) $\frac{108-52x}{x(3-x)^{2}} - \frac{4}{3-x} - \frac{12}{x} + (\frac{1+x}{3-x})^{2}$

8) $\frac{(a+b)^{2}}{(x-a)(x+a+b)} - \frac{a+2b+x}{2(x-a)} + \frac{(a+b)x}{x^{2}+bx-a^{2}-ab} + \frac{1}{2}$

9) $\frac{3(x^{2}+x-2)}{x^{2}-x-2} -\frac{3(x^{2}-x-2)}{x^{2}+x-2} - \frac{8x}{x^{2}-4}$

More later,

Nalin Pithwa

## A magic trick!

You may have heard of a magic trick that goes like this. Take any number. Add 5. Double the result. Subtract 6. Divide by 2. Subtract 2. Now tell me your answer, and I will tell you what you have started with.

Pick a number and try it.

You can see what is going on if you let x be your original number, and follow the steps to make a formula $f(x)$ for the number you end up with.

Have fun !!!

Shared by Nalin Pithwa,

Ref: Calculus and Analytic Geometry by G B Thomas, 9th edition.

## The importance of lines and slopes

1. Light travels along straight lines. In fact, the shortest distance between any two points is the path taken by a light wave to travel from the initial point to the final point. In other words, it is a straight line. (A slight detour: using this elementary fact, can you prove the triangle inequality?)
2. Bodies falling from rest in a planet’s gravitational field do so in a straight line.
3. Bodies coasting under their own momentum (like a hockey puck gliding across the ice) do so in a straight line. (Think of Newton’s First Law of Motion).

So we often use the equations of lines (called linear equations) to study such motions.

Many important quantities are related by linear equations. Once we know that a relationship between two variables is linear, we can find it from any two pairs of corresponding values just as we find the equation of a line from the coordinates of two points.

Slope is important because it gives us a way to say how steep something is (roadbeds, roofs, stairs, banking of railway tracks). The notion of a slope also enables us to describe how rapidly things are changing. (To philosophize, everything in the observable universe is changing). For this reason, slope plays an important role in calculus.

More later,

Nalin Pithwa.

PS: Ref: Calculus and Analytic Geometry by G B Thomas and Finney; or any other book on calculus.

PS: I strongly recommend the Thomas and Finney book : You can get it from Amazon India or Flipkart:

https://www.amazon.in/Thomas-Calculus-George-B/dp/9353060419/ref=sr_1_1?crid=36S3685TG7OYF&keywords=thomas+calculus&qid=1561503390&s=books&sprefix=Thomas+%2Caps%2C259&sr=1-1

or Flipkart:

https://www.flipkart.com/thomas-calculus-1/p/itmebug5kzrnttfj?pid=9789332547278&lid=LSTBOK9789332547278CHN4GH&marketplace=FLIPKART&srno=s_1_23&otracker=AS_Query_OrganicAutoSuggest_2_9&otracker1=AS_Query_OrganicAutoSuggest_2_9&fm=SEARCH&iid=fdc8327b-756c-4f6d-aa10-b45acc900e12.9789332547278.SEARCH&ppt=sp&ppn=sp&ssid=uz7zckp71c0000001561503474614&qH=2488f76736a10369