Monthly Archives: December 2015

A solution to a S L Loney Part I trig problem for IITJEE Advanced Math

Exercise XXVII. Problem 30.

If a, b, c are in AP, prove that $\cos{A}\cot{A/2}$, $\cos{B}\cot{B/2}$, $\cos{C}\cot{C/2}$ are in AP.

Proof:

Given that $b-a=c-b$

TPT: $\cos{B}\cot{B/2}-\cos{A}\cot{A/2}=\cot{C/2}\cos{C}-\cos{B}\cot{B/2}$. —— Equation 1

Let us try to utilize the following formulae:

$\cos{2\theta}=2\cos^{2}{\theta}-1$ which implies the following:

$\cos{B}=2\cos^{2}(B/2)-1$ and $\cos{A}=2\cos^{2}(A/2)-1$

Our strategy will be reduce LHS and RHS of Equation I to a common expression/value.

$LHS=(\frac{2s(s-b)}{ac}-1)(\frac{\sqrt{\frac{(s)(s-b)}{ac}}}{\sqrt{\frac{(s-a)(s-c)}{ac}}})-\cos{A}\cot{(A/2)}$

which is equal to

$(\frac{2s(s-b)}{ac}-1))(\frac{\sqrt{\frac{(s)(s-b)}{ac}}}{\sqrt{\frac{(s-a)(s-c)}{ac}}})-(2\cos^{2}(A/2)-1)\frac{\cos{A/2}}{\sin{A/2}}$

which is equal to

$(\frac{2s(s-b)}{ac}-1))(\frac{\sqrt{\frac{(s)(s-b)}{ac}}}{\sqrt{\frac{(s-a)(s-c)}{ac}}})-(\frac{2s(s-a)}{bc}-1)\frac{\sqrt{\frac{s(s-a)}{bc}}}{\sqrt{\frac{(s-b)(s-c)}{bc}}}$

which is equal to

$(\frac{2s(s-b)}{ac}-1))\sqrt{s(s-b)}{(s-a)(s-c)}-(\frac{2s(s-a)}{bc}-1)\sqrt{\frac{s(s-a)}{(s-b)(s-c)}}$

which in turn equals

$\sqrt{\frac{s}{(s-a)(s-b)(s-c)}}((\frac{2s(s-b)}{ac}-1)(s-b)-(\frac{2s(s-a)}{bc}-1)(s-a))$

From the above, consider only the expression, given below. We will see what it simplifies to:

$\frac{2s(s-b)^{2}}{ac}-(s-b)-\frac{2s(s-a)^{2}}{bc}+(s-a)$

$=\frac{2s(s-b)^{2}}{ac}-\frac{2s(s-a)^{2}}{bc}+b-a$

$=(\frac{2s}{c})(\frac{(s-b)^{2}}{a}-\frac{(s-a)^{2}}{b})+b-a$

$=\frac{2s(s-b)^{2}}{ca}-\frac{2s(s-a)^{2}}{bc}+c-b$ —- Equation II.

Now, consider RHS of Equation I. Let us see if it also boils down to the above expression after simplification.

$RHS=\cot{(C/2)}\cos{C}-\cos{B}\cot{(B/2)}$

$=(2\cos^{2}{(C/2)}-1)\cot{(C/2)}-(2\cos^{2}({B/2})-1)\cot{(B/2)}$

$=(\frac{2s(s-c)}{ab}-1)\frac{\sqrt{\frac{s(s-c)}{ab}}}{\sqrt{\frac{(s-b)(s-a)}{ab}}}-(\frac{2s(s-b)}{ac})\frac{\sqrt{\frac{s(s-b)}{ac}}}{\sqrt{\frac{(s-a)(s-c)}{ac}}}$

$= \sqrt{\frac{s}{(s-a)(s-b)(s-c)}}((\frac{2s(s-c)}{ab}-1)(s-c)-(\frac{2s(s-b)}{ac}-1)(s-b))$

From equation II and above, what we want is given below:

$\frac{2s(s-c)^{2}}{ab}-(s-c)-\frac{2s(s-b)^{2}}{ac}+(s-b)=\frac{2s(s-b)^{2}}{ac}-\frac{2s(s-a)^{2}}{bc}+c-b$

that is, want to prove that $c(s-c)^{2}+a(s-a)^{2}=2b(s-b)^{2}$

but, it is given that $a+c=2b$ and hence, $c=2b-a$, which means $a+c-b=b$ and $b-a=c-b$

that is, want to prove that

$c(a+b-c)^{2}+a(b+c-a)^{2}=2b(a+c-b)^{2}=2b^{3}$

i.e., want: $c(a+b-c)^{2}+a(b+c-a)^{2}=2b^{3}$

i.e., want: $(2b-a)(a+b-2b+a)^{2}+a(b-a+2b-a)^{2}=2b^{3}$

i.e., want: $(2b-a)(2a-b)^{2}+a(3b-2a)^{2}=2b^{3}$

Now, in the above, $LHS=(2b-a)(4a^{2}+b^{2}-4ab)+a(9b^{2}+4a^{2}-12ab)$

$= 8a^{2}b+2b^{3}-8ab^{2}-4a^{3}-ab^{2}+4ab^{2}+9ab^{2}+4a^{3}-12a^{2}b$

$= 2b^{3}$.

Hence, $LHS+RHS$.

QED.

Stanislaw Ulam

It is still an unending source of surprise for me to see how a few scribbles on a blackboard or on a sheet of paper could change the course of human affairs. — Stanislaw Ulam.

This remark of Stanislaw Ulam’s is particularly appropriate to his own career. Our world is very different today because of Ulam’s contributions in mathematics, physics, computer science, and the design of nuclear weapons.

While still a schoolboy in Lwów, then a city in Poland, he signed his notebook “S. Ulam, astronomer, physicist and mathematician.”

Of these early interests perhaps it was natural that the talented young Ulam would eventually be attracted to mathematics; it is in this science that Poland has made its most distinguished intellectual contributions in this century. Ulam was fortunate to have been born into a wealthy Jewish family of lawyers, businessmen, and bankers who provided the necessary resources for him to follow his intellectual instincts and his early talent for mathematics. Eventually Ulam graduated with a doctorate in pure mathematics from the Polytechnic Institute at Lwów in 1933. As Ulam notes, the aesthetic appeal of pure mathematics lies not merely in the rigorous logic of the proofs and theorems, but also in the poetic elegance and economy in articulating each step in a mathematical presentation. This very fundamental and aristocratic form of mathematics was the concern of the school of Polish mathematicians in Lwów during Ulam’s early years.

The pure mathematicians at the Polytechnic Institute were not solitary academic recluses; they discussed and defended their theorems practically every day in the coffeehouses and tearooms of Lwów. This deeply committed community of mathematicians, in pursuing their work through collective discussion in public, allowed talented young students like Ulam to observe the intellectual excitement and creativity of pure mathematics. Eventually young Ulam could participate on an equal footing with some of the most distinguished mathematicians of his day. The long sessions at the cafes with Stefan Banach, Kazimir Kuratowski, Stanislaw Mazur, Hugo Steinhaus, and others set the tone of Ulam’s highlyverbal and collaborative style early on. Ulam’s early mathematical work from this period was in set theory, topology, group theory, and measure. His experience with the lively school of mathematics in Lwów established Ulam’s lifelong, highly creative quest for new mathematical and scientific problems.

As conditions in prewar Poland deteriorated, Ulam welcomed opportunities to visit Princeton and Harvard, eventually accepting a faculty position at the University of Wisconsin. As United States involvement in World War II deepened, Ulam’s students and professional colleagues began to disappear into secret government laboratories. Following a failed attempt to contribute to the Allied war effort by enlisting in the U. S. military, Ulam was invited to Los Alamos by his friend John von Neumann, one of the most influential mathematicians of the twentieth century. It was at Los Alamos that Ulam’s scientific interests underwent a metamorphosis and where he made some of his most far reaching contributions.

By virtue of his defense work at the Los Alamos Laboratory, Ulam enjoyed many advantages not available to academic scientists. Chief among these was his early access to the most powerful and fastest computers in existence. For several decades after the war, the computing facilities at the national weapons labs far exceeded those available to university scientists working on non classified research. This was an advantage that Ulam exploited in a variety of remarkable ways.

The growth of powerful computers was initially driven by the war effort. At the beginning of World War II there were no electronic computers in the modern sense,only a few electromechanical relay machines. During the war, scientists at the University of Pennsylvania and at the Aberdeen Proving Ground in Maryland developed the ENIAC, the Electronic Numerical Integrator and Computer, which had circuitry specifically designed for computing artillery firing tables for the Army. By modern standards, this early computer was extremely slow and elephantine: the ENIAC operating at the University of Pennsylvania in 1945 weighed thirty tons and contained about eighteen thousand vacuum tubes with 500,000 soldered connections. While on a visit to the University of Pennsylvania in 1944, John von Neumann was inspired to design an electronic computer that could be programmed in the modern sense, one which could be instructed to perform any calculation and would not be restricted to computing artillery tables. The new computer would have circuits that could perform sequences of fundamental arithmetic operations such as addition and multiplication. Von Neumann desired a more flexible computer to solve the mathematically difficult A bomb implosion problem being discussed at Los Alamos. The first electronic computer at Los Alamos, however, known as the MANIAC(Mathematical Analyzer, Numerical Integrator and Computer), was not available until 1952.

One of Ulam’s early insights was to use the fast computers at Los Alamos to solve a wide variety of problems in a statistical manner using random numbers, a method which has become appropriately known as the Monte Carlo method. It occurred to Ulam during a game of solitaire that the probability of various outcomes of the card game could be determined by programming a computer to simulate a large number of games. Newly selected cards could be chosen from the remaining deck at random, but weighted by the probability that such a card would be the next selected. The computer would use random numbers whenever an unbiased choice was necessary. When the computer had played thousands of games, the probabilty of winning could be accurately determined. In principle the probability of solitairesuccess could be rigorously calculated using probabilty theory rather than computers. However, this approach is impossible in practice since it would involve too many mathematical steps and exceedingly large numbers. The advantage of the Monte Carlo method is that the computer can be efficiently programmed to execute each step in a particular game according to known probabilities and the final outcome can be determined to any desired precision depending on the number of sample games computed. The game of solitaire is an example of how the Monte Carlo method can be used to solve otherwise intractable problems with brute computational power.

Stanislaw Ulam had formidable memory power and laser like concentration. He would do all the deep, complicated, esoteric math in his head not requiring the use of paper and pencil also. He was endowed with exceptional charisma. In his autobiography (Adventures of a Mathematician) (also, this article is based on this source), he mentions that as a child, he was mesmerized by pictures in an astronomy book, bought some more astronomy books, even bought a telescope, listened to public lectures on relativity, and believed that the thirst to go deeper into astronomy brought him to mathematics.

More later,

Nalin Pithwa

A Note for the Teachers of Mathematics

Ref: CBSE VI Mathematics

Mathematics has an important role in our life, it not only helps in day-to-day situations

but also develops logical reasoning, abstract thinking and imagination. It enriches life

and provides new dimensions to thinking. The struggle to learn abstract principles develops

the power to formulate and understand arguments and the capacity to see interrelations

among concepts. The enriched understanding helps us deal with abstract ideas in other subjects

as well. It also helps us understand and make better patterns, maps, appreciate area and

volume and see similarities between shapes and sizes. The scope of Mathematics includes

many aspects of our life and our environment. This relationship needs to be brought out at all

possible places.

Learning Mathematics is not about remembering solutions or methods but knowing

how to solve problems. We hope that you will give your students a lot of opportunities to

create and formulate problems themselves. We believe it would be a good idea to ask them

to formulate as many new problems as they can. This would help children in developing an

understanding of the concepts and principles of Mathematics. The nature of the problems

set up by them becomes varied and more complex as they become confident with the ideas

they are dealing in.

The Mathematics classroom should be alive and interactive in which the children should

be articulating their own understanding of concepts, evolving models and developing

definitions. Language and learning Mathematics have a very close relationship and there

should be a lot of opportunity for children to talk about ideas in Mathematics and bring in

their experiences in conjunction with whatever is being discussed in the classroom. There

should be no obvious restriction on them using their own words and language and the shift

to formal language should be gradual. There should be space for children to discuss ideas

amongst themselves and make presentations as a group regarding what they have

understood from the textbooks and present examples from the contexts of their own

experiences. They should be encouraged to read the book in groups and formulate and

express what they understand from it.

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Nalin Pithwa

Some questions on progressions for IITJEE Mains

(1) If $\log_{2}(5.2^{x}+1)$, $_{4}(2^{1-x}+1)$ and 1 are in AP, then x is equal to

(a) $\frac{\log (5)}{\log (2)}$

(b) $\log_{2}(2/5)$

(c) $1-\frac{\log (5)}{\log (2)}$

(d) $\frac{\log (2)}{\log (5)}$

(2) Values of the positive integer m for which $n^{m}+1$ divides $1+n+n^{2}+ \ldots+n^{127}$ are

(a) 8 (b) 16 (c) 32 (d) 64

(3) If p, q, r are positive and are in AP, the roots of the quadratic equation $px^{2}+qx+r=0$ are all real for

(a) $|\frac{r}{p}-7| \geq 4\sqrt{3}$ (b) $|\frac{q}{r}-4| \geq 2\sqrt{3}$

(c) $|\frac{p}{r}-7| \geq 4\sqrt{3}$ (d) $|\frac{q}{p}-4| \geq 2\sqrt{3}$

(4) If sum of the GP p,1, $\frac{1}{p^{2}}$, $\frac{1}{p^{3}}$, … is 9/2, the value of p is

(a) 3

(b) 2/3

(c) 3/2

(d) 1/3

(5) The roots of $x^{3}+bx^{2}+cx+d=0$ are

(a) in AP if $2b^{3}-9bc+27d=0$

(b) in GP if $b^{3}d=c^{3}$

(c) in GP if $27d^{3}=9bcd^{2}-2c^{3}d$

(d) equal if $c^{3}=b^{3}+3bc$

Nalin Pithwa

Kinds of people

There are 10 kinds of people in the world; those who understand binary numerals, and those who don’t.

More fun later, happy holidays 🙂

Nalin Pithwa

Purpose of this blog and references I used

The purpose of this blog is to present some lecture notes freely to aspirants/students of IITJEE Main and Advanced and RMO and INMO (conducted by TIFR and Homi Bhabha Center). I have found that putting lecture notes as blogs helps students revise on their own later as per their convenient time and mood. I am only a coach/mentor/tutor/teacher/guide for these exams and do not claim any glory/credit as a real mathematician, whose books/literature I use to prepare my lecture material.

I myself continue to read/study/learn mathematics from classics and other books available from Amazon India. The publishers I like are Dover Books, Springer India/international, Hindustan Book Agency and MAA and AMS.

To prepare my lectures, I use several references, old and recent, and some of which I have used are given below:

a) For Number Theory: I like to give a tangible feeling/experimental/programming approach to basic Number Theory ( for this I use the book, “A Friendly Introduction to Number Theory” by Dr. Joseph Silverman. For example, in this blog, I have reproduced “Pythagorean Triples”, “Pythagorean Triples on Unit Circle” and a few other articles verbatim from this book. Dr. Joseph Silverman is a giant mathematician, whose other books on elliptic curves I use to learn/implement cryptography codes.

I thank Mr. Nigel Childs for pointing out my error in not acknowledging this earlier. Elsewhere, in my blogs/Lectures I have acknowledged the sources, and sometimes I have forgotten (Sorry!!).

b) For Number Theory : other references I have used for these blog articles are as follows: Elementary Number Theory by David Burton and some books of Prof Titu Andreescu.

c) For Fun with Mathematics: I have used Prof Ian Stewart’s book(s). I also plan to use Martin Gardner’s literature.

d) For Calculus based articles: I have used the book “Understanding Mathematics” by C. Musili et al.

e) For Combinatorics: I have used books of Laslzo Lavasz and Sharad Sane.

This also applies to my other blog https://madhavamathcompetition.wordpress.com

Regards,

Nalin Pithwa

Ramanujan’s gift: solutions to elliptic curves

(The following article appeared in The Hindu, Nov 30, 2015 authored by Shubashree Desikan)

As numbers go by, 1729, the Hardy-Ramanujan number is not new to math enthusiasts. But, now this number has triggered a major discovery — on Ramanujan and the theory of what are known as elliptic curves.

The anecdote goes that once when Hardy visited Ramanujan who was sick, Hardy remarked: “I had ridden in taxivab number 1729, and it seems to me a rather dull number. I hope it was not an unfavourable omen.” To this, Ramanujan had replied, “No, it is a very interesting number. it is the smallest number expressible as the sum of two cubes in tow different ways.”

“Yes, $1729=9^{3}+10^{3}=12^{3}+1^{3}$ .”

This  story is often narrated to explain Ramanujan’s familiarity with numbers but not more than that. Recent discoveries have brought to light that it was far from coincidence that Ramanujan knew the properties of 1729. There are now indications that he had, in fact, been looking at more general structures of which this number was but an example.

Mathematicians Ken Ono and Andrew Granville were leafing through Ramanujan’s manuscripts at the Wren Library in Cambridge University, two years ago, when they came across the equation $9^{3}+10^{3}=12^{3}+1^{3}$, scribbled in a corner. Recognizing the representations of the number 1729, they were amused at first; then they looked again and found that there was another equation on the same page that indicated Ramanujan had been working even then, on a famous seventeenth century problem known as Fermat’s Last Theorem (proved by Andrew Wiles in 1994).

“I thought I knew all of the papers there, but to my surprise,we found one page with near misses to the Fermat equations,” writes Dr. Ono, who is also a Ramanujan scholar in an email to this correspondent. Having a sneaking suspicion that Ramanujan had a secret method that gave him his amazing formulas, Dr. Ono returned to Emory University and started working on these leads with his PhD student Sarah Trebet Leder.

“Together, we worked backwards through Ramanujan’s notes, and we figured out his secret. …[Ramanujan] arrived at the formulae on this page by producing a much more general identity. One which I recognized as a K3 surface ( a concept that mathematician Andrew Wiles used for solving Fermat’s Last Theorem), an object that mathematicians did not discover until the 1960s,” notes Dr. Ono.

Ramanujan died in 1920, long before mathematicians discovered the K3 surfaces, but from research done by Dr. Ono and Trebat Leder, it transpires that he knew these functions long before. Dr. Ono continues, ” Ramanujan produced so many mysterious formulae, which can be misunderstood at first glance. We have come to learn that Ramanujan was perhaps the greatest anticipator of mathematics. His bizarre methods and formulas have repeatedly offered hints of the future in mathematics. In this case, we have added to Ramanujan’s legend.”

Commenting on their own work on this, he says, “Ramanujan anticipated the theory of K3 surfaces before anyone had the merest glimpse. These surfaces are now at the forefront of research in mathematics and physics. In addition to adding to Ramanujan’s legacy Sarah and I were able to apply his formulas to a problem in number theory (finding large rank elliptic curves) and his formulae immediately set the record on the problem. We hardly had any work to do. Ramanujan’s formula was a gift to us.”

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