Thursday, November 06, 2008

Prize for students


In case you read (even parts of) the book, interact with me, suggest this book to your friends and teachers, ask me really good questions, and in general take part in useful discussions on this page...

And you are a student... studying anywhere in grades 6-12th...I will award free subscriptions to mathguru.com (and maybe more websites).

So do feel free to interact here.

Errata

Here is a list of errata that have crept into the book due to one reason or the other. I'm sorry, and I hope this errata sheet will help you learn from the book without too many problems. 

Here is the errata (Last updated October 9, 2010):

Errata for Get Smart! Maths Concepts, by Gaurav Bhatnagar

If you find more errors in the book that are not mentioned in the errata list, please do leave a comment on this site. 


Post solutions/answers here

If you wan't to discuss the answers to the problems in the DO AND DISCUSS section, this is the place to do so. Submit answers by adding a comment here, or send me email at gaurav.bhatnagar at mathguru dot com.

Thursday, October 21, 2004

Oxie - A Limerick on Vikas Verma

A post-modern guy, Oxie (alias Vikas),
Is a sophisticated guy, not at all crass,
Ask him how many children—he says: "Tough to keep a tally.
But if you are in Europe, and see kids playing in a gali,
It may be something to do with me, Vikas."

~~
Vikas is no more. He died of a massive heart attack, when he was about 40 years old. He was a friend from my high school. The name Oxie was given to him in his college. 

I wrote this limerick in better times. He was a fun loving guy, and loved making people around him have fun. I will at some point write an obituary. Right now (even after 3-4 years, on 27-4-2011) I am too affected by his death to write much about him. 

Friday, August 27, 2004

Itching for a run anyone? A limerick



Just came back from a 45 min run,
Read MC’s words on running the ton,
Prevent friction; apply lubricant, before you try,
He says: You know where to apply—
So that is why I feel like scratching my buns!

~
MC is Maneesh Chandra, a school friend. His comments on running inspired this one.

Joy - A limerick



A foodie among foodies: his name is Joy,
Runs for fitness and health—this guy,
Sprints to the kitchen,
And from there to the John,
This keeps him looking like a young boy.

~
This limerick is on my high school friend Joy Basu. He made this joke about himself in our school newsgroup, and I loved it so much that I made it into a limerick. So its joint work.  

Thursday, February 26, 2004

Bhatnagar and Mishra

By Gaurav Bhatnagar 

Bhatnagar and Mishra,
began their collaboration
in Modern High School.
They wrote limericks,
about their friends,
and thought:
That is cool.

As they grew older,
they continued their work
on the CPT Framework.
On using technology
for Math education—
to make learning fun.

Over email they argued,
discussed, wrote
Back and forth,
and finished their Note.

Research in education
Was plenty of fun.
On the one hand they wrote:
Solving math problems—
they are tedious forced marches
through hostile territory;
and, in the very next breath
contradicted themselves
Saying:
By solving math problems
learners can savour
the joy of mathematical discovery!




Bhatnagar-Mishra,
Modernites,
Continued to argue,
discuss, gain insights.
Published paper after paper,
books and more—
until they passed
into popular lore.

There were three great educationists
in those times.
One was Mishra,
Another was Bhatnagar,
And the third was called
Bhatnagar-Mishra.

Thursday, December 25, 2003

Santa: A limerick


There was a guy called Santa,
Brought gifts everywhere, even in Atlanta,
But here's the thing,
Nobody heard a thing,
Because he didn't believe in unnecessary Banta.

Saturday, April 05, 2003

On Returning To Desh


Book Review: A Long Day’s Night, by Pradip Ghosh, Srishti Publishers and Distributors (2002) Rs. 295/- Reissued by Rupa (2009). Available on Amazon.


As a grad student, whenever I was with a few of my friends, three pegs down, discussing this and that—work, courses, the Buddha (I mean, of course, the advisor), teaching, research, the job scene, or whatever—we often brought up the topic of returning to desh.

We talked about the systems in oosa, and the lack of them in desh; of the good times we had in the hostel, the fundu guys we grew up with (now all in the US) and the Ajit jokes they had invented. Those who had “worked” in India were determined not to return, the others were not so sure. The nature of ABCDs was discussed, and horrors of having ABCDs as children listed. (The children were yet to come: most of us were not even married.)

I read a book recently that reminded me of these discussions.

The book is “A Long Day’s Night”, written by Professor Pradip Ghosh—formerly of IIT-Kanpur, and currently in the University of Hawaii. It is about being a research scientist in India. One day in the life of Professor Virendra Chauhan, our hero, should convince most people that the probability of returning to India AND hoping to have a research career—without considerable misery—is vanishingly small.



***


The setting of the story is the campus of an engineering school. IITK junta might find it somewhat familiar. The story is about an experimental research scientist who has bought a major piece of equipment for the lab, and found that it never met the specs. An engineer from the company is visiting to solve all problems, and the story is all about the day of this visit.

It was a long day. The visiting engineer first tried to convince Professor Chauhan that the company had made no mistakes, and the equipment meets the specs. But he dodged these arguments and constrained the engineer to admit the lapses of the company. Work began on the equipment.

And then they began facing other difficulties, all of which had to be overcome in a few hours, since the engineer had to take a flight back in the evening. Professor Chauhan doggedly refused to allow the visitor any excuse to leave the task undone. He even undertook a trip to the city in the hot afternoon, to get work done by a capable machinist.

You will have to read the book to find out whether the piece of equipment worked or not, and for the reasons why the faulty equipment was bought in the first place.

And about the night that followed the long day.



***

Pradip Ghosh’s book is just delectable. Once I began the book, I couldn’t stop, and as soon as I finished it, I just HAD to re-read the end.

I found myself really respecting the character of Professor Chauhan. So much so, that I feel calling him Virendra in this review might offend him, though I know he is a big enough guy to know that addressing him by his first name has nothing to do with the respect I have for him. This shows the skill of the author, in creating such an impression about his characters.



***

Reading about Professor Chauhan’s hot, tiring day, I came out with a lot of respect for those who are able to do good work under our desi system. I bailed out after only one year in an Indian research institution—and that too when my work was theoretical, only requiring pencil, paper, TeX and Mathematica. And here was this scientist who goes on and on. Even after buying the equipment his research requires, he is unable to make it work for years. Still, he goes on.

I have no doubt that most scientists working in Indian institutions face these difficulties. It is remarkable that so many still have the energy to continue doing good work.



***

The book also has advice for budding scientists—about choosing an area of study, an advisor, a research problem, etc., etc. Gyan dispensed by the wise Professor Chauhan, while having chai with his US bound students.

There are many other issues there that I have not described, but you may find interesting. Consider the story of Harjinder Singh, a colleague of Professor Chauhan:

Harjinder Singh, on the other hand, was a totally different kind of character. An electrical engineer specializing in circuits, thoroughly domesticated with a family of wife, two daughters, and a son, involved and astute in family matters, socially conscious, professionally active, almost perpetually unhappy about the milieu in which he lived. He struggled in a set-up in which he found that it was the system that determined and limited his professional accomplishment and not his innate and acquired abilities. He rejected, he rebelled, but totally because of social and economic constraints, could not kick and leave the system. Beyond this anguish, which was not limited to Harjinder but to many colleagues of Virendra, Harjinder was a person of much sensitivity.


***
Harjinder once told Virendra that he wanted to study history formally, but there were social forces that worked against it. Relatives, even his father, first asserted and then ruled that he should study engineering, because of better job prospects. He did well in the entrance examination to a regional engineering college, and that sealed his fate forever.
How many Harjinders do you know? I can identify many from my own friends.


***

After reading the book, you may wonder if anyone will want to return to desh. But still people do. Why do they return? Why did I return?

People keep asking me why I returned to India. Other people whom I meet, who have studied or worked in the US and have returned, are also asked this question all the time.

I really don’t know how to answer this question. Sometimes I answer: I wonder, man, I wish I had more sense. Sometimes I throw it right back and say: Why do you wish to stay in another country? Many times I just stay quiet, and leave the question unanswered.

There is only one thing that I am sure of. In our long discussions with other desi friends, we always missed the point.


***

Here is an extract from Eric Segal’s The Class explaining the urge to return home. In the words of Professor Finley, discussing Odysseus’ decision to return home from the enchanted isle of the nymph Calypso:
"Imagine our hero is offered an unending idyll with a nymph who will remain
forever young. Yet he forsakes it all to return to a poor island and a woman who, Calypso explicitly reminds him, is fast approaching middle age, which no cosmetic can embellish. A rare, tempting proposition, one cannot deny. But what is Odysseus’ reaction?”

“Goddess, I know that everything you say is true and that clever Penelope is no match for your face and figure. But she is after all a mortal and you divine and ageless. Yet despite all this I yearn for home and for the day of my returning.”


“Here,” he said, at a whisper that was nonetheless audible in the farthest corner, “is the quintessential message of the Odyssey…”


A thousand pencils poised in readiness to transcribe the crucial words to come.


“In, as it were, leaving an enchanted—and one must presume pleasantly tropical—isle to return to the cold winter winds of, shall we say, Brookline, Massachusetts, Odysseus forsakes immortality for—identity.”


***
All in all, I don’t think Professor Chauhan is unhappy in India, despite facing long days as a research scientist. This is where he belongs. He enjoys the natural beauty around him, and enjoys the company of his students and colleagues. This is evident from the author’s description of the world around him. The author is silent about the happiness that Professor Chauhan’s family brings him. But they are there. As the author says: “His is a mixed lot, like everybody else’s.”

***

I think that is enough about Professor Chauhan. Lets let him be for now.

But maybe the next occasion where you are with friends, three pegs down, discussing this and that—the invasion of Iraq, black and white, colonialism, the linear or cyclic nature of time—you will bring up A Long Day’s Night for discussion.



Pradip Ghosh, A long day's night, book review

Book Review: A Long Day’s Night, by Pradip Ghosh, Srishti Publishers and Distributors (2002) Rs. 295/- Reissued by Rupa (2009). Available on Amazon.

Friday, March 14, 2003

Experience Mathematics #29 -- Abstraction


The first time you encountered abstraction in mathematics was when you associated the number (say $5$) with five oranges and five apples. When we begin to learn mathematics, we associate numbers with specific objects. Soon we realize that we can think of the number $5$ as a concept removed from the apples and oranges. This is abstraction. Now we can apply the concept of $5$ (and also other numbers) in counting any set of objects.

Mathematics students become used to abstracting concepts into symbols that we can apply in many situations. The same skill in abstract thought helps in other domains also. For example, object oriented programming—the most useful of the programming paradigms is all about abstraction.

In Object-oriented programming, we think of everything as an object. For example, the button you press in most applications is an object. Now a button looks and behaves in much the same way in any application. So we would naturally wish to program it just once. So most language environments (like Java or Visual Basic) give us a “Class” that represents the object that is the button. (A Class is like the set we encounter in mathematics.)

However, when we use the button in a particular program, we may wish to add certain properties of our own. For example, we may like to put the word “OK” as a label on the button we wish to use. So we “instantiate” a button and set its properties that include a label “OK” that will appear on it. Further, when a user clicks on the button, the button performs an “Action”. You have to code this “Event” to tell the button what to do.

The Classes contain data (or “properties”) that are used to describe a particular member (or “instance”) of the class, functions (or “methods”) that do certain tasks, and have the ability to process messages (or “events”) that the rest of the application uses to tell the class to perform its tasks.

Friday, February 28, 2003

Experience Mathematics #28 -- How fast do functions grow?

In the last column, we talked about the explosive growth of the exponential function. The number of computers infected by the SQL Slammer worm increased dramatically, bringing the Internet crashing down in a couple of hours.

Computer scientists measure the speed of computer algorithms by comparing them to functions.
Some of the functions they use are logarithmic: $\log(n)$, linear: $n$; the power functions: $n^2, n^3, n^4,\dots $; and the exponential functions: $2^n, 10^n,$ etc.

Usually $n$ is the size of the input. Computer scientists make statements such as: The “order” of an algorithm is $n^2$. For example, if you have to sort $n$ numbers, the algorithm is of order $n^2$. This means that the computer has to make approximately $n^2$ calculations. To get an idea of which algorithms are faster, consider when $n=1000$. $\log(n)$ is just $3$. The linear function $n$ is also manageable, at $1000$. However, $n^2$ is $1,000,000$ (one million) and $n^3$, is one billion. And $10^n$ is a huge number, 1 followed by a thousand zeros. This number of calculations is more than what Deep Junior had to perform to defeat Kasparov in Chess.

All these functions go to infinity as $n$ goes to infinity. That is to say, they become bigger and bigger as $n$ becomes bigger. What matters (to computer scientists) is how fast or slow this increase is. The slower the increase, the faster the algorithm.

Logarithmic, linear and Polynomial time algorithms the only algorithms that are fast enough to work in practical situations.

Computer scientists are continuously finding faster and faster algorithms. Recently, Maninder Agarwal, Neeraj Kayal, and Nitin Saxena, of IIT, Kanpur, found a deterministic polynomial-time algorithm to determine whether a number is a prime number.

This solved a problem that mathematicians have been trying to solve for centuries.

Friday, February 07, 2003

Experience Mathematics # 27 -- The exponential function



“It only took 10 minutes for the SQL Slammer worm to race across the globe and wreak havoc on the Internet two weeks ago” is what newspapers reported on February 7, 2003. The number of computers doubled every 8.5 seconds in the first minute of the worm’s existence. So how many computers got infected in the first minute?

The number of computers infected in $t$ seconds can be modelled by the function $N(t)$, where
$$N(t)=2^{t/8.5}.$$

This is a reasonable model. In the beginning (when $t=0$) we assume that the worm has infected only $1$ computer, and indeed, $N(0)=1$. In 8.5 seconds, the number becomes $N(8.5)=2$, so it doubles. In another $8.5$ seconds, $N(17)=4$. This doesn’t sound like very fast growth, but at the end of one minute ($t=60$) the number becomes more than $133$. In two minutes, $28995$ computers are infected, and in $5$ minutes, the number of computers infected is in the billions. Which means that the rate of growth must have slowed down, because there aren’t so many computers in the world! I hope this helps you understand why it caused the slowdown of the Internet traffic in Korea.

$N(t)$ is an example of an exponential function, which increases very fast. The prototypical example is THE exponential function, $f(x)=e^x$. Here the number e is an irrational number (just like the famous $\pi$, whose value is approximately $2.7182818\dots$. The graph of the function (made using desmos.com) is as follows.


The exponential function is not symmetric across the $y$-axis, nor across the origin. That is to say, it is not an even function or an odd function. However, consider the functions:
$$E(x)=(e^x+e^{-x})/2,$$ and $$O(x)=(e^x-e^{-x})/2.$$ $E$ is an even function, and $O$ is an odd function, and the exponential function is the sum of these functions. Draw the function $E(x)$ from $x=-5$ to $x=5$ using MS-Excel (update: try www.desmos.com), and see what the graph looks like. Does it look like a clothesline secured at its two ends?

Friday, January 31, 2003

Experience Mathematics # 26 -- Symmetries


There are two kinds of symmetries in a function. A function may be symmetric across the $y$-axis, or symmetric across the origin. (If a curve is symmetric across the $x$-axis, it is not a function. Can you tell why?)

For example, the function $f(x)= x^2$ is an example of a function that is symmetric across the $y$-axis.


 This symmetry is obvious from the graph. An algebraic way to see that the function $f(x)= x^2$ is symmetric across the $y$-axis, is to replace $x$ by $–x$ in the formula, and note that:
$f(–x) =f(x)$ (since $(–x)^2=x^2$).
For example, the $y$-coordinate corresponding to the point $–2$ is the same as that corresponding to $2$.
The function $f(x)= x^3$ is an example of a function that is symmetric across the origin.


Each point (for example the point $(2, 8)$) maps to a symmetric point (the point $(-2, -8)$) in the graph. An algebraic way to notice that this function is symmetric across the origin is to note that
$f(–x) =–f(x)$ (because $(–x)^3= –x^3$).

Functions symmetric across the $y$-axis are called even functions, and functions symmetric across the origin are called odd functions.

What is remarkable is that any function defined on the set of real numbers can be written as a sum of an odd and an even function. Can you figure out a way to write the exponential function $f(x)=e^x$ as the sum of an even and an odd function? The curve formed by a hanging clothesline  appears in the answer to this question.

Thursday, January 23, 2003

Experience Mathematics # 25 - Functions


A ball thrown in the air follows the path of a parabola. Parabolas are modelled by a function of the form $p(x)=ax^2+bx+c$, where $a$, $b$ and $c$ are real numbers. This kind of function—a polynomial of degree 2—is called a Quadratic Function. While we will not formally define functions, it is helpful to get an intuitive idea of functions from several points of view.

One point of view is to think of functions as a rule. For example, consider the quadratic function:
$f(x)=1-x^2$. Every real number $a$ corresponds to a unique real number denoted by $f(a)$ obtained by replacing $x$ by $a$ in the above equation. For example,
$f(0)=1, f(1)=0, f(-2)=-3.$

This suggests that we can also think of a function as an input-output machine. For each input $a$ we have a unique output $f(a)$. The set of possible input values (in this case the set $R$ of real numbers) is called the domain of the function.

Imagine making a table of all the input-output values of the function. (There are an infinite number of elements in the domain, so you can only imagine making a table!) All these values can be plotted on the coordinate plane. The input values are the $x$-coordinates and the output values are the $y$-coordinates.

If we do this, we will get a graph of the function. We denote the graph by $y=f(x)$, (or $y= 1-x^2$).

This is the third way of thinking about a function: as a graph. The graph is shown below.


Note that this parabola is symmetric about the $y$-axis. It meets the $x$-axis when $x=1$ and when $x=-1.$ These are (graphically speaking) the solutions of the equation $1-x^2=0$. The function has a maximum when $x=0$, corresponding to the highest point a ball reaches, when it is thrown in the air.

Friday, January 10, 2003

Experience Mathematics # 24 -- The Calculus


Happy New Year. The earth has finished another revolution around the sun, taking a little more than 365 days to do so. Meanwhile, the moon continues to rotate around the earth, the planets around the sun, and the same forces that make these things move in an elliptical path ensure that a ball thrown up in the air always falls down, or that a ball thrown in the air (towards a friend) takes a parabolic path.

Over this and the next few columns, I will discuss these natural motivations that are behind the notions that you encounter as you study the Calculus.

The first concept is that of a function. Mathematicians were already familiar with curves from Euclidean and coordinate geometry by 1600 or so A.D. It was natural to begin modelling various physical phenomena with functions. For example, $y=1-x^2$ models the parabola. For each value of the input $x$, we get a unique output $y$. If you plot the curve in the coordinate plane, you obtain a parabola.

It was natural to do two things. To figure out laws that can explain why a ball thrown in the air follows a path traced by such a curve. This led to the laws of Gravitation. And the other thing is to use these laws to predict the answers to common questions that arise. For example: How high will the ball go? How far will the ball go? Given the curve, when does the curve go up (increase)? And when does it come down (decrease)? We will consider such questions and relate them to what you encounter in Calculus.

Curves such as the circle ($x^2+y^2=1$) are not functions since there is not one output $y$ for each input $x$. For example, for $x=0, y$ can be $1$ or $–1$. So, every curve does not give rise to a function.