Showing posts with label Why I love math so much. Show all posts
Showing posts with label Why I love math so much. Show all posts

Monday, April 25, 2011

Identities and Mathematical Intuition: Talk in DPS - Dwarka to DPS Math Teachers

On April 18th, I gave a talk on Identities to Delhi Public School (DPS) Math teachers  attending a training conference/workshop. The teachers were from DPSs all over the country and teach in senior school (XIth-XIIth).


The overall idea of the talk was to organize information about identities according to the three kinds of mathematical intuition I have spoken about earlier. The three kinds of mathematical intuition are: Symbolic, graphical or physical intuition, and structural intuition. These are motivated by the following quote:
…some mathematicians are more endowed with the talent of making pictures, others with that of juggling symbols and yet others with the ability of picking a flaw in an argument.
~Gian Carlo Rota 

Wednesday, February 09, 2011

In Praise of an Elementary Identity of Euler

After many years, a new math paper. Its mostly a survey of my favorite identities, but has some new identities too. The new results have been checked (as typeset in the paper) using Maxima.  I have tried to write the first few sections so that  anyone can read and appreciate it.
I would appreciate any comments, typos, etc.

Update (March 16, 2011):
Presentation from: Georgia Southern q-Series conference, March 15. Here is a link.

Update (June 11, 2011): The paper is published by the Electronic J. of Combinatorics, Vol 18 (2), P13 44 pp. Download.


Keywords:
Telescoping, Fibonacci Numbers,  Pell Numbers, Derangements, Hypergeometric Series, Fibonacci Polynomials,  q-Fibonacci Numbers,  q-Pell numbers, Basic Hypergeometric Series, q-series, Binomial Theorem, q-Binomial Theorem, Chu--Vandermonde sum, q-Chu--Vandermonde sum, Pfaff--Saalschutz sum, q-Pfaff--Saalschutz sum, q-Dougall summation, very-well-poised 6 phi 5 sum, Generalized Hypergeometric Series, WZ Method

Sunday, February 06, 2011

My Mathematical Forefathers

From time to time, I look at the Mathematics Genealogy Project, and search for my own mathematical tree. I was happy to note that I am a direct descendant of Gauss and of Leibnitz. What I noticed today, was that I am a mathematical cousin of Saroj Malik, my teacher in Hindu College, who taught me abstract algebra and elementary number theory. We branch out at Gauss.

Here is the complete list of my mathematical forefathers.

  • Friedrich Leibnitz
  • Jakob Thomasius
  • Otto Mencke
  • Johann Christoph Wichmannshausen
  • Christian August Hausen
  • Abraham Gotthelf Kästner
  • Johann Friedrich Pfaff
  • Carl Friedrich Gauß
  • Christoph Gudermann
  • Karl Theodor Wilhelm Weierstraß
  • Leo Königsberger
  • Georg Alexander Pick
  • Charles Loewner
  • Adriano Mario Garsia
  • Stephen Carl Milne
  • Gaurav Bhatnagar


Monday, August 02, 2010

Math Problem Book - Grade 6

Here is a book made from some problem sets I used to give Tejasi and her friends in my garage. The explanations have been added later on. Most of the kids who took these problem sets benefited...in the sense that they began doing well in exams. Here is a collection of the problem sets for grade 6.

Click here to download/view Math Problem Book - Grade 6

Sunday, August 01, 2010

Experience Mathematics

A book I wrote long ago. Recently, I re-edited it based on comments made by Professor Dick Askey. Looking for a publisher, but meanwhile here it is for my friends and their kids...

Click here to view the PDF File

Friday, June 04, 2010

The q-disease

Special Functions,
by Andrews, Askey and Roy.
Here's a belated review,
and a thank you.

~*~*~*~*~*~

Beauty in mathematics,
said Polya,
is seeing the truth
without effort.

Everything
in The Book
is as elegant,
as could be.

Everything
as simple,
as effortless,
as should be.

Everything
as beautiful,
as it is.

~*~*~*~*~*~

Thursday, February 12, 2009

Math Poettary - Infinite

Check out the post here.

A brief explanation

The post mentions the sphere as a "one-point compactification" of the (complex) plane (by adding a point at infinity). The property of the sphere being compact somehow makes it a little closer to being "finite" and therefore easier to handle. But to understand more precisely what all this means you need to take a good course in Complex Analysis or Topology.

When studying complex analysis, I thought that the theorems are simpler, more beautiful, and closer to the finite case than analogous theorems in Real Analysis. I don't know whether that is due to the relationship with the sphere, but I suspect it is so.

Here is an example: You know that a polynomial p(x) with real coefficients (and a finite degree) can be written as a product of factors of the form (x-a) where a is a zero of the polynomial. (The root a is of-course a complex number). Turns out, under certain conditions, we can write a function (which can be viewed as an infinite series) as an (infinite) product of its zeros. For example, consider this formula:

Euler's Product:

Euler Product

(The formula above taken from Wikipedia's entry on the Wallis Product.)

The formula looks nicer if you replace x by (pi)x. Then the expression on the left has zeros at +1, +2, +3, ... and -1, -2, -3, .... And on the right you get factors of the form (1-x/n)(1+x/n) which is zero for x = +n and -n.
In fact, the way we write the product is something to do with making the product "converge" (or make sense).

This formula is definitely something I will write about one day. I think I need to pick up a complex analysis book again...its been too long...and have almost forgotten the beautiful stuff I used to see everyday.

Infinite


Infinite
Originally uploaded by punyamishra

Infinite plane
made compact
becomes a sphere.

Still infinite,
but compact,
somehow closer to
being finite.

GB # 23 (also Math Poettary)

Look here too.

Friday, January 30, 2009

Discover


Discover
Ambigram by punyamishra



The scientific training,
teaches how to discover.

The artistic training
teaches how to uncover.

The mathematician uncovers,
only to discover --
new things to uncover.

GB #18 (Also Math Poettary)

Wednesday, January 28, 2009

Isosceles Triangle

An Isosceles Triangle
looks in the mirror,
and finds itself un-reversed.
"My base angles are equal,"
it says.

Math Poettary #2

Saturday, January 10, 2009

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.

Friday, December 20, 2002

Experience Mathematics #23 -- Ambigrams (by Punya Mishra)

Symmetry is important in mathematics and in art. Today we will look at a special kind of wordplay based on ideas of symmetry and figure and ground. Consider the word below: 



Can you read it? Now turn the page you are holding upside down and try reading it that way. The word stays the same. This image/word has rotational symmetry—essentially, it stays the same when rotated 1800.

Such visual wordplays are called ambigrams. The word “ambigram” was first coined by the cognitive scientist Douglas Hofstadter. Here is an ambigram of the word ambigram itself.


Ambigrams can be of many different kinds. For instance consider the word “logical” below.



This word has reflection symmetry i.e. it will read the same even when reflected in a mirror.

Some ambigrams are not about symmetry as much as they are about reading words in multiple ways. Here is one titled “good-evil” Can you see both words? Look carefully. This is similar to figure-ground paintings by M. C. Escher.



Creating ambigrams is great fun. Why don’t you try creating some yourself? If you want to see more examples of such wordplay you can search on Google or go to my wordplay gallery: http://punya.educ.msu.edu


This guest column has been written by Professor Punya Mishra, College of Education, Michigan State University, USA. You can email him at punya@msu.edu

Friday, December 13, 2002

Experience Mathematics # 22: The mobius strip

Take a long, thin strip of paper, give it a half twist, and paste the two ends together. What you get is a mobius strip (see picture).



Compare the mobius strip with the cylinder, which you get if you don’t give a half twist. A mobius strip has only one surface. Can you see why? Draw a line along the edge and keep going on. Eventually, you will arrive at the starting point. A cylinder, on the other hand, has an inside and an outside surface. The artist Escher portrayed this idea in Mobius Strip II (woodcut, 1963) (see picture).



If you cut the paper cylinder in half, you will get two cylinders. However, if you cut the mobius strip, you will get something very similar to a mobius strip. How many half-twists does the cut mobius strip have?

The mobius strip is one of the many surfaces that appear in Topology, a branch of mathematics. Topologists have been described as the mathematicians who cannot tell the difference between a coffee mug and a doughnut. This is because a mug can be “continuously deformed” to become a doughnut. A doughnut (which is a tyre tube, topologically speaking) cannot be transformed into a sphere. So, according to topologists, the tyre tube is not the same as a sphere, but a coffee mug is homotopic to a doughnut.

Pic credits: Both were stolen off the web, I don't know from where. 

Thursday, December 05, 2002

Experience Mathematics # 21 -Euclid's fifth axiom


Euclid’s fifth axiom says that given a line $l$ and a point $P$ not on the line, there is exactly one line parallel to $l$ passing through the point $P$. For centuries people thought that Euclid’s fifth axiom was “obvious”. But some mathematicians did not find it obvious. 

Finally, Reimann and Lobachevsky, both modified the axiom and tried to derive a new geometry. 

Reimann began with the axiom: Given a line $l$ and a point $P$ not on the line, there is no line parallel to $l$ passing through the point $P$. Reimann derived many geometrical theorems that are applicable on the surface of a sphere. For example, he showed that the sum of angles of a triangle is always greater than 180 degrees. Try drawing a triangle on a sphere and see why this has to be true.

Similarly, if we take a hyperbola ($y=1/x$) and rotate it around the y-axis, then we obtain a surface where Lobachevsky’s geometry holds. Lobachevsky’s geometry contains the axiom: Given a line $l$ and a point $P$ not on the line, there is more than one line parallel to $l$ passing through the point $P$. In this geometry, the sum of angles of a triangle is always less than $180$ degrees.

There is a property of the surface (known as curvature) that determines the geometry. Only surfaces with curvature zero follow the Euclidean geometry. Another example of a surface is that of a saddle (of a horse). Can you tell which geometry is applicable on this surface?