Maths Concepts

Math Poettary - Infinite

noreply@blogger.com (Gaurav Bhatnagar) — Fri, 13 Feb 2009 07:47:00 +0000

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:

(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.

The Modulus Function

noreply@blogger.com (Gaurav Bhatnagar) — Mon, 09 Feb 2009 16:47:00 +0000

Every differentiable function
is also continuous.
"This does not mean,"
the Modulus Function
points out,
"that every continuous function
is differentiable too."

Math Poettary #5

Not differentiable

noreply@blogger.com (Gaurav Bhatnagar) — Mon, 09 Feb 2009 03:21:00 +0000

A continuous function
Going along nicely
without lifting pencil
from paper.

Ouch!
This sharp point
got me,
its not differentiable.

Math Poettary #4

Continuous function

noreply@blogger.com (Gaurav Bhatnagar) — Sun, 08 Feb 2009 06:16:00 +0000

A continuous function,
draw it without
picking up pencil
from paper.

At all points,
the left hand limit
the same as
the right hand limit,
the same as
the value of the function
at the point.

At all points
its value
what it should be.

Math Poettary #3

Isosceles Triangle

noreply@blogger.com (Gaurav Bhatnagar) — Wed, 28 Jan 2009 17:32:00 +0000

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

Math Poettary #2

Ambigram Poettery

noreply@blogger.com (Gaurav Bhatnagar) — Sat, 10 Jan 2009 13:15:00 +0000

I started another blog with some "poettery" along with Punya Mishra's ambigrams. These are comments on ambigrams he has made over the years. You can find them here: Ambigram Poettery.

I call it poettery, because it is the product of my pottering around the poetry format. Really, its not poetry in the sense of the word-- can't really say I don't really understand the sense of the word.

Ambigrams, though, are something that are really interesting, and people interested in math are likely to like them.

Binary Pascal's Triangle

noreply@blogger.com (Gaurav Bhatnagar) — Sat, 10 Jan 2009 13:11:00 +0000

Binary Pascal's Triangle
Originally uploaded by GauravBhatnagar

"Bit by bit,
I understand,
the triangle of Sierpinski"
says Pascal?

Math Poettary #1

Get Smart! Maths Concepts

noreply@blogger.com (Gaurav Bhatnagar) — Mon, 08 Dec 2008 07:13:00 +0000

The book Get Smart! Maths Concepts has been released by Penguin Books, India. Get information about the book in its blog: http://getsmartmathconcepts.blogspot.com/

The book is a survey of middle school math, and covers concepts that I feel will be useful for all students to know. It may even give a quick review of things you need to know in pre-algebra and algebra, before you encounter them again in a calculus class. The emphasis is on thinking mathematically, rather than just learning the procedures to solve problems.