Saturday, December 18, 2010

Going deeper underground.

Mathematical analysis draws me in like a black hole sucking light, but everybody argues: "why the hell do you need this", help!!!

I have written about sqrt(2) more than once, and how it provided many motivations in the past.
It turns out, you can also use it to motivate interest in 'mathematical analysis'.


Take a right triangle with 2 sides of length 1, and place it on a 2D coordinate system as show in the figure. Now let us try to estimate the length of the hypotenuse using limits.
We will cut the hypotenuse into tiny segments, which is a standard practice of estimating lengths of curves, and compute the total length of the segments. If we do this we end up with the answer of 2 and not sqrt(2)! What is wrong with this?
But Jad! you say, what you just calculated is the Manhattan distance and not the length of the hypotenuse. And you would be right. But, can't we make the same claim about this not working for our summation of small rectangles for definite integrals? what exactly is the difference? and how can it be formalized? Mathematical analysis can tell you.
For more fun examples, I recommend page 3,4,5 from: http://www.math.ucla.edu/~tao/resource/general/131ah.1.03w/week1.pdf


Another topic for this post is 'telescoping series'.
For discrete integrals, our current topic in the math project, sums of series becomes very important, it is also a brain enlarging topic and a lot of fun.
One technique for summing infinite series is called 'telescoping series'.
It works on sums of series where terms cancel each other out, and what is left is a finite and easy to calculate number.
The nice thing about this for me was that it ties to the discrete math toy problem of finding the sum of i's where i goes from 1 to n.
I have given a geometric proof of this I came up with in a previous post.
One can also do it by induction.

It turns out, it is also possible using telescoping as shown in the figure.
The interesting thing is that both induction and telescoping are very indirect, in that, if you wanted to find the answer to this problem, you would need intuition first, you would get the answer and THEN try to prove it.
The more we dig into math the more we see that this is how most of the important theories get discovered. This is one of the reasons why learning about this is great for one's brain.

Usually, one tries to proceed sequentially and directly from problem to solution, and that only works for the simpler problems. After a certain level, laziness is simply not an option anymore!

Richard Bellman had the following remark once:
"Each individual problem was an exercise in ingenuity, much like plane geometry. Change one small feature, and the structure of the solution was strongly altered. There was no stability!"


5 comments:

madjestic said...

Hey, Jad, Vlad is here:

http://www.math.ucla.edu/~tao/resource/general/131ah.1.03w/week1.pdf

All went well till the page 14, or the 'loose axiom' :) How can one live with that! How come that even such simple things of the system are fundamentally unprovable, without involving elements of the super-system? It almost seems preferable to attempt to proov things not from the bottom-up, but up to bottom. It is impossible to converge to infinity, but is possible to converge to zero...

madjestic said...

Page 16, par.4:
"...But there's no point doing so; we only need one number system in order to do mathematics"

Where is that statement derived from? Who are "we", what does "no point" mean - those are all elements, alien to the proof! How can one build a proof, doing that?! That's not a proof - that's a belief system!

It's interesting to read this guy just to observe his cognition. However his proof does not prove much to me, starting from page 14, the loose axiom.

madjestic said...

"We will not prove the above assumption and it will be the only assumption we will ever make. "

Somewhere in the future that should be engraved on the tomb-stone of modern science.

madjestic said...

Sopped reading after page 16. Maybe will continue later, for the sake of entertainment.

Jad said...

Hi Vlad,

I feel I can very well live with induction and do accept it as an axiom, I suggest you try one exercise based on a proof by induction, and then decide if it feels satisfactory or not, you have to think about hat is n, if it is any number, then ... induction is pretty convincing IMHO, but you definitely do not have to agree, and then it is time for discussion. Ultimately math is about convincing other people.
If I am not mistaken even constructivists (you will read about them later in our course) accept induction, but refuse proofs by contradiction.
How would YOU go about it?

I agree that there is no point, but what I would have said instead is that they are all the same system, only with different graphical representations, but same mathematical properties.
How would YOU go about it?

It is already engraved on the tomb of science by Goedel!

Glad you liked it!