Density of rational numbers in R

This theorem states that between any two real numbers we can always find a rational number.
Consider a Real number line, starting with 0 and having two points a and b on the number line
0--------------------a-------b-------

Our objective here is to find a rational number (lets say m/n) between the interval (a,b)
Lets divide this line into subsections, each of which is less than the width of interval (b-a). Using Archimedian property we can always find a natural number n such that 1/n < (b-a)

0-1/n-2/n--(m-1)/n-a-(m/n)---b---

Thus we see that smallest value of m which makes the ratio a > (m)/n is (m-1)/n < a . Now we have to make sure that ratio m/n is smaller than b. Substituting for the value of a in the equation 1/n < (b-a) we get
b > 1/n+(m-1)/n
b > m/n
hence proved

Archimedian Property and Axiom of completeness

Yesterday we discussed Archimedian Property. It says that Natural numbers are not bounded above i.e there is no maximum value of natural number. On discussion with Andy he gave me a very good way to visualize this property. Think of a Real number line starting from zero to infinity. Now on this number line you choose any number, how so big, I can find a Natural number bigger than that. Similarly choose a number on this Real line, how so small, I can find a Natural number whose reciprocal will be smaller than this number.

Formally Archimedian property is stated as
(i) Given any number x belongs to R, there exists an n belonging to N satisfying n > x
(ii) Given any number y > 0, there exists an n belonging to N satisfying 1/n < y.

Axiom of completness says that any bounded above set of real number has a sup, i.e if we have a set of real numbers which is bounded above we can always assume that it has a sup. This axiom has a wide implications and one of the important application is in proving "Nested Interval Property".

The nested interval property says that the intersection of nested closed intervals is NON EMPTY. How this comes ? Well its easy to see that Axiom of completeness says that whenever we have a bounded above set we will have a sup. In case of nested interval we have upper bounds, which in turn implies that we have a sup (recall that all bn's are upper bound

Proving Square root of 3 is irrational

Anyone who has ever taken high school mathematics at one point or other must have encountered the famous theorem of square root of 2 is irrational number. In fact this is considered to be one of the most famous and elegant proof in mathematics because of its simplicity. It also demonstrates the "method of contradiction", one of the most used method for proving, when all else doesn't seem to work.
The irrationality of square root of 3 can be proven similarly. The catch is when one does (p/q)^2 =3, we get p^2 = 3*q^2 which implies that either p^2 and q^2 are both even or both are odd. So we have two cases instead of one.
Case 1: Assuming both are even. But then if two numbers are even one can always divide them by 2 and so our initial assumption that p/q have no common factor is wrong. Which so implies that p and q cannot be both even.
Case 2: Now assuming both as odd. Thus p can be written as (2*m+1) and q can be written as (2*n+1) and our equation p^2= 3*q^2 can be written as
(2*m+1)^2 = 3 *(2*n+1)^2
4m^2 +4m+1 = 3(4n^2+4n+1)
4m^2+4m+1 = 12n^2+12n+3
4m^2+4m = 12n^2+12n+2
2(m^2+m) = (6n^2+6n+1)
Now left side is always even for all values of m, right side is always odd for all values of n, so this equality is never possible for any integer values of m and n and hence our assumption that p and q are odd is invalid.
Check this and some other goodies I found on the NASA website. There is one more recreational mathematics stuff that you will enjoy at this website. It is like take any 3 digit number. Exchange the digit at unit position with the one at 100 position . Find the difference between the two numbers . Now do the same with the number so obtained and add to it, you will get 1089. Its cool. Check this here.

Configuring Browser plugins for Firefox on Linux

Once we have Firefox on Linux obviously the next thing one has in mind to be able to use it as a normal browser. The three main plugins that should be configured are Macromedia, real player and xine. Macromedia for all the flash stuff, real player and helix player are perhaps the best choice if you are thinking of watching streaming video for example BBC newscast and xine for all other media type. The trick is to have the following files in your plugins directory under the firefox-installer directory. The files to be copied are libflashplayer.so, nphelix.so, xineplugin.so

Poem on Analysis

I found this at TAMU's website for analysis course. I don't know who is the author of this. But its fun and making me excited about my analysis course

Throw at me an epsilon,
A delta I shall find;
Infinite series and sequences
Put me in a bind.
Oh, what fun it is to have
Analysis on my mind, (Chorus)
Oh, what fun it is to have
Analysis on my mind.

A function may have limits, though
Continuous it mayn't be;
But even global continuity
Can't buy differentiability.
Oh, what fun it is to have
Analysis on my mind, (Chorus)
Oh, what fun it is to have
Analysis on my mind.

Here's Lagrange espousing
His famous law of the mean;
Hail to Cauchy and Riemann,
Heroes I've never seen.
Oh, what fun it is to have
Analysis on my mind, (Chorus)
Oh, what fun it is to have
Analysis on my mind.

And now comes this Frenchman,
Lebesgue is his name;
He ought to have been a tailor, for
Measure is his game.
Oh, what fun it is to have
Analysis on my mind, (Chorus)
Oh, what fun it is to have
Analysis on my mind.

In spite of all my troubles
I've also seen some gains.
The royal blood of analysis
Now courses through my veins.
Oh, what fun it is to have
Analysis on my mind, (Chorus)
Oh, what fun it is to have
Analysis on my mind.

Lots of classes today

So my holidays are finally over and I am back to school today. It was hectic as I needed to find out which all class to take. I am finally taking Analysis with Dr. Budzban. The way he introduced the class makes me feel that it will be lot of fun and learning. Today was just a review. The electrical engineering math should allow me to review my complex analysis, linear algebra and statistics. The abstract algebra with Dr. Yucas is one other class which will widen my horizon. I have always been interested in Rings and Groups and with the new book I am very excited to learn about it. Today he went over some example which I should be reviewing shortly. The course in Graph Theory and Numerical Analysis look equally appealing. Dr Porter is going to take it and I think I will take discrete mathematics sometime especially when he offers it. His enthusiasm was infectious. The numerical analysis is going to be taught by Dr. Xu. He is so soft spoken and I look forward to take a class with him. Finally the last class which I attended today is offered by Dr. Kocik and I am very sure that this class will turn out to be one very exciting class. For that I should be reviewing my Linear Algebra and watch the strang's lecture. I have to differ my plan of taking Linear Algebra this time even though it looks that it will be very interesting with Dr. Fitzgerald. Thus now I am sure I will be going ahead with Math Methods for Engineers, Intro to Analysis, Abstract Algebra and Applied Matrix Theory.

Scilab Installed,

Well my Linux Box is now rocking. With Xine installed and configured. I was able to watch the videos and it was definitely great. It plays all the formats including Xvids. My mp3 player on it is also configured after I installed the patch from guru website. Now some 10 minutes ago I got my Scilab installed too. So on the whole its now more than a entertainment system. Once my new DSL connection start it will become my serious desktop.

Linux Installed, Sound card configured

Few day back I installed Red Hat Linux on one Dell Optiplex Gx1 computer given to me by my sister. After installing everything I needed, there was still problem with the sound. I opened the computer and found that the sound card in my computer was on board. So a quick search on internet about Dell Optiplex Gx1 gave the description of sound card as cs4236 and a hardware check option in Linux gave the same hardware for audio. So the only problem I had was to configure it. Another search on linux cs4236 got me this website http://www.topology.org/Linux/sound.html and I followed the instruction ( which is " modprobe cs4232 io=0x530 irq=5 dma=1 dma2=0 mpuio=0x330 mpuirq=9 synthirq=5 synthio=0x388 " and now its playing awesome. I am now looking forward to install some matlab clone and hook it to DSL connection.

Ray Charles

Its amusing that I never got to know about Ray Chales until I saw his movie 10 minutes ago and reading about him in wikipedia. Ray was amazingly talented artist and a iconic figure in music. His story is indeed good. After going blind at an age of 7 how he reached the pinnacle of success. With numerous hits, a drug problem he overcame which continued for 17 yrs and his approach to life. He was surely a genious who knew his music and was smart enough to market it too. A truly wonderful movie and a superb performance. If you haven't seen it yet then you are missing on one of the finest movie and a great inspirtional movie. Two thumbs up from me.

Sleepless Night

I am now awake for more than 15 hrs. Today I went to cycle for around 1 hr and I believe it must be around 10 miles of Joy ride and a work out. Then I give myself a haircut which I think I did a pretty good job ! Then from 9- 9:30 I went for swimming. Did some laps and then straight to library. Caught up with emails and news, took a nap and now at 11:02 here I am updating this blog.

New Bicycle and movies review

I got a new bicycle. Its again a "Next Bicycle". Its grey color, 21 gears with 26 inch wheels. Other than that I have moved to new apartment and its cool with more furniture plus a very long study table. I hope to revise my Chinese before the school starts and read couple of more books. I watched "Taxi", "Meet the fockers" and "March of the Penguins". All three are good. My favorite is "March of the Penguins". The scenery is breathtaking. The story is about how the Penguins raise their children in such harsh condition. How the parent penguins juggle the responsibility while one goes to hunt for food and the other protects the sibling from Nature and Predators. Engrossing with pretty good music. Meet the fockers is comedy and is fun to watch its about a couple starring Ben Stiller who wants to get married but the bride's parents are not comfortable with Ben's parents. Its just plain funny. The Taxi is ok starring Queen Latifa being a daring Taxi Driver. Its again a comedy and not bad.

Semester Over, so what next ??

Finally I am done with all exams and got my results. Glad I took Statistics this semester. Never thought I will enjoy it so much. Though I had slightly different thoughts a week ago, when I was about to be overwhelmed with so many things to finish. Now the big priority should be to catch up with movies, friends and study something new. I am especially thinking of modelling with Differential equations, Curl and Divergence theorms and some history of mathematics!