Each Subfield of the field of Complex Numbers contains every rational number

Some Linear Algebra stuff after a long time. Hope to continue the steam if possible.

Problem : Prove that each subfield of the field of complex numbers contains every rational number

Proof :
Let C be the field of complex numbers and let F be any subfield of C
Now, let p/q is one such rational number that does not belong to F
Therefore, -p/q also does not belong to F

Now, let x belongs to F.

Since x belongs to F and p/q does not, hence (x+p/q) does not belong to F
Hence, (x+p/q) + (-p/q) also does not belong to F
=> x + (p/q +(-p/q)) does not belong to F
=> x + 0 does not belong to F
=> x does not belong to F
which is a contradiction.

Hence Proved.

Fascinating Fact about Pythagorean Triplets

Everybody knows what a Pythagorean triplet is, it is the tuple (a,b,c) such that a² + b² = c². It turns out that if we restrict ourselves to the set of primitive integral pythagorean triplets, every such triplet where a, b and c are coprime can be represented by integers m, n where

a = m² - n²
b = 2mn
c = m² + n²

Besides being of mathematical interest, this result also gives us an efficient method to enumerate integral pythagorean triplets.

Visit this link for the proof.

UPDATE: The link seems to imply that the relation holds for all a,b,c. But I think there is a flaw in the proof when the author assumes that since a² is square, c-b and c+b must also be square. It seems to me that this is not necessarily true unless c-b and c+b are coprime. This is why I added the qualification that a, b and c must be coprime.

All feedback is welcome. :-)

Draper Award to Dr. Rudolph Kalman

2008 Draper Award from NAE has been presented to Dr. Rudolph Kalman of the Kalman filter fame.

http://www.nae.edu/nae/awardscom.nsf/weblinks/JMAN-7A4HZV?OpenDocument

http://www.cs.unc.edu/~welch/kalman/media/draper2008.html

LTE vs Wimax :: A comparative Study

http://www.comsysmobile.com/pdf/LTEvsWiMax.pdf

Sad Day :: SiRF Getting Sued

Over the past few weeks I have been following the news on how SiRF is doing bad, how its revenues are down etc.

But a series of Class-Action suits? That was somehow suprising and I was not able to understand why the heck should the company be sued by investors?

Here is a very nice article I read in GPS world and I agree with the points the author has brought up. Please read it.

In Defense of SiRF

I really wish that no harm happens to SiRF and it continues bring in new qualities in the domain of GPS just as it has done all these days.

Countability of Algebraic Numbers

I have attempted to solve the Countability of Algebraic numbers. I have to attach in the form of image as I do not know the optimum Latex incorporation into blogger. I wish we had the full fledged latex compiler in Blogger :-)


Please do comment and let me know if you think this is wrong because my head has been spinning trying to find out the way to show the countability. I am sure I get better with time. :-) Once again, looking for feedback.

LTE or WiMAX : The Road to 4G

I found this interesting article. Good way to get started and understand the subtle differences.

http://www.ucstrategies.com/detail.aspx?id=2730