Bhaskara's proof of the pythagoras's theorem

Bhaskara's Proof

Bhaskara's proof (1114 AD) of the theorem follows from the figure




So simple, elegant and beautiful. I wish I was aware of this proof as a student

Crazy mathematics

As I was shifting my books from one location to another, I found a big heap of books, I had not even started reading or looked at in detail. Being me, I got distracted by them and decided to at-least give them a chance to fulfill their destiny (to be read cover to cover by their owners or the friends of the owners).

I started with Tom Apostol's Mathematical Analysis. It is a wonderfully terse book and I found myself struggling by the time I reached the topic on "Point Set Topology". I had been through a quick journey of the past - real and complex numbers, sets and cardinality, prime numbers and many more things.

I also ran into the famous Cauchy-Schwarz inequality. I remember having proved the inequality in the exercise of Don Knuth's Art of computer programming volume I, but by now I had forgotten it all -- the sum manipulation, the whole of chapter 1 as hiding somewhere in my memory and seeking it was a hard/impossible task. This time, I found myself taking the help of a symbolic computation package - maxima. My dialog was as follows

(%i2) expand((a1^2 + a2^2)*(b1^2 + b2^2));
(%o2) a2^2*b2^2+a1^2*b2^2+a2^2*b1^2+a1^2*b1^2
(%i4) expand((a1b1 + a2b2)^2);
(%o4) a2b2^2+2*a1b1*a2b2+a1b1^2

(%i31) (a1^2 + a2^2)*(b1^2 + b2^2) - ((a1b1 + a2b2)^2);
(%o31) (a2^2+a1^2)*(b2^2+b1^2)-(a2b2+a1b1)^2
(%i32) expand(%);
(%o32) a2^2*b2^2+a1^2*b2^2+a2^2*b1^2+a1^2*b1^2-a2b2^2-2*a1b1*a2b2-a1b1^2

To cut the long output short, I ended up with Lagrange's inequality.

This is just a small part of the mathematics I have to (re)learn, there is also graph theory, game theory, probability theory, Markov chains and theory of computation.

All the mathematics is very exciting, but time consuming. I have loads of other information to read and digest. Let's see where all this I need to learn this crazy math attitude takes me.

Using maxima -- baby steps

The use of symbolic computing makes it easy to visualize and solve problems that were otherwise dry and would put you to sleep. To test the simplicity of a freely available tool called maxima, I tried to solve a common problem with it.

The problem is quite simple, remember high school physics? Well I don't :-). Consider a tuning fork, which oscillates and produces a sound (depending on it's frequency).

We start by entering the mathematical model of the fork

'm*diff(x(t), t, 2)=-k*x(t);

Maxima prints


We now request maxima to solve the equation for us by

desolve([%],[x(t)]);

Maxima asks

Is k * m positive, negative, or zero? We say positive

Maxima prints



We ask maxima to simplify the result for us

radcan(%);

Maxima outputs



That's it, we have our solution.