What's up Math?

Hi Friends This is a sub section of my site containing topics somehow related to Mathematics. I want to stress that I am only interested in Mathematics.So I am neither an expert nor a Graduate in Mathematics. So your corrections on anything wrong will be highly appreciated.

Mathematics is so pure and so great and it will never fail to amuse us with its complexity and a sense of conformity. The current human knowledge of Mathematics is enormous For example Andrew Wiles' proof of Fermat Last Theorem is more than 100 pages.

So I have written these pages just for amusement rather than trying to be complete by discussing everything which in the first place I dont know , and which can not be done by any human being. These were simple ideas but may be fascinating ideas.

Links external to my site was marked by this logo . But Click those links only after seeing all of my pages :-).

Here my two favourites.One is more of a logical paradox.

Russell's paradox

The Classic paradox from Bertrand Russell.

A class is a collection of objects. For example a book is a member of class of books . But class of books is not a book , so it is not a member of itself. So there are classes which are not a member of themselves like class of books .

A Class of All classes is a member of itself as it's members were classes. There is also a class of all classes which are not a member of themselves (e.g class of books), and we call it A.

Now the question is whether the class A is a member of itself?

Analyse deeply and check the answer.

Possibility 1: Class A is a member of itself.

But All its members(i.e classes) should satisfy the condition that it should not be a member of itself. So If it is a member of itself then it loses the requirement for a member it is not a member of itself.

Possibility 2: Class A is a not a member of itself.

Now it satisfy the condition for being a member so it is a member of itself.

Reductio ad Absurdum

There are many ways to prove a theorem like Proof by induction (using Domino effect), reductio ad Absurdum etc. Reductio ad absurdum is proof by contradiction. Here is a classic proof for square root of 2 is irrational by reductio ad absurdum.

We will start it with the opposite which is square root of 2 is rational which means it can be expressed as p/q. so Öp  

2 ^1/2 = p/q

where p and q have no common denominator.Now squaring both sides.

2=p²/q²

p² = 2 q².

as q² is multiplied by 2. p² is even. when p² itself is even, p is also even.

Now p² is divisible by 4 as it is a square of an even number. That means q² is even and q is even with the same logic followed for p.

Which tell us that p and q is even so having a common denominator as 2. Which is a contradiction from our initial assumption that it has no common denominator.

So what is wrong? Our initial assumption Which is root 2 is rational is wrong. SO root 2 must be irrational. Hence the proof.

Quite extraordinary