March 19, 2005

My simplest proof

I wrote the below theorem

Theorem
a squared + b squared > 2ab (if a not equals b)

Proof
Proving by disproving the opposite
i.e. Disproving the statement a squared + b squared < 2ab

Consider a squared + b squared < 2ab
Subtracting 2ab on both sides
a squared + b squared -2ab < 0
ie. (a-b) whole squared < 0
It is absurd. Never can a square of a number be less than zero.
Disproved. Hence proved

Unofficial Copyright

15 comments:

phoe6 said...

a squared + b squared -2ab
ie. (a-b) whole squared

How??

Ramesh said...

Whats the problem?

(a-b)^2 = a^2 + b^2 - 2ab

is a formula. Right?

Kannappan said...

Senthil and Ramesh
Thanks for your comments guys. My major questions are Has anyone written this formula before. Have you come across this before. If not (which I highly doubt) I can very well claim it as my own.

Kannappan said...

The title for this post was influenced by My Simplest Theorem

phoe6 said...

ok, the intial comment:
a squared + b squared -2ab
ie. (a-b) whole squared

was on the doubt that, is not it taking a unproved theorem(In this case only!!) ( or something of the sort that, we assume something neglecting some theorem, then use the same neglected theorem to prove the assumption)
I hope you are getting what I am pointing at.

I did'nt have much time go more in detail,but I shall try.

phoe6 said...

another friend of mine found it something like straight forward/ or found this as expression in some different way, but did not have time to get more into this. But there was mixed feelings

Can you please explain in detailed way whats amusing in this, so that we can have greater depth in this subject.

Kannappan said...

The amusing thing is you have two different quantities Apples and Bananas. Say for example 10 Apples(5 male and 5 female) and 12 oranges (6 male and 6 female). Imagine that the fruits have the ability to mate. And You are a small time fruit seller. You want to increase the number of fruits by letting the fruits mate. Any normal person will wonder which kind of mating either a Cross(Apple-orange) or a Normal (Apple-Apple) mating will yield a large quantity of new fruits. As we had discussed several months back, it is the Normal mating that succeeds. Thats each of the 5 male apples mate with all the 5 female apples to provide 25 new fruits. And each of the six male oranges mate with six of the female oranges to yield 36 new oranges. So we have a total of 61 new fruits. If we had done a cross breed.. Say each of the 5 male apples mate with all the 6 female oranges to provide 30 new hybrid fruits. And each of the 6 male oranges mate with all the 5 apples to provide 30 more new hybrid fruits. So we have only 60 new fruits using Cross Mating as compared to 61 fruits in the Normal Mating.

This is the illustration. I just proved the same beyond any doubt. Now its an universal truth. The next time a tutor tells you that a*a plus b*b is less than a*b plus b*a,Dont hesitate to kick him out.

Amuses me for sure. I really got a kick...Oh my god kind of...where was it hiding....Its trivial though..very trivial...which makes me wonder why the hell didnt I or anyoneelse for that matter, prove it the moment we saw the formula (a-b) whole squared equals a squared + b squared + 2ab for the first time in life.

Thats how it is. It looks very simple while seeing but one doesnt see it until one really sees it.

It is a genuine theorem. Not even the slightest doubt about it. Am trying to trace as to see who has proved it for the first time ever.

This theorem will be helpful in reduction of scenarios below...Say in some rocket science research...You come across an expression that says 23*c plus 56*d is greater than 43*(a squared) plus 43*(b squared). You can quickly simplify this expression to 23*c plus 56*d is greater than 86*a*b and proceed to the next step. You have reduced an expression that had a power of 2 to an expression whose maximum power is 1. You dont need to take any squares or squareroots any more on that expression. This sounds pretty intrestinn to me. :)

Kannappan said...

Senthil,

I still dint get your initial comment. Are you asking how (a-b) whole squared equals a squared + b squared - 2ab?

If so, its a nice question. It made me think for sometime. However it seems simple.

(a-b) whole squared is equal to (a-b) * (a-b) "as per the definition of square of a term", which in turn is equal to a*a - b*a - a*b + b*b which in turn is equal to a squared - 2ab + b squared.

phoe6 said...

thanks.
I have one more suggestion:
Document this entire this and properly explain the claims and publish it in any of the Mathematical Peer Review Journals.
That would bring up a good review and criticisms. Finding some place where a good number of math's people hang around and letting them know to comment upon will be a good idea.

phoe6 said...

(a-b) whole squared is equal to (a-b) * (a-b) "as per the definition of square of a term", which in turn is equal to a*a - b*a - a*b + b*b which in turn is equal to a squared - 2ab + b squared.

And Square a number can never be negative.
So,
(a-b)^2 > 0
a^2 +b^2 - 2ab > 0
a^2 + b^2 > 2ab

Anything wrong in this?

Kannappan said...

So,
(a-b)^2 > 0
a^2 +b^2 - 2ab > 0
a^2 + b^2 > 2ab

Anything wrong in this?


Nothing wrong. Infact it sounds more refined.

Kannappan said...

In general

(a^2n)+(b^2n) > 2*(a^n)*(b^n)

using the same logic.

Anonymous said...

Your proof is only valid for rational numbers.

Anonymous said...

i^2=-1

Google for i^2 for confirmation

Kannappan said...

Thanks Anonymous. Let me check that out.