Skip to main content

HOW TO PROVE?

      


                        ADVT:  REFURBISHED MOBILES IN AMAZON

       In a stadium, 50000 strong spectators are watching a cricket game. There must be at least two people who were born on the same day of the same year. Are we certain? Can we prove it? Yes, there is a way.

     Let us assume that everyone in the crowd is aged between 0 and 80. In each year there are 365 days, so, in 80 years there are no more than 29200 different birth dates. But the crowd is bigger than 29200, then there is certain to be a duplicate. We can be 100% sure.

    This is called "pigeon hole proof". That is, there are 29200 pigeon holes (dates), but there are 50000 pigeons(people). You can not put all the pigeons in a given number of holes. We have to reuse some of the holes. This proves the statement, "there must be at least two people who were born on the same day of the same year".

    1. How many people do you need before you can be certain that two of them have the same number of hairs on their head?"

    2. How many books will you need before you can be certain that two of them had exactly the same number of words?"

     Statements like these can be proved using the pigeon hole principle.

PROOF BY CONTRADICTION:

   Either multiplicand or multiplier must be greater than 8 to get the product 72. That is, in x*y = 72, either x or y must be greater than 8. Is it true?

     If both are equal to 8, the product is 64 (8*8=64) - a contradiction. Hence one number must be greater than 8 to get 72. Another example: A member of parliament argues in these lines. "The Right Honourable gentleman claims that he will increase public spending. The only way in which he can achieve this is by increasing taxes -which he has already ruled out (a contradiction). I, therefore, pronounce his argument to be in tatters".

    Hence proving is art also Math.

Comments

Post a Comment

Popular posts from this blog

THE EARTH, A SUPER ORGANISM

     JOIN MY COURSE: "Become a programmer in a day with python"       A man called 'love lock' (what a name) proposed a theory called Gaia theory, named after Greek Goddess.      It says, "Earth is a self-regulating organism like a human being.  The organic life in it interacts with in-organic matter and maintains atmosphere, temperature and environment".  Hence the earth is still suitable for the life to thrive.      Imagine, in a particular place, there are lot of flowers.  Some flowers are white and some are darkly coloured.  We know, white reflects light and heat while dark absorbs the same.  White flowers can thrive in hot climate.  But dark flowers requires cold climate.  The absorption and reflection balances and the environment reaches average, warm temperature at which both the flowers can co-exist.  This is the essence of "Gaia" theory.      On our earth, ...
3 comments
Read more

THE PARABOLA

          A jet of water shooting from a hose pipe will follow a parabolic path.  What is the so special about parabola.    Y= x^2 Draw a graph for the above equation.  It will result in a parabola.  This parabola is also called unit parabola.  Any equation involving square will yield a parabola. Example:  Y = 2x^2 +3x+3 (also called quadratic equation)    X= 2 and -2, both  satisfies the equation 4 = X^2.  Parabolic equations always have two solutions.     Any motion taking place freely under gravity follows parabolic path. Examples:   An object dropped from a moving train,   A bomb dropped from flying plane,  A ball kicked upwards.      If a beam of light rays fall on the parabolic shaped mirror, they will be reflected and brought to focus on a point.  This fact is made use of in Dish Antenna, Telescope mirrors, etc.   ...
Post a Comment
Read more

DISORDER IS THE "ORDER OF THE DAY"

         Imagine a balloon full of air.  The air molecules are moving randomly inside the balloon.  Let us pierce the balloon with a pin.  The air rushes out.  Why should not the air molecules stay inside the balloon safely and ignore the little hole?  That is not the way the world works.  The molecules always "want to occupy as many states as possible".  Hence the air goes out in the open to occupy more volume.   The things always goes into disorder (entropy) and the disorder increases with time.  The above statement is what we call "second law of thermodynamics".      Consider a cup of coffee on the table. Suppose the heat from entire room flows to your cup of coffee, the coffee will boil and the rest of the room will freeze.  Freezing means bringing things to order and arrangement.  It violates the second law.  Hence it will never happen.  Hence heat must flow from high ...
Post a Comment
Read more