MONTE CARLO TECHNIQUE

Imagine a square of unit side. Inscribe a quarter circle within the square. The area of the square is 1 unit. The area of the quarter circle is pi/4 unit.

Let us randomly select N points within the square. Out of N points, let C points fall within the circle. Logically, the area of the square is proportional to N points and the area of the circle is proportional to C points
.
Circle: pi/4 proportional C
Square: 1 proportional N.
Dividing,
( pi/4)/1= C/N
pi = 4*C/N

So we can estimate pi in this method. As the number of random points increases, the accuracy of the pi increases. A computer program is highly suitable to implement this technique. This method is named monte carlo technique because monte carlo is famous for casinos based on random numbers.

WHAT THE PROGRAM SHOULD DO?
For each point; up to N points
1. Select a point randomly within the square.
LOOP 2. Check if a point falls within the circle.
3. Count the point falling in the circle in C.
Next point
4.Find the ratio C/N and multiply by 4 to get pi.

APPLICATIONS
This is the versatile problem solving method. When all other methods fail, this "random method" may come to our rescue. It used in many fields. For example: numerical integration, solving system of equations, searching, area of irregular shapes etc,.

Science update: The diameter of the observable universe is known to be about 93 billion light years. To calculate the circumference of a circle with such diameter accurately only 39 decimal places of "pi" are needed.

Comments

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, ...

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. ...

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 ...

sciencecapsules

Search This Blog