A GIFT THAT A KING CANNOT DELIVER

There was a chess competition in a king's regime. The king promised gift of winner's choice. After the game, the winner just asked some grains of rice as his price. He added one condition.
He wanted to put one grain of rice on the first square of chess board; 2 grains on the 2nd square; 4 grains on the 3rd; 8 grains on the 4th one. That is, doubling the number of grains each time till reaching 64th square. The king was surprised by the humble request and ordered to execute the prize. The servants brought sack full of rice and begum to arrange the grains on the board as instructed. Halfway through the work, the king and the servants realized that all rice grains in his kingdom would not be enough to fill all the 64 squares. The king submitted to the wisdom of the winner. Then the winner said, "I do not want the price. Just give rice food to the needy daily". And he went away.

This is a good example for the 'geometric series' which is of the form,

= a+ar+ar^2+ar^3+............+ar^n

our example,
=1+1*2+1*4+1*8+............
=1+1*2+1*2^2+1*2^3+.......+1*2^64
The sum of series = a[1-r^n/1-r]
hence, no.of grains= 1*[1-2^64/1-2]
= [1-2^64]/-1 = -1+2^64
= 2^64-1
= 18446744073709551615 grains
=460 billion tones of rice
Moral of the story: the mathematics can kill our ego.

Is 0.999......equal to 1:

0.999....... = 0.9+0.09+0.009+0.0009+..............
= 9/10+9/100+9/1000+.....................
=9/10[1+1/10+1/10^2+1/10^3+.........]-------A
so the series in the bracket is geometric series with a=1,r=1/10 and n goes to infinity
we know,The sum of series = a[1-r^n/1-r]
= 1*[1-(1/10)^infinity/1-1/10]
1/10 to the power infinity logically becomes zero.
than, sum = 1/[1-1/10]
= 1/[9/10] = 10/9
multiply the sum 10/9 with 9/10 which was outside the bracket in equation A
FINALLY,
0.999.......= [9/10]*[10/9] = 1
0.999........definitely is equal to 1

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 oxygen constitute 20% of the atmosphere. The oxygen level is always mai

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. Inverted parabola shape is used in the construction of buildings and bridges. Because the shape is able to bear more weight. A plane

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 temperature to low temperature and not the other way. The air molecules in y

sciencecapsules

Search This Blog