WHEN THE NUMBERS EXPLODE!

"The greatest shortcoming of the human race is our inability to understand the exponential function". said a great physicist.
This can be illustrated by a simple story. Once a king a offered a reward to a poet. He asked the poet what he desired to have? The poet replied,"chess board filled up with grains". The king exclaimed,"what"?. The poet said, "I want a chess board filled up with grains in this way. Put one grain on first square, two on second square, four on third square. Every time double the number of grains and fill up all the 64 squares". Anyway, the king ordered the servants to carry out the poet's wish.
Half way through the process the numbers exploded. The king realized the reward could not be fulfilled even if he used all the grains in the kingdom. That is the power of exponential function. Here the function is

Y =2^x, 2^0=1, 2^1=2, 2^2=4, 2^3=8.......2^64= 1.8 followed by 19 zeros

Moore's law is the consequence of this principle. It says,"Number of transistors in a silicon chip doubles every 18 months". That is why we spent 25 years with Bulky desk-top and then rapidly switched over to smart phones.
The spread of viruses often works in a similar way. One ill person infects a few others, who in turn each infect a few more, until it becomes an epidemic. Immunization is necessary to control the break out.
If you want to save for retirement, start early. Even a little delay will cost you so much. If you save 250 dollars every month from age 18, you will get more than a million dollars at 65. But, if you start the savings at 30, you will only reach 478 thousand dollars at 65. Here compound interest comes into play.
If we fail to understand exponential growth, our debts can rapidly spiral out of control too. Exponential function is both the creator and destroyer. Chain reaction in Nuclear bomb, Laser action in a laser device, snow balling, avalanche, growth of bacterial population....all examples of exponential explosion.

FOOT NOTE:
Y =e^x
Where e=2.71... Euler number. It is a special function used for natural growth and decay. You can say 'e' is the unit of growth or decay. "If one unit grows continuously or exponentially for unit time, it will grow from 1 to maximum 2.71... units, that is e.
Hence numbers do explode.
--------------------------------------------------------------------------------------------------

Comments

LISSAJOUS FIGURES

Definition: "When a particle is subjected to two sine wave motion or two oscillatory motion at right angles, the particle describes lissajous figures". We know sine wave motion and circular motion is basically same. Hence we draw two circles A and B perpendicular to each other. The circle B rotates twice faster than circle A. That is, frequency of circle B is two times than that of A. A particle at the intersection of two circles is subjected to two sine wave motion A and B at 90 degree simultaneously. The particle will describe figures depending on the frequency and phase of A and B . In our case, the ratio of frequency is 1:2 and the two waves are in phase. To draw lissajous figures : A moving point in both the circles are chosen. Here we should remember; during the time taken by the circle A to complete one rotation, circle B completes two. Hence the points are marked on the circles according to their speed. Then straight lines

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

CASINO'S GAME

Let us find out how the casino survives with mathematics. Say, your friend invite you for a game of dice. You must bet (wager) 2 dollars. If you roll 'six' you will get back 8 dollars. The game will go on for 30 rounds. All sounds good. The probability of rolling 'six' is 1/6. Since the game will be played for 30 times, the 'expected win' is 30*1/6 = 5. That is, you are expected to win 5 rounds out of 30. Hence your gain will be 5 * 8 =40 dollars. ok. This also implies that you will loose 25 rounds. Hence your loss will be 25*2 =50 dollars. Your net gain will be gain-less = 40-50 = -10 dollars. For 30 rounds, the loss is -10 dollars, Hence, for one round =-10/30 = -1/3 dollars. There will be a loss of -1/3 or 0.33 dollars per round. It is not a fair game. Let us make a simple formula to calculate 'Pay out per round\. The probability for a win = p The pay-out in case of win = V No. of rounds = n The expect

sciencecapsules

Search This Blog