HOW THE SPACE WORKS?

We know, just 2 numbers (x,y) are enough to mark a point in a graph sheet or in any surface. Can we find the distance of the point from the origin just using these two co-ordinates? Here the Pythagoras theorem comes to our help. Since x and y co-ordinates indicate horizontal and vertical distance of the point from the origin, square both of them; add them and take square root. We get the diagonal distance from the origin.
r^2 = x^2 +y^2
Pythagoras theorem is the first topic in any math book. It still reigns in math and physics world. It's importance will never fade with time.
Suppose we want to find inner diagonal in a rectangular box, it can be found using the formula given below. We should remember the box is a 3 dimensional one.

Diagonal^2 = Length ^2 + breadth^2+height^2
Hence Pythagoras theorem can be extended to 3 dimension or any dimension by just adding square of one more quantity.

5^2 =3^2+4^2
25= 9+ 16
The 3,4,5 satisfies Pythagoras theorem. Hence they are called Pythagoras triplets. Finding Pythagoras triplets (whole numbers) is difficult. But you can make them using the following simple algorithm.
Pick any two whole numbers and call them p and q. Make A,B,C using the formulas given below.

A =p^2+q^2
B = 2pq
C = p^2-q^2
A,B and C definitely will be Pythagoras triplets.

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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

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