SOLIDIFYING THE CONCEPT OF NUMBERS

First, will understand, why we have numbers. Say, you are the manager of a restaurant. People sitting in the table order for 4 burgers. In the 'numberless world' , you will pass the order to the kitchen by calling out 'burger, burger, burger, burger'. We know saying 'four burgers' is easy. Four is a number. So, numbers are a shortcut for counting by ones'. Number is abstract form. Four may mean four butterflies or four stones. Numbers are concepts like beauty and intelligence.
Let us use pebbles to represent the numbers. Four can be arranged into 2*2 square form. Take 8. It can be represented by 8 pebbles arranged into 2*4 rectangular form. Many numbers can be represented by a square or a rectangle formed using pebbles. Examples 6,8,9,10,12,14....
But some numbers cannot be made into rectangle. Example 1,3,5,7,11..... These numbers which cannot subdivided into other numbers are called prime numbers. The prime numbers are 'atoms' of the number world. We can construct any number using them. How many prime numbers are there? Is there a limit for prime numbers? We do not know the answer.
Numbers should have base. We use numbers of base 10 (0.....9). Clocks use numbers of base 60. Computers manipulate binary numbers (0,1) of base 2. Base 16 numbers are also used in electronics. Even you can create your own system of numbers, if need be.
Let us solve one simple problem using pebbles to represent numbers.
Sum the numbers from 1 to 10. The answer is 55. No wonder. Let us solve it in a different way.

Put one pebble for number 1 in the first row. Put two pebbles for 2 in the second row. Repeat the process up to number 10. You will get a right angled triangle. That is half of a square, cut along the diagonal. Fill the missing 'half' with pebbles. You get 10*11 matrix of pebbles. Total elements (pebbles) in the square is 110. Half of it is 55. Hence the answer sum is 55. Hence if you picture and solidify the numbers, you can easily attack the numerical problems.
Numbers are the foundation on which Fort of mathematics is built.
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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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