WHY HALF TIMES HALF IS QUARTER?

Let us understand the fractional multiplication visually. You may feel it is elementary. But it gives new perspective to fractional multiplication.
First, let us take 4*3. Let us take a vertical line equally divided into 4 parts and horizontal line into 3 parts.

The area enclosed is the result of 4*3. There are 12 unit squares. Hence the answer is 12 units.

Now, how to visualize (1/2)*(1/2). Take a vertical line, divide into 2 equal parts and name it 1/2 and 1. Repeat for horizontal line.
Here 1/2 and 1/2 bounds one square which is one fourth (1/4) area of the whole unit square. So 1/2*1/2=1/4.

In multiplication, the answer need not increase but diminish also.
Now let us prove 1/3*6/2=1.
Draw vertical line and graduate it in 1/3. Mark horizontal line in terms of 1/2.

1/3 and 6/2 bounds 6 squares. The area of each square 1/6 (since unit square contains 6 squares).
Hence the total area of six squares is 6 *1/6 =1. Therefor 1/3*6/2=1. Now you can realize why 6 in 1/6 is called Denominator because it indicates denomination.

We got the answer as 'one' in the above case. That can also be proved visually. Move the last four shaded squares to the unit square and fill it up -as illustrated below. Now the shaded region occupies one unit area.

Hence 4/7 means 4 times 1/7 (denomination). If the fraction is exactly divisible like 1/2 =0.5, it is rational number. If it is not divisible like 1/3 =0.333... It is irrational number. The value of pi = 22/7 is an irrational number. (We know that the circle of circumference 22 units has the diameter of 7 units). People still find more digits in pi using super computers. By the way, all natural constants are irrational numbers like e = 2.71....root 2 =1.414...
Dealing with the fractions is a tricky business. Another article tells why we add the fractions the way we add.
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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

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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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