REAL USE OF IMAGINARY NUMBER

What is the solution of X^2 + 1 = 0 ?
X^2 = -1
X = ROOT OF -1 = i

Root of -1 is meaningless and does not have any physical or real attribute. So we call it imaginary number i. Now a new number system is born 1i, 2i, 3i,...

Normally, positive numbers are represented in positive X - axis and negative numbers are represented in negative X - axis. That is, 180 degree opposite to the positive numbers .Since i = (-1)^1/2 , the imaginary number should be represented half way between positive and negative numbers. That is in Y - axis perpendicular to the X-axis. So finally i means 90 degree rotation.

Now we take one real example. Let there be a square table on the floor. Let us assume that you and your friend push the table at the two sides. that is, both of you push the table perpendicular to each other. Commonsense says that table will move in a inclined direction.

Let your weight be 60 kg and your friend weight be 40 kg. Hence the force on the table is 60+40i, i indicates one force is at 90 degree to another force. By the way, this number is called complex number because it is real + imaginary.

The net force on the table is root of (60^2+40^2)= 72.11 (Pythagoras theorem)
Direction of movement of table is Inverse tan (40/60) = 33.7 degree angle to the horizontal(trigonometry)
Suppose I want to rotate the direction of movement of the table by 90 degree, what should I do ?
Since i = 90 degree rotation, multiply the force by i.

= i*(60+40i)
= 60i+40i^2
= 60i-40 since i^2 = -1
= -40+60i

This means, 40 kg person should push the table from negative X-axis and the 60 kg person should push the table from positive Y -axis. Then the direction of the movement of the table will rotate by 90 degree as illustrated below.

direction = inverse tan(-60/40)
= 123.7 = 33.7+90 degrees

Hence you can rotate force or any quantity by any angle by using suitable complex numbers.

The complex numbers like 10+20 i packs lot of info. in compact form. It is easier to do mathematical operations with them. No wonder it is found everywhere in physical science and engineering.

1+i represents 45 degree rotation. ponder over it.

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