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.

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