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Rotation of Complex number and significance of complex number for Class 11 , IITJEE maths and other exams




Rotation of Complex Number

Multiplying i is a rotation by 90 degrees counter-clockwise
Multiplying by -i is a rotation of 90 degrees clockwise
Example
z=1
If we multiply it by i, it becomes
z=i so that it has rotated by the angle 90 degrees


What is the significance of Complex Numbers? Why they are required?

Real numbers such as natural number,rational number , irrational number are invented in the history as and when we encounter various mathematical needs. Same happen with the complex numbers.
We had no solution for the problem
x2=-1
Eular was the first mathematicain to introduce solution to this problem,he introduced the symbol i
$ i=\sqrt{(-1)}$
So i2=-1
So solution to the problem becomes
x=i or -i
He called the symbol i as imaginary unit.
Just like all the other number ,this number was added to our Number vocabulary. This like other numbers is useful in explaining where physical explanation.
It is very useful in the field Electrical and electronics.

Examples:
1. Find the modulus and amplitude for the complex number
z=-1-i
Solution:
We have already given the steps for modulus and arg
Modulus
|z|=$\sqrt{(x^2+y^2)}$
How to find the arg
Step 1) for z=a+ib , find the acute angle value of $\theta=tan^{-1}|\frac{y}{x}|$
Step 2) Look for the values of a ,b
if (a,b) lies in First quadrant then Argument=$\theta$
if (a,b) lies in second quadrant then Argument =$\pi-\theta$
if (a,b) lies in third quadrant then Argument =$-\pi+\theta$
if (a,b) lies in Fourth quadrant then Argument =$-\theta$
So
|z|=$\sqrt{2}$
Acute angle
$\theta=tan^{-1}|\frac{y}{x}|$
$\theta=\frac{\pi}{4}$
Now the complex lies in third quadrant
So $arg=\frac{-3\pi}{4}$
2) Find the polar coordinate equation for the above complex number
Solution: we know the polar coordinate equation is given by
Polar form = |z|[cos (arg) + sin (arg)]
so
$z=



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