- What is complex numbers
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- Properties Of complex Numbers
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- Conjugate of Complex Numbers
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- Modulus of complex numbers
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- Graphical Representation of Complex Number
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- Polar Representation of the complex number
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- Rotation of Complex Number
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- What is the significance of Complex Numbers

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

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

x^{2}=-1

Eular was the first mathematicain to introduce solution to this problem,he introduced the symbol i

$ i=\sqrt{(-1)}$

So i^{2}=-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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