- 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

Complex number z=a+ib can be represented by a point P(a,b) on a xy plane. The x -axis represent the real part while the y-axis represent the imgainary part

Such a diagram is called Argand diagram or Complex plane

Points to Note

1) The length OP is called the module of z and is denoted by |z|

|z|=$\sqrt{(x^2+y^2)}$

OP=$\sqrt{(x^2+y^2)}$

2) Purely real number lies on X-axis while purely imaginary number lies on y-axis

3) The Line OP makes an angle $\theta$ with the positive direction of x-axis in anti-clockwise sense is called the argument or amplitude of z

It is given by

$\theta=tan^{-1}\frac{y}{x}$

4) Argument of a complex number is not unique since if $\theta$ is the value of argument then $2n\pi + \theta$ (n=0, $\pm $1, $\pm $2, ....) are also values of the argument. Thus, argument of complex number can have infinite number of values which differ from each other by any multiple of $2\pi$

5) The unique value of $\theta$ such that $\pi < \theta <= \pi $ is called the principal value of the amplitude or principal argument

6) The principal argument of the complex number is find using the below steps

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 =$-\theta$

7) The conjugate of the complex number lies symmetrically about the x-axis

8) All the real number can be represent as complex number with zero imaginary part. like a +i(0) . So we can term real numbers as subset of bigger set of complex numbers

Just like a vector,A complex number on the argand plane for two things modulus and arg(z) which is direction. So all the complex number can be defined as position vector on the argand plane.

The addition ,substraction of complex number can happen just like vector addition and substraction

We can represent complex number is terms of polar coordinates

Let r be any non negative number and $\theta$ any real number. If we take $x=rcos\theta$ and $y=rsin\theta$ then, $r=\sqrt{x^2+y^2}$which is the modulus of z and $\theta=tan^{-1}\frac{y}{x}$ which is the argument or amplitude of z and is denoted by arg.z

we also have $x+iy=r(cos\theta+isin\theta)=r[cos(2n\pi+\theta)+isin(2n\pi+\theta)]$ , where n=0, $\pm $1, $\pm $2, ....

The unique value of $\theta$ such that $\pi < \theta <= \pi $ is called the principal value of the amplitude or principal argument

So we will be writing the polar coordinates form of complex number in principal argument form

**How to convert any complex number in Polar form**

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 =$-\theta$

Step 3) Polar form = |z|[cos (arg) + sin (arg)]

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