- What is complex numbers
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- Algebra Of complex Numbers
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- Conjugate of Complex Numbers
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- Modulus of complex numbers
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- Argand Plane
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- Polar Representation of the complex number
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- Rotation of Complex Number
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- Identities for Complex Numbers
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- Eulers formula and De moivre's theorem
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- Cube Root of unity

Such a diagram is called Argand Plane or Complex plane

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

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

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

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)]

z=-1-i

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 $arg=\frac{-3\pi}{4}$

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

so

$z=

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