Representation of complex numbers

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On an Argand Plane complex numbers can be represent from following three ways:

 1. Co-ordinate representation
2. Vector representation
3. Polar Representation

 1. In co-ordinate representation any complex number z = x + iy can be represented as (x,y) where x is real part of complex number and y is imaginary part of complex number

 2. In Vector representation any complex number z = x + iy is represented by a vector from origin to the point z i.e. (x,y)

 Any vector has two things
1. magnitude 
2. direction

 here |z| is the magnitude of complex number 
and \theta  is the Argument of the complex number (direction)

1. Modulus of Complex number |z|:

 Modulus of complex number is defined as the length of vector from origin to the point z i.e. (x,y)

 i.e. |z| = \sqrt {{x^2} + {y^2}}

in polar representation, |z| = r = \sqrt {{x^2} + {y^2}}

2. Argument of complex number arg(z):

 Argument of a complex number is the angle between the line joining z to the origin with positive direction of x-axis

 Argument has infinite values as if \theta  is an argument then 2n\pi + \theta  will also be a valid argument. So, we have to take a principal argument of z to determine a unique value for a given complex number.

 The principal Argument value of z \in ( - \pi ,\pi ]. 

 If complex number is above x-axis we consider the argument in [0,pi] anticlockwise direction and if below the x-axis consider the argument in       (-pi,0] in clockwise direction.


Cartesian representation, z = x + iy or (x,y)
Polar representation, z = (r,\theta ) where, |z| = r = \sqrt {{x^2} + {y^2}}  and \arg (z) = \theta


Addition & Subtraction of Complex numbers

 Addition & Subtraction of Complex Numbers follow the law of addition and subtraction of vectors.

If, {z_1} = {x_1} + i{y_1}
and {z_2} = {x_2} + i{y_2}

 Then, z = {z_1} + {z_2} is addition of vector {z_1} and {z_2}

 z = {x_1} + {x_2} + i({y_1} + {y_2})

 And, z = {z_1} - {z_2} is addition of vector {z_1} and  - {z_2}
z = {x_1} - {x_2} + i({y_1} - {y_2})

 Negative of z (Reflection about Origin)
i.e. -z = -x - iy 

 and Conjugate of z ( Reflection about x-axis)
i.e.\overline z = x - iy


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