Table Of Content
Brush up Basics
The complex number z in geometrical form is written as z = x + iy.In geometrical representation complex number z is represented by a point P(x, y) on the complex plane or the argand plane where OA =x is x-intecept and AP=y is y-intercept.
Geometrical representation of complex number
The length of OP is called modulus of complex number and is denoted by |z|. Applying Pythagoras Theorem in ΔOAP we get
In polar representation a complex number z is represented by two parameters ‘r’ and ‘θ’. Parameter ‘r’ is the modulus of complex number and parameter ‘θ’ is the angle which the line OP makes with the positive direction of x-axis. It is also called argument of complex number and is denoted by arg(z).Finding the value of these two parameters from parameters x and y will help us convert the complex number to polar form.
Polar representation of complex number
The complex number in polar form is written as
Four Steps to convert
Step 1 : Plot the complex number
Plot the complex number x + iy represented by point P(x,y) in the x-y plane.
Point P(x,y) plotted on argand plane
Step 2 : Calculate the distance from Origin
Calculate the distance (OP) of the point P from the origin. Let this distance be denoted by ‘r’ such that
Point at a distance 'r' from origin
Step 3 : Find corresponding quadrant of a complex number
Determine the quadrant in which complex number x + iy lies.This can be done by determining the sign of x and y.Depending upon sign of x and y , following figure shows corresponding quadrant of a complex number.
Determine Quadrant of Point
Step 4 : Calculate the argument of complex number
To calculate argument of complex number find the smallest angle which the line OP makes with the x-axis.Let us call this angle ‘α’.Mathematically,
We need to calculate argument of z denoted by ‘θ’ .Argument of z is calculated using knowledge of quadrant of complex number.
Case 1 : Complex number in first quadrant
When point P belongs to first quadrant, then argument of z, arg(z) is equal to α.
Argument of Complex Number in first quadrant
Case 2 : Complex number in second quadrant
When point P belongs to second quadrant, then argument of z, arg(z) is equal to π – α.
Argument of Complex Number in second quadrant
Case 3 : Complex number in third quadrant
When point P belongs to third quadrant, the n argument of z, arg(z) is equal to –( π – α).
Argument of Complex Number in third quadrant
Case 4 : Complex number in fourth quadrant
When point P belongs to fourth quadrant, then argument of z, arg(z) is equal to –α.
Argument of Complex Number in fourth quadrant
- Put the value of ‘r’ and ‘θ’ calculated in step 2 and step 5 respectively in equation.
This is required polar form.
Example to clear it all
Convert complex number -1 + i into polar form.
Step 1 Therefore number -1 + i lie in second quadrant. Here, x = -1 < 0 and y = 1 > 0. Step 2 Plot the point P(-1,1) in argand plane.
Step 3 Calculate OP.
Step 4 Mark angle α.
Now,
Calculate argument
Since complex number lies in second quadrant.
Step 5 Put the values of ‘r’ and ‘θ’ in
is required polar form.