## How to convert complex number in geometrical form to polar form?

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.

#### 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.

[Refer case 2, step 5]

Step 5 Put the values of ‘r’ and ‘θ’ in

is required polar form.