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

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

Modulus of Complex number

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

Polar representation of complex number

The complex number in polar form is written as

image0031

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

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

Distance OP

Point at a distance 'r' from origin

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

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,

Tan alpha

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

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

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

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

Argument of Complex Number in fourth quadrant

  • Put the value of ‘r’ and ‘θ’ calculated in step 2 and step 5 respectively in equation.

image015

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.

image0151

Step 3 Calculate OP.

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Step 4 Mark angle α.

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Now,

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Calculate argument

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Since complex number lies in second quadrant.

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[Refer case 2, step 5]

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

image022

is required polar form.