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Python Program to convert complex numbers to Polar coordinates

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  • Difficulty Level : Expert
  • Last Updated : 19 Jul, 2022
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Before starting with the program, let’s see the basics of Polar Coordinates and then use Python’s cmath and abs module to convert it. Polar coordinates are just a different way of representing Cartesian coordinates or Complex Numbers. A complex number  z is defined as :


It is completely determined by its real part x and imaginary part y. Here, j is the imaginary unit.

The polar coordinates (r , φ) is completely determined by modulus r and phase angle φ.


  • r: Distance from z to origin, i.e., 

r = \sqrt{x^{2}+y^{2}}

  •   φ: Counterclockwise angle measured from the positive x-axis to the line segment that joins z to the origin.

The conversion of complex numbers to polar co-ordinates are explained below with examples.

Using cmath module

Python’s cmath module provides access to the mathematical functions for complex numbers. It contains several functions that are used for converting coordinates from one domain to another. 

Out of them, some are explained as-

1. cmath.polar(x):

Return the representation of x in polar coordinates. cmath.polar() function is used to convert a complex number to polar coordinates. 


# Python code to implement
# the polar()function
# importing "cmath"
# for mathematical operations
import cmath
# using cmath.polar() method
num = cmath.polar(1)


(1.0, 0.0)

Time Complexity: O(1)

Auxiliary space: O(1)

2. cmath.phase (z): This method returns the phase of complex number z(also known as the argument of z).


import cmath
x = -1.0
y = 0.0
z = complex(x,y);
# printing phase of a complex number using phase()
print ("The phase of complex number is : ",end="")
print (cmath.phase(z))


The phase of complex number is : 3.141592653589793

Using abs()

abs(): This method returns the modulus (absolute value) of complex number z.


num1 = 3 + 4j
print('Absolute value of 3 + 4j is:', abs(num1))


Absolute value of 3 + 4j is: 5.0

You are given a complex number z and your task is to convert it to polar coordinates.

Let us consider a complex number as 1+5j, and we need to convert it to Polar coordinates. 


import cmath
c = complex(1+5j)



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