Python program to convert complex number to polar coordinate values

Converting a complex number to polar coordinates involves finding the magnitude (radius) and phase angle. For a complex number x + yj, the radius is sqrt(x² + y²) and the angle is measured counter-clockwise from the positive x-axis.

Python's cmath module provides built-in functions: abs() for magnitude and phase() for the angle in radians.

Syntax

import cmath

magnitude = abs(complex_number)
angle = cmath.phase(complex_number)

Example

Let's convert the complex number 2+5j to polar coordinates ?

import cmath

def convert_to_polar(c):
    return (abs(c), cmath.phase(c))

c = 2+5j
polar_coords = convert_to_polar(c)
print(f"Complex number: {c}")
print(f"Polar coordinates: {polar_coords}")
print(f"Magnitude: {polar_coords[0]:.4f}")
print(f"Angle (radians): {polar_coords[1]:.4f}")

Complex number: (2+5j)
Polar coordinates: (5.385164807134504, 1.1902899496825317)
Magnitude: 5.3852
Angle (radians): 1.1903

Converting Multiple Complex Numbers

Here's how to convert multiple complex numbers at once ?

import cmath

def convert_to_polar(c):
    return (abs(c), cmath.phase(c))

complex_numbers = [2+5j, 3+4j, 1-2j, -2+3j]

for c in complex_numbers:
    magnitude, angle = convert_to_polar(c)
    print(f"{c} ? (r={magnitude:.3f}, ?={angle:.3f})")

(2+5j) ? (r=5.385, ?=1.190)
(3+4j) ? (r=5.000, ?=0.927)
(1-2j) ? (r=2.236, ?=-1.107)
(-2+3j) ? (r=3.606, ?=2.159)

Converting Angle to Degrees

To get the angle in degrees instead of radians ?

import cmath
import math

def convert_to_polar_degrees(c):
    magnitude = abs(c)
    angle_radians = cmath.phase(c)
    angle_degrees = math.degrees(angle_radians)
    return (magnitude, angle_degrees)

c = 2+5j
magnitude, angle = convert_to_polar_degrees(c)
print(f"Complex number: {c}")
print(f"Magnitude: {magnitude:.4f}")
print(f"Angle: {angle:.2f}°")

Complex number: (2+5j)
Magnitude: 5.3852
Angle: 68.20°

Key Points

• abs() returns the magnitude (distance from origin)

• cmath.phase() returns the angle in radians (-? to ?)

• Use math.degrees() to convert radians to degrees

• Negative angles indicate clockwise rotation from positive x-axis

Conclusion

Python's cmath module makes converting complex numbers to polar coordinates straightforward using abs() and phase(). The magnitude represents distance from origin while the phase angle shows direction in the complex plane.