**π‘ Problem Formulation:** This article addresses the conversion of complex numbers, which are expressed as a combination of a real and an imaginary part (a + bi), into polar coordinates, which represent the same number as a radius (r) and an angle (ΞΈ). For example, converting the complex number 3 + 4i should result in a radius of 5 and an angle of 0.927 radians.

## Method 1: Using ‘cmath’ module

This method utilizes Pythonβs ‘cmath’ module which provides access to mathematical functions for complex numbers. The `cmath.polar()`

function directly converts a complex number object into a polar coordinate pair, returning a tuple with the magnitude (radius) and the phase (angle in radians).

Here’s an example:

import cmath # Convert a complex number to polar coordinates complex_num = 3 + 4j polar_coordinates = cmath.polar(complex_num) print("Polar Coordinates:", polar_coordinates)

Output:

Polar Coordinates: (5.0, 0.9272952180016122)

This method is straightforward and concise. In a single line, the `cmath.polar()`

function performs the conversion, making it an excellent choice for simplicity and readability.

## Method 2: Manual calculation using ‘math’ module

For those who want to understand the underlying mathematics, manual calculation using the ‘math’ module is an option. This involves computing the magnitude using Pythagoras’ theorem and the phase using the arctangent function `math.atan2()`

.

Here’s an example:

import math # Define the complex number real_part = 3.0 imaginary_part = 4.0 # Calculate magnitude and phase magnitude = math.sqrt(real_part**2 + imaginary_part**2) phase = math.atan2(imaginary_part, real_part) print("Polar Coordinates:", (magnitude, phase))

Output:

Polar Coordinates: (5.0, 0.9272952180016122)

This approach allows full control over the conversion process and is useful for educational purposes or environments where ‘cmath’ might not be available, though it’s a bit more involved than Method 1.

## Method 3: Using ‘numpy’ library

The ‘numpy’ library is commonly used for mathematical and scientific computing in Python. It can handle arrays of complex numbers and includes a function `numpy.angle()`

to compute the phase and `numpy.abs()`

for magnitude.

Here’s an example:

import numpy as np # Define the complex number as a numpy array complex_num = np.array(3 + 4j) # Calculate magnitude and phase magnitude = np.abs(complex_num) phase = np.angle(complex_num) print("Polar Coordinates:", (magnitude, phase))

Output:

Polar Coordinates: (5.0, 0.9272952180016122)

This method benefits from numpy’s performance optimizations, especially when processing large arrays of complex numbers. However, it requires numpy to be installed, which might not always be available.

## Method 4: Using ‘scipy’ library

The ‘scipy’ library extends ‘numpy’ with additional utilities, including the `scipy.spatial.transform.rotation`

module which provides tools for working with rotations in n dimensions, although itβs a bit of an overkill for simple conversions.

Here’s an example:

# Scipy does not directly provide an easier method than numpy # This is mostly included for academic interest.

Explanation of why ‘scipy’ might not be the best tool for a simple complex-to-polar conversion can go here.

## Bonus One-Liner Method 5: Using Lambda Function

For a quick, on-the-fly conversion, you can define a lambda function that wraps the functionality of the `cmath.polar()`

function into a one-liner that can be reused easily.

Here’s an example:

to_polar = lambda complex_num: cmath.polar(complex_num) print("Polar Coordinates:", to_polar(3 + 4j))

Output:

Polar Coordinates: (5.0, 0.9272952180016122)

This one-liner is concise and Pythonic, offering the simplicity of `cmath.polar()`

with the flexibility of a lambda function.

## Summary/Discussion

**Method 1:**Using cmath. Straightforward and concise. Requires no extra libraries.**Method 2:**Manual calculation. Offers a deeper understanding of the mathematics. Slightly more complex and verbose.**Method 3:**Using numpy. Optimized for large arrays of complex numbers. Requires numpy to be installed.**Method 4:**Using scipy. More features than necessary for this task, and less straightforward than numpy or cmath methods.**Method 5:**Lambda function. Pythonic and convenient for quick conversions, but not as clear for beginners.