**π‘ Problem Formulation:** Python developers frequently need to work with complex numbers, particularly when dealing with fields such as signal processing or electronics. The need arises to convert these complex numbers into polar coordinates, which represent a complex number in terms of its magnitude and angle relative to the positive x-axis. For example, a complex number `(3+4j)`

would be represented in polar coordinates as roughly `(5.0, 0.927)`

, indicating a magnitude of 5.0 and an angle of 0.927 radians.

## Method 1: Using the `cmath`

Module

To convert a complex number to polar coordinates in Python, one can use the `cmath`

module, which offers the `polar()`

function specifically for this purpose. The function returns a tuple with the magnitude (absolute value) first, followed by the phase angle (argument).

Here’s an example:

import cmath complex_num = 3 + 4j polar_coordinates = cmath.polar(complex_num) print(polar_coordinates)

The output:

(5.0, 0.9272952180016122)

This code imports the `cmath`

module and then uses its `polar()`

function to convert the complex number 3 + 4j into polar coordinates, resulting in a magnitude of 5.0 and an angle of approximately 0.927 radians.

## Method 2: Manual Conversion

If for educational purposes or due to constraints, one needs to convert a complex number to polar coordinates manually, the Pythagorean theorem and the `math.atan2()`

function can be utilized. These give the magnitude and angle respectively using the real and imaginary parts of the number.

Here’s an example:

import math complex_num = 3 + 4j magnitude = math.sqrt(complex_num.real**2 + complex_num.imag**2) angle = math.atan2(complex_num.imag, complex_num.real) print((magnitude, angle))

The output:

(5.0, 0.9272952180016122)

This snippet calculates the magnitude using the square root of the sum of squares of the real and imaginary parts, and the angle using the `atan2()`

function from the `math`

module, yielding the same result as Method 1.

## Method 3: Using numpy and scipy

The `numpy`

and `scipy`

libraries offer functions that can be used for numerical computations like converting complex numbers to polar coordinates. By utilizing `numpy.abs()`

for the magnitude and `numpy.angle()`

for the phase angle, one can achieve the conversion succinctly.

Here’s an example:

import numpy as np complex_num = 3 + 4j magnitude = np.abs(complex_num) angle = np.angle(complex_num) print((magnitude, angle))

The output:

(5.0, 0.9272952180016122)

This code employs the `numpy`

library to compute the magnitude and angle directly using its built-in functions, which are particularly optimized for array operations and can be more efficient for a large set of complex numbers.

## Method 4: Using Python Properties

Python’s complex number type has built-in properties to directly access its real and imaginary parts. This knowledge can be leveraged with Math functions to manually compute polar coordinates without explicitly importing cmath.

Here’s an example:

import math complex_num = 3 + 4j magnitude = abs(complex_num) angle = math.atan2(complex_num.imag, complex_num.real) print((magnitude, angle))

The output:

(5.0, 0.9272952180016122)

This code utilizes Pythonβs ability to obtain the absolute value and the `imag`

and `real`

properties of a complex number alongside the `math`

module to calculate the polar coordinates in a more Pythonic way.

## Bonus One-Liner Method 5: Lambda Function

A one-liner using a lambda function can be crafted for quick conversions or inline operations. It employs the same logic as prior methods but condensed into a single line of code.

Here’s an example:

polar = lambda c: (abs(c), math.atan2(c.imag, c.real)) print(polar(3 + 4j))

The output:

(5.0, 0.9272952180016122)

This snippet demonstrates a concise lambda function storing the logic to convert a complex number to polar coordinates, showcasing Python’s capability for compact code.

## Summary/Discussion

**Method 1:**Using`cmath`

module. Strengths: Simple and direct, with the function designed for this exact purpose. Weaknesses: Requires importing an additional module.**Method 2:**Manual Conversion. Strengths: Educational value. No additional modules required. Weaknesses: More verbose and prone to human error.**Method 3:**Using`numpy`

and`scipy`

. Strengths: Efficient for array operations, which is good for processing multiple complex numbers. Weaknesses: Overhead of large library imports when working with individual numbers or simple scripts.**Method 4:**Using Python Properties. Strengths: Pythonic and straightforward, no need for the`cmath`

module. Weaknesses: Not as immediately clear as using the`cmath.polar()`

function.**Method 5:**Lambda Function. Strengths: Extremely concise, good for one-time or inline use. Weaknesses: Can become unreadable if overused or when working within larger, more complex codebases.