Data Modelling & AI Data Structure & Algorithm

Convert given Complex Numbers into polar form and perform all arithmetic operations

30 July 2024

2

Given two Complex Numbers Z1 and Z2 in the Cartesian form, the task is to convert the given complex number into polar form and perform all the arithmetic operations ( addition, subtraction, multiplication, and division ) on them.

Examples:

Input: Z1 = (2, 3), Z2 = (4, 6)
Output:
Polar form of the first Complex Number: (3.605551275463989, 0.9827937232473292)
Polar form of the Second Complex Number: (7.211102550927978, 0.9827937232473292)
Addition of two Complex Numbers: (10.816653826391967, 0.9827937232473292)
Subtraction of two Complex Numbers: (3.605551275463989, -0.9827937232473292)
Multiplication of two Complex Numbers: (25.999999999999996, 1.9655874464946583)
Division of two Complex Numbers: (0.5, 0.0)

Input: Z1 = (1, 1), Z2 = (2, 2)
Output:
Polar form of the first Complex Number: (1.4142135623730951, 0.7853981633974482)
Polar form of the Second Complex Number: (2.8284271247461903, 0.7853981633974482)
Addition of two Complex Numbers: (4.242640687119286, 0.7853981633974482)
Subtraction of two Complex Numbers: (1.4142135623730951, -0.7853981633974482)
Multiplication of two Complex Numbers: (4.000000000000001, 1.5707963267948963)
Division of two Complex Numbers: (0.5, 0.0)

Approach: The given problem can be solved based on the following properties of Complex Numbers:

A complex number Z in Cartesian form is represented as:

$Z = a+ i\times b$ ,
where a, b € R and b is known as the imaginary part of the complex number and $i = \sqrt(-1)$

The polar form of complex number Z is:

$Z = r(sin(\theta) + i*cos(\theta))$

$r = \sqrt{a^{2}+ b^{2}}$
$\theta = tan^{-1}(b/a)$
$\theta = Sin^{-1} (b/r)$
$\theta = Cos^{-1} (a/r)$

where, r is known as modules of a complex number and
$\theta$ is the angle made with the positive X axis.

In the expression of complex number in polar form taking r as common performing the expression turn into:
- $Z = r*e^{i\theta}$ , which is known as the Eulerian form of the Complex Number.
- The eulerian and polar forms both are represented as: $(r, \theta)$ .
The multiplication and divisions of two complex numbers can be done using the eulerian form:

For Multiplication:

$Z = (r_{1}*e^{\theta_{1}})*(r_{2}*e^{\theta_{2}})$
=> $Z = (r_{1}*r_{2})*e^{ \theta_{1} + \theta_{2}}$

For Division:

$Z = (r_{1}*e^{\theta_{1}})\div (r_{2}*e^{\theta_{2}})$
=> $Z = (r_{1}\div r_{2})*e^{ \theta_{1} - \theta_{2}}$

Follow the steps below to solve the problem:

Convert the complex numbers into polar using the formula discussed-above and print it in the form for $(r, \theta)$ .
Define a function say Addition(Z1, Z2) to perform addition operation:
- Find the real part of the complex number by adding two real parts Z1 and Z2, and store it in a variable say a.
- Find the imaginary part of the complex number by adding two imaginary parts of the complex numbers Z1 and Z2 and store it in a variable say b.
- Convert the Cartesian form of the complex to polar form and print it.
Define a function say Subtraction(Z1, Z2) to perform subtraction operation:
- Find the real part of the complex number by subtracting two real parts Z1 and Z2, and store it in a variable say a.
- Find the imaginary part of the complex number by subtracting two imaginary parts of the complex numbers Z1 and Z2 and store it in a variable say b.
- Convert the Cartesian form of the complex to polar form and print it.
Print the multiplication of two complex number Z1 and Z2 as $Z = (r_{1}*r_{2})*e^{ \theta_{1} + \theta_{2}}$
Print the Division of two complex number Z1 and Z2 as $Z = (r_{1}\div r_{2})*e^{ \theta_{1} - \theta_{2}}$

Below is the implementation of the above approach:

C++

#include <iostream>
#include <cmath>
 
using namespace std;
 
// Function to find the polar form
// of the given Complex Number
pair<double, double> get_polar_form(double re, double im){
    // Z is in cartesian form
 
    // Stores the modulo of complex number
    double r = sqrt(re * re + im * im);
     
    // If r is greater than 0
    if(r){
        double theta = asin(im / r);
        return make_pair(r, theta);
    }
 
    // Otherwise
    else{
        return make_pair(0, 0);
    }
}
   
 
// Function to add two complex numbers
pair<double, double> Addition(pair<double, double> z1, pair<double, double> z2){
    // Z is in polar form
    double r1 = z1.first;
    double theta1 = z1.second;
    double r2 = z2.first;
    double theta2 = z2.second;
 
    // Real part of complex number
    double a = r1 * cos(theta1) + r2 * cos(theta2);
     
    // Imaginary part of complex Number
    double b = r1 * sin(theta1) + r2 * sin(theta2);
 
    // Find the polar form
    return get_polar_form(a, b);
}
   
 
 
// Function to subtract two
// given complex numbers
pair<double, double> Subtraction(pair<double, double> z1, pair<double, double> z2){  
    // Z is in polar form
    double r1 = z1.first;
    double theta1 = z1.second;
    double r2 = z2.first;
    double theta2 = z2.second;
 
    // Real part of the complex number
    double a = r1 * cos(theta1) - r2 * cos(theta2);
     
    // Imaginary part of complex number
    double b = r1 * sin(theta1) - r2 * sin(theta2);
 
    // Converts (a, b) to polar
    // form and return
    return get_polar_form(a, b);  
}
 
 
// Function to multiply two complex numbers
pair<double, double> Multiplication(pair<double, double> z1, pair<double, double> z2){
    // z is in polar form
    double r1 = z1.first;
    double theta1 = z1.second;
    double r2 = z2.first;
    double theta2 = z2.second;
 
    // Return the multiplication of Z1 and Z2
    return make_pair(r1 * r2, theta1 + theta2);
}
 
// Function to divide two complex numbers
pair<double, double> Division(pair<double, double> z1, pair<double, double> z2){
    // Z is in the polar form
    double r1 = z1.first;
    double theta1 = z1.second;
    double r2 = z2.first;
    double theta2 = z2.second;
     
    // Return the division of Z1 and Z2
    return make_pair(r1 / r2, theta1 - theta2);
}
 
// Driver Code
int main() {
    pair<double, double> z1 = make_pair(2.0, 3.0);
    pair<double, double> z2 = make_pair(4.0, 6.0);
 
    // Convert into Polar Form
    auto z1_polar = get_polar_form(z1.first, z1.second);
    auto z2_polar = get_polar_form(z2.first, z2.second);
 
    cout << "Polar form of the first Complex Number: " << z1_polar.first << " " << z1_polar.second << endl;
    cout << "Polar form of the Second Complex Number: " << z2_polar.first << " " << z2_polar.second << endl;
 
    cout << "Addition of two Complex Numbers: " << Addition(z1_polar, z2_polar).first << " " << Addition(z1_polar, z2_polar).second << endl;
 
    cout << "Subtraction of two Complex Numbers: " << Subtraction(z1_polar, z2_polar).first << " " << Subtraction(z1_polar, z2_polar).second << endl;
 
    cout << "Multiplication of two Complex Numbers: " << Multiplication(z1_polar, z2_polar).first << " " << Multiplication(z1_polar, z2_polar).second << endl;
 
    cout << "Division of two Complex Numbers: " << Division(z1_polar, z2_polar).first << " " << Division(z1_polar, z2_polar).second << endl;
 
    return 0;
}
 
// This code is contributed by phasing17.

Java

import java.util.*;
 
class GFG {
 
  // Function to find the polar form of the given Complex Number
  public static double[] GetPolarForm(double re, double im) {
    // Z is in cartesian form
 
    // Stores the modulo of complex number
    double r = Math.sqrt(re * re + im * im);
 
    // If r is greater than 0
    if (r > 0) {
      double theta = Math.asin(im / r);
      return new double[] {r, theta};
    }
 
    // Otherwise
    else {
      return new double[] {0.00, 0.00};
    }
  }
 
  // Function to add two complex numbers
  public static double[] Addition(double[] z1, double[] z2) {
    // Z is in polar form
    double r1 = z1[0];
    double theta1 = z1[1];
    double r2 = z2[0];
    double theta2 = z2[1];
 
    // Real part of complex number
    double a = r1 * Math.cos(theta1) + r2 * Math.cos(theta2);
 
    // Imaginary part of complex Number
    double b = r1 * Math.sin(theta1) + r2 * Math.sin(theta2);
 
    // Find the polar form
    return GetPolarForm(a, b);
  }
 
  // Function to subtract two given complex numbers
  public static double[] Subtraction(double[] z1, double[] z2) {
    // Z is in polar form
    double r1 = z1[0];
    double theta1 = z1[1];
    double r2 = z2[0];
    double theta2 = z2[1];
 
    // Real part of the complex number
    double a = r1 * Math.cos(theta1) - r2 * Math.cos(theta2);
 
    // Imaginary part of complex number
    double b = r1 * Math.sin(theta1) - r2 * Math.sin(theta2);
 
    // Converts (a, b) to polar form and return
    return GetPolarForm(a, b);
  }
 
  // Function to multiply two complex numbers
  public static double[] Multiplication(double[] z1, double[] z2) {
    // z is in polar form
    double r1 = z1[0];
    double theta1 = z1[1];
    double r2 = z2[0];
    double theta2 = z2[1];
 
    // Return the multiplication of Z1 and Z2
    return new double[] {r1 * r2, theta1 + theta2};
  }
 
  // Function to divide two complex numbers
  public static double[] Division(double[] z1, double[] z2) {
    // Z is in the polar form
    double r1 = z1[0];
    double theta1 = z1[1];
    double r2 = z2[0];
    double theta2 = z2[1];
 
    // Return the division of Z1 and Z2
    return new double[] {r1 / r2, theta1 - theta2};
  }
 
  public static void main(String[] args) {
    double[] z1 = {2.0, 3.0};
    double[] z2 = {4.0, 6.0};
 
    // Convert into Polar Form
    var z1_polar = GetPolarForm(z1[0], z1[1]);
    var z2_polar = GetPolarForm(z2[0], z2[1]);
 
    System.out.println(
      "Polar form of the first Complex Number: "
      + z1_polar[0] + " " + z1_polar[1]);
    System.out.println(
      "Polar form of the Second Complex Number: "
      + z2_polar[0] + " " + z2_polar[1]);
 
    System.out.println(
      "Addition of two Complex Numbers: "
      + Arrays.toString(Addition(z1_polar, z2_polar)));
    System.out.println(
      "Subtraction of two Complex Numbers: "
      + Arrays.toString(Subtraction(z1_polar, z2_polar)));
    System.out.println(
      "Multiplication of two Complex Numbers: "
      + Arrays.toString(Multiplication(z1_polar, z2_polar)));
    System.out.println(
      "Division of two Complex Numbers: "
      + Arrays.toString(Division(z1_polar, z2_polar)));
  }
}
 
// This code is contributed by phasing17.

Python3

# Python program for the above approach
import math
 
# Function to find the polar form
# of the given Complex Number
def get_polar_form(z):
   
    # Z is in cartesian form
    re, im = z
 
    # Stores the modulo of complex number
    r = (re * re + im * im) ** 0.5
 
    # If r is greater than 0
    if r:
        theta = math.asin(im / r)
        return (r, theta)
       
    # Otherwise
    else:
        return (0, 0)
 
# Function to add two complex numbers
def Addition(z1, z2):
   
    # Z is in polar form
    r1, theta1 = z1
    r2, theta2 = z2
 
    # Real part of complex number
    a = r1 * math.cos(theta1) + r2 * math.cos(theta2)
     
    # Imaginary part of complex Number
    b = r1 * math.sin(theta1) + r2 * math.sin(theta2)
     
    # Find the polar form
    return get_polar_form((a, b))
 
# Function to subtract two
# given complex numbers
def Subtraction(z1, z2):
   
    # Z is in polar form
    r1, theta1 = z1
    r2, theta2 = z2
 
    # Real part of the complex number
    a = r1 * math.cos(theta1) - r2 * math.cos(theta2)
     
    # Imaginary part of complex number
    b = r1 * math.sin(theta1) - r2 * math.sin(theta2)
 
    # Converts (a, b) to polar
    # form and return
    return get_polar_form((a, b))
 
# Function to multiply two complex numbers
def Multiplication(z1, z2):
   
    # z is in polar form
    r1, theta1 = z1
    r2, theta2 = z2
 
    # Return the multiplication of Z1 and Z2
    return (r1 * r2, theta1 + theta2)
 
 
# Function to divide two complex numbers
def Division(z1, z2):
   
    # Z is in the polar form
    r1, theta1 = z1
    r2, theta2 = z2
 
    # Return the division of Z1 and Z2
    return (r1 / r2, theta1-theta2)
 
 
# Driver Code
if __name__ == "__main__":
   
    z1 = (2, 3)
    z2 = (4, 6)
 
    # Convert into Polar Form
    z1_polar = get_polar_form(z1)
    z2_polar = get_polar_form(z2)
 
    print("Polar form of the first")
    print("Complex Number: ", z1_polar)
    print("Polar form of the Second")
    print("Complex Number: ", z2_polar)
 
    print("Addition of two complex")
    print("Numbers: ", Addition(z1_polar, z2_polar))
     
    print("Subtraction of two ")
    print("complex Numbers: ",
           Subtraction(z1_polar, z2_polar))
     
    print("Multiplication of two ")
    print("Complex Numbers: ",
           Multiplication(z1_polar, z2_polar))
           
    print("Division of two complex ")
    print("Numbers: ", Division(z1_polar, z2_polar))

Javascript

// JavaScript program for the above approach
 
// Function to find the polar form
// of the given Complex Number
function get_polar_form(z){
    // Z is in cartesian form
    let re = z[0];
    let im = z[1];
 
    // Stores the modulo of complex number
    let r = (re * re + im * im) ** 0.5;
     
    // If r is greater than 0
    if(r){
        let theta = Math.asin(im / r);
        return [r, theta];
    }
 
    // Otherwise
    else{
        return [0, 0];
    }
         
}
   
 
// Function to add two complex numbers
function Addition(z1, z2){
    // Z is in polar form
    let r1 = z1[0];
    let theta1 = z1[1];
    let r2 = z2[0];
    let theta2 = z2[1];
 
    // console.log(r1, r2, theta1, theta2)
    // Real part of complex number
    let a = r1 * Math.cos(theta1) + r2 * Math.cos(theta2);
     
    // Imaginary part of complex Number
    let b = r1 * Math.sin(theta1) + r2 * Math.sin(theta2);
    console.log(a, b)
    // Find the polar form
    return get_polar_form([a, b]);
}
   
 
 
// Function to subtract two
// given complex numbers
function Subtraction(z1, z2){  
    // Z is in polar form
    let r1 = z1[0];
    let theta1 = z1[1];
    let r2 = z2[0];
    let theta2 = z2[1];
 
    // Real part of the complex number
    let a = r1 * Math.cos(theta1) - r2 * Math.cos(theta2);
     
    // Imaginary part of complex number
    let b = r1 * Math.sin(theta1) - r2 * Math.sin(theta2);
 
    // Converts (a, b) to polar
    // form and return
    return get_polar_form([a, b]);  
}
 
 
// Function to multiply two complex numbers
function Multiplication(z1, z2){
    // z is in polar form
    let r1 = z1[0];
    let theta1 = z1[1];
    let r2 = z2[0];
    let theta2 = z2[1];
 
    // Return the multiplication of Z1 and Z2
    return [r1 * r2, theta1 + theta2];
}
 
// Function to divide two complex numbers
function Division(z1, z2){
    // Z is in the polar form
    let r1 = z1[0];
    let theta1 = z1[1];
    let r2 = z2[0];
    let theta2 = z2[1];
 
    // Return the division of Z1 and Z2
    return [r1 / r2, theta1-theta2];
}
   
 
// Driver Code
z1 = [2, 3];
z2 = [4, 6];
 
// Convert into Polar Form
z1_polar = get_polar_form(z1)
z2_polar = get_polar_form(z2)
 
console.log("Polar form of the first");
console.log("Complex Number: ", z1_polar);
console.log("Polar form of the Second");
console.log("Complex Number: ", z2_polar);
 
console.log("Addition of two complex");
console.log("Numbers: ", Addition(z1_polar, z2_polar));
 
console.log("Subtraction of two ");
console.log("complex Numbers: ",
       Subtraction(z1_polar, z2_polar));
 
console.log("Multiplication of two ");
console.log("Complex Numbers: ",
       Multiplication(z1_polar, z2_polar));
 
console.log("Division of two complex ");
console.log("Numbers: ", Division(z1_polar, z2_polar));
 
// The code is contributed by Gautam goel (gautamgoel962)

C#

// C# code to implement the approach
 
using System;
using System.Collections.Generic;
 
class GFG {
    // Function to find the polar form of the given Complex
    // Number
    public static Tuple<double, double>
    GetPolarForm(double re, double im)
    {
        // Z is in cartesian form
 
        // Stores the modulo of complex number
        double r = Math.Sqrt(re * re + im * im);
 
        // If r is greater than 0
        if (r > 0) {
            double theta = Math.Asin(im / r);
            return Tuple.Create(r, theta);
        }
 
        // Otherwise
        else {
            return Tuple.Create(0.00, 0.00);
        }
    }
 
    // Function to add two complex numbers
    public static Tuple<double, double>
    Addition(Tuple<double, double> z1,
             Tuple<double, double> z2)
    {
        // Z is in polar form
        double r1 = z1.Item1;
        double theta1 = z1.Item2;
        double r2 = z2.Item1;
        double theta2 = z2.Item2;
 
        // Real part of complex number
        double a
            = r1 * Math.Cos(theta1) + r2 * Math.Cos(theta2);
 
        // Imaginary part of complex Number
        double b
            = r1 * Math.Sin(theta1) + r2 * Math.Sin(theta2);
 
        // Find the polar form
        return GetPolarForm(a, b);
    }
 
    // Function to subtract two given complex numbers
    public static Tuple<double, double>
    Subtraction(Tuple<double, double> z1,
                Tuple<double, double> z2)
    {
        // Z is in polar form
        double r1 = z1.Item1;
        double theta1 = z1.Item2;
        double r2 = z2.Item1;
        double theta2 = z2.Item2;
 
        // Real part of the complex number
        double a
            = r1 * Math.Cos(theta1) - r2 * Math.Cos(theta2);
 
        // Imaginary part of complex number
        double b
            = r1 * Math.Sin(theta1) - r2 * Math.Sin(theta2);
 
        // Converts (a, b) to polar form and return
        return GetPolarForm(a, b);
    }
 
    // Function to multiply two complex numbers
    public static Tuple<double, double>
    Multiplication(Tuple<double, double> z1,
                   Tuple<double, double> z2)
    {
        // z is in polar form
        double r1 = z1.Item1;
        double theta1 = z1.Item2;
        double r2 = z2.Item1;
        double theta2 = z2.Item2;
 
        // Return the multiplication of Z1 and Z2
        return Tuple.Create(r1 * r2, theta1 + theta2);
    }
 
    // Function to divide two complex numbers
    public static Tuple<double, double>
    Division(Tuple<double, double> z1,
             Tuple<double, double> z2)
    {
        // Z is in the polar form
        double r1 = z1.Item1;
        double theta1 = z1.Item2;
        double r2 = z2.Item1;
        double theta2 = z2.Item2;
 
        // Return the division of Z1 and Z2
        return Tuple.Create(r1 / r2, theta1 - theta2);
    }
 
    // Driver Code
    public static void Main(string[] args)
    {
        Tuple<double, double> z1 = Tuple.Create(2.0, 3.0);
        Tuple<double, double> z2 = Tuple.Create(4.0, 6.0);
 
        // Convert into Polar Form
        var z1_polar = GetPolarForm(z1.Item1, z1.Item2);
        var z2_polar = GetPolarForm(z2.Item1, z2.Item2);
 
        Console.WriteLine(
            "Polar form of the first Complex Number: "
            + z1_polar.Item1 + " " + z1_polar.Item2);
        Console.WriteLine(
            "Polar form of the Second Complex Number: "
            + z2_polar.Item1 + " " + z2_polar.Item2);
 
        Console.WriteLine(
            "Addition of two Complex Numbers: "
            + Addition(z1_polar, z2_polar));
        Console.WriteLine(
            "Subtraction of two Complex Numbers: "
            + Subtraction(z1_polar, z2_polar));
        Console.WriteLine(
            "Multiplication of two Complex Numbers: "
            + Multiplication(z1_polar, z2_polar));
        Console.WriteLine(
            "Division of two Complex Numbers: "
            + Division(z1_polar, z2_polar));
    }
}
 
// This code is contributed by phasing17

Output:

Polar form of the first
Complex Number:  (3.605551275463989, 0.9827937232473292)
Polar form of the Second
Complex Number:  (7.211102550927978, 0.9827937232473292)
Addition of two complex
Numbers:  (10.816653826391967, 0.9827937232473292)
Subtraction of two 
complex Numbers:  (3.605551275463989, -0.9827937232473292)
Multiplication of two 
Complex Numbers:  (25.999999999999996, 1.9655874464946583)
Division of two complex 
Numbers:  (0.5, 0.0)

Time Complexity: O(1)
Auxiliary Space: O(1)

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Convert given Complex Numbers into polar form and perform all arithmetic operations

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