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

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:

, 
where a, b € R and b is known as the imaginary part of the complex number and 

• The polar form of complex number Z is:

where, r is known as modules of a complex number and
 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:
  • , which is known as the Eulerian form of the Complex Number.
  • The eulerian and polar forms both are represented as: .
• The multiplication and divisions of two complex numbers can be done using the eulerian form:

For Multiplication:

=> 

For Division:

=> 

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 .
• 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 
• Print the Division of two complex number Z1 and Z2 as 

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)