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

• Last Updated : 14 Jun, 2022

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 .
• 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:

## Python3

 # Python program for the above approachimport math # Function to find the polar form# of the given Complex Numberdef 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 numbersdef 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 numbersdef 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 numbersdef 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 numbersdef 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 Codeif __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 Numberfunction get_polar_form(z){    // Z is in cartesian form    let re = z;    let im = z;     // 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 numbersfunction Addition(z1, z2){    // Z is in polar form    let r1 = z1;    let theta1 = z1;    let r2 = z2;    let theta2 = z2;     // 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 numbersfunction Subtraction(z1, z2){     // Z is in polar form    let r1 = z1;    let theta1 = z1;    let r2 = z2;    let theta2 = z2;     // 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 numbersfunction Multiplication(z1, z2){    // z is in polar form    let r1 = z1;    let theta1 = z1;    let r2 = z2;    let theta2 = z2;     // Return the multiplication of Z1 and Z2    return [r1 * r2, theta1 + theta2];} // Function to divide two complex numbersfunction Division(z1, z2){    // Z is in the polar form    let r1 = z1;    let theta1 = z1;    let r2 = z2;    let theta2 = z2;     // Return the division of Z1 and Z2    return [r1 / r2, theta1-theta2];}    // Driver Codez1 = [2, 3];z2 = [4, 6]; // Convert into Polar Formz1_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)

Output:

Polar form of the first
Complex Number:  (3.605551275463989, 0.9827937232473292)
Polar form of the Second
Complex Number:  (7.211102550927978, 0.9827937232473292)
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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