Product of Complex Numbers using three Multiplication Operation

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Given four integers a, b, c, and d which represents two complex numbers of the form (a + bi) and (c + di), the task is to find the product of the given complex numbers using only three multiplication operations.
Examples:

Input: a = 2, b = 3, c = 4 and d = 5
Output: -7 + 22i
Explanation:
Product is given by:
(2 + 3i)*(4 + 5i) = 2*4 + 4*3i + 2*5i + 3*5*(-1)
= 8 – 15 + (12 + 10)i
= -7 + 22i
Input: a = 3, b = 7, c = 6 and d = 2
Output: 4 + 48i

Naive Approach: The naive approach is to directly multiply given two complex numbers as:

=> (a + bi)*(c + di)
=> a(c + di) + b*i(c + di)
=> a*c + ad*i + b*c*i + b*d*i*i
=> (a*c – b*d) + (a*d + b*c)*i

The above operations would required four multiplication to find the product of two complex number.
Efficient Approach: The above approach required four multiplication to find the product. It can be reduced to three multiplication as:
Multiplication of two Complex Numbers is as follows:

(a + bi)*(c + di) = a*c – b*d + (a*d + b*c)i

Simplify real part:

real part = a*c – b*d
Let prod1 = a*c and prod2 = b*d.
Thus, real part = prod1 – prod2

Simplify the imaginary part as follows:

imaginary part = a*d + b*c
Adding and subtracting a*c and b*d in the above imaginar part we have,
imaginary part = a*c – a*c + a*d + b*c + b*d – b*d,
On rearranging the terms we get,
=> a*b + b*c + a*d + b*d – a*c – b*d
=> (a + b)*c + (a + b)*d – a*c – b*d
=> (a + b)*(c + d) – a*c – b*d
Let prod3 = (a + b)*(c + d)
Then the imaginary part is given by prod3 – (prod1 + prod2).

Thus, we need to find the value of prod1 = a * c, prod2 = b * d, and prod3 = ( a + b ) * ( c + d ).
So, our final answer will be:

Real Part = prod1 – prod2
Imaginary Part = prod3 – (prod1 + prod2)

Below is the implementation of the above approach:

C++

// C++ program for the above approach
#include <bits/stdc++.h>
using namespace std;
 
// Function to multiply Complex
// Numbers with just three
// multiplications
void print_product(int a, int b,
                   int c, int d)
{
    // Find value of prod1, prod2 and prod3
    int prod1 = a * c;
    int prod2 = b * d;
    int prod3 = (a + b) * (c + d);
 
    // Real Part
    int real = prod1 - prod2;
 
    // Imaginary Part
    int imag = prod3 - (prod1 + prod2);
 
    // Print the result
    cout << real << " + " << imag << "i";
}
 
// Driver Code
int main()
{
    int a, b, c, d;
 
    // Given four Numbers
    a = 2;
    b = 3;
    c = 4;
    d = 5;
 
    // Function Call
    print_product(a, b, c, d);
    return 0;
}

Java

// Java program for the above approach
class GFG{
 
// Function to multiply Complex
// Numbers with just three
// multiplications
static void print_product(int a, int b,
                          int c, int d)
{
     
    // Find value of prod1, prod2 and prod3
    int prod1 = a * c;
    int prod2 = b * d;
    int prod3 = (a + b) * (c + d);
 
    // Real Part
    int real = prod1 - prod2;
 
    // Imaginary Part
    int imag = prod3 - (prod1 + prod2);
 
    // Print the result
    System.out.println(real + " + " +
                       imag + "i");
}
 
// Driver Code
public static void main(String[] args)
{
     
    // Given four numbers
    int a = 2;
    int b = 3;
    int c = 4;
    int d = 5;
     
    // Function call
    print_product(a, b, c, d);
}
}
 
// This code is contributed by Pratima Pandey

Python3

# Python3 program for the above approach 
 
# Function to multiply Complex 
# Numbers with just three 
# multiplications 
def print_product(a, b, c, d):
     
    # Find value of prod1, prod2 
    # and prod3
    prod1 = a * c
    prod2 = b * d
    prod3 = (a + b) * (c + d)
 
    # Real part
    real = prod1 - prod2
 
    # Imaginary part
    imag = prod3 - (prod1 + prod2)
 
    # Print the result
    print(real, " + ", imag, "i") 
 
# Driver code
 
# Given four numbers
a = 2
b = 3
c = 4
d = 5
 
# Function call
print_product(a, b, c, d) 
 
# This code is contributed by Vishal Maurya. 

C#

// C# program for the above approach
using System;
class GFG{
 
// Function to multiply Complex
// Numbers with just three
// multiplications
static void print_product(int a, int b,
                          int c, int d)
{
    // Find value of prod1, prod2 and prod3
    int prod1 = a * c;
    int prod2 = b * d;
    int prod3 = (a + b) * (c + d);
 
    // Real Part
    int real = prod1 - prod2;
 
    // Imaginary Part
    int imag = prod3 - (prod1 + prod2);
 
    // Print the result
    Console.Write(real + " + " + imag + "i");
}
 
// Driver Code
public static void Main()
{
    int a, b, c, d;
 
    // Given four Numbers
    a = 2;
    b = 3;
    c = 4;
    d = 5;
 
    // Function Call
    print_product(a, b, c, d);
}
}
 
// This code is contributed by Code_Mech

Javascript

<script>
 
// Javascript program for the above approach
 
// Function to multiply Complex
// Numbers with just three
// multiplications
function print_product( a, b, c, d)
{
    // Find value of prod1, prod2 and prod3
    let prod1 = a * c;
    let prod2 = b * d;
    let prod3 = (a + b) * (c + d);
 
    // Real Part
    let real = prod1 - prod2;
 
    // Imaginary Part
    let imag = prod3 - (prod1 + prod2);
 
    // Print the result
    document.write(real + " + " + imag + "i");
}
 
 
// Driver Code
 
let a, b, c, d;
 
// Given four Numbers
a = 2;
b = 3;
c = 4;
d = 5;
 
// Function Call
print_product(a, b, c, d);
 
</script>

Output:

-7 + 22i

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

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