# Product of Complex Numbers using three Multiplication Operation

• Last Updated : 29 Nov, 2021

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 ``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

 ``

Output:

`-7 + 22i`

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

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