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// multiplicationsvoid 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 Codeint 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 approachclass GFG{// Function to multiply Complex// Numbers with just three// multiplicationsstatic 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 Codepublic 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 numbersa = 2b = 3c = 4d = 5# Function callprint_product(a, b, c, d) # This code is contributed by Vishal Maurya. |
C#
// C# program for the above approachusing System;class GFG{// Function to multiply Complex// Numbers with just three// multiplicationsstatic 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 Codepublic 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// multiplicationsfunction 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 Codelet a, b, c, d;// Given four Numbersa = 2;b = 3;c = 4;d = 5;// Function Callprint_product(a, b, c, d);</script> |
-7 + 22i
Time Complexity: O(1)
Auxiliary Space: O(1)
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