Complex Number Multiplication in C++

In this tutorial, we will learn complex number multiplication in C++.

Complex Numbers

Complex numbers are a combination of a (real part)  and b ( imaginary part )

written in the form a + bi

a, b are real numbers.

i²=-1, as no real number satisfies this equation so i is called an IMAGINARY number

Complex Numbers Multiplication:

Any two complex numbers can be multiplied to yield another complex number.

Let z1=a+bi  and z2=c+di be two complex numbers where

On multiplying z1 and z2:

z1.z2=(a+bi)(c+di)

       =ac+adi+bci+bdi²

       =ac+(ad+bc)i-bd                (as  i²=-1)

        =( ac – bd ) + ( ad + bc )  i            //which is also a complex number

Complex Number Multiplication Properties:

Commutative Property

z1 = a+ ib

z2 = c + id

z ₁ . z ₂ = z1 . z2

Associative Property

z1 = a+ ib

z2 = c + id

z3 = e + if

(z₁ . z₂ ). z₃ = (ace -bde – adf -bcf) + i(ade + bce + acf -bdf) = z ₁ .(z ₂ . z ₃ )

Example

(1+4i)(5+i)   ​= 1*5+1i+4i*5+4i*i    =   5+i+20i+4i²    =  5+21i+4i²

Complex Number Multiplication C++ Code

The class complex is created, each object of the class has two attributes- real and imag which are used for storing the real part and imaginary part of the complex number.

Each object of the class also has a function- called display( ), used to print the complex number in the form (real part)+ i(complex part).

#include<bits/stdc++.h>
using namespace std;

//C++ Program for multiplication of two complex nos

typedef struct Complex{
  float real;
  float imag;
  void display();

}Complex;

void Complex::display()
{
   cout<<real<<"+"<<imag<<"i"<<endl;
}

Complex Multiply(Complex , Complex);

int main(){
  Complex z1 , z2 , ans;
  
  cout<<"Enter the real and imaginary parts of the first complex number:";
  cin>>z1.real>>z1.imag;
  cout<<"z1 = ";
  z1.display();
  //z1=a+bi
  
  cout<<"Enter the real and imaginary parts of the second complex number:";
  cin>>z2.real>>z2.imag;
  cout<<"z2 = ";
  z2.display();
  //z2=c+di
  
  //ans=(ac-bd)+i(ad+bc)
  ans.real=(z1.real*z2.real)-(z1.imag*z2.imag);
  ans.imag=(z1.real*z2.imag)+(z1.imag*z2.real);
  cout<<"The result after the multiplication of z1 and z2 is: "<<endl;
  cout<<"z1.z2 = ";
  ans.display();
  return 0;
}

Output

Enter the real and imaginary parts of the first complex number: 4   6

z1 = 4+6i

Enter the real and imaginary parts of the second complex number: 3   8

z2 = 3+8i

The result after the multiplication of z1 and z2 is:

z1.z2 = -36+50i