Complex number Multiplication and Division

The arithmetic operation like multiplication and division over two Complex numbers is explained

Complex number Multiplication

multiply two complex numbers z1 and z2. note i^2 = -1

       z1 =  a + bi
       z2 =  c + di

      z1*z2   =   (a+bi)  * (c+di)
              =   a*c + a*di  + bi*c + bi*di
              =   a*c + a*di  + bi*c + b*d*(i^2)
              =   a*c + a*di  + bi*c  + b*d*(-1)
              =   a*c + a*di  + c*bi - b*d
              =  (a*c - b*d)  + (a*di + c*bi)

Complex number Division

divide two complex numbers z1 and z2. note i^2= -1

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

  multiply numerator and denumerator by conjugate  z2
  
      conjugate z2  =  c  + (-1)* di
                          =  c  - di

      z1 / z2   =   (a+bi) (c-di)  /  (c+di ) (c-di)   

     (a+bi) (c-di)   =  a*c - a*di + c*bi   -bi*di
                     =  a*c - a*di + c*bi   -b*d*(i^2)
                     =  a*c - a*di + c*bi   -b*d*(-1)  
                     =  a*c - a*di + c*bi   +b*d
              =  (a*c + b*d)  - a*di + c*bi  
   
   (c+di) (c-di)  =  c*c  - c*di + c*di - di*di
                  =  c^2  - d^2 i^2
                  = c^2  - d^2 (-1)
                  = c^2  + d^2

  z1 / z2   =      (a*c + b*d)  + (c*di - a*bi)   /  c^2  + d^2

Java code - Complex multiplication and division

The following Java code implements arithmetic operation multiplication and division on two Complex numbers

package math.complex;

public class MulDiv {

 public static Complex multiply(Complex C1,Complex C2) 
     {
  
     Complex tmp = new Complex();     
     tmp.real = C1.real * C2.real;
     tmp.imag = C1.real * C2.imag;     
     
     tmp.imag += C1.imag * C2.real;
     tmp.real += C1.imag * C2.imag*(-1);
     
     return tmp;
    }

  public static Complex divide(Complex C1,Complex C2){
  
    Complex conjc2 = C2.conjucate();  
    Complex numr  = MulDiv.multiply(C1, conjc2);
    Complex demr  = MulDiv.multiply(C2, conjc2);    
      
    double dval =  demr.real + demr.imag;  
    Complex tmp = new Complex ( numr.real/dval,  numr.imag/dval );
    return tmp;
  
 }

 
  public static void main(String[] args) {
  
  
 System.out.println("Complex Number Arithmetic Operation in java n");
  
  System.out.println("Complex Number Multiplication n");        
  Complex C1 = new Complex(-1,2);
  System.out.println("Complex  C1: " +C1);  
  Complex C2 = new Complex(2,-1);
  System.out.println("Complex  C2: " +C2);
  
  Complex C3= MulDiv.multiply(C1, C2);
  System.out.println("C3 = C1 x C2 : " + C3 );
  
  
  System.out.println("Complex Number Division  n");
    
  System.out.println("Complex  C1: " +C1);    
  System.out.println("Complex  C2: " +C2);
  Complex C4= MulDiv.divide(C1, C2);
  System.out.println("C4 = C1 / C2 : " +C4);

 }

}

Output

Complex Number Arithmetic Operation in java 

Complex Number Multiplication 

Complex  C1: -1.0+2.0i
Complex  C2: 2.0-1.0i
C3 = C1 x C2 : 0.0+5.0i

Complex Number Division  

Complex  C1: -1.0+2.0i
Complex  C2: 2.0-1.0i
C4 = C1 / C2 : -0.8+0.6i