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
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