public class Complex {

  public double x,y;

  public Complex() {
    x = 0.0;
    y = 0.0;
  }

  public Complex(double x) {
    this.x = x;
    this.y = 0.0;
  }

  public Complex(double x, double y) {
    this.x = x;
    this.y = y;
  }
  
  public String toString() {
    return x+"+"+y+"i";
  }

  
  // find the inner product when the numbers are treated as vectors
  public double innerProduct (Complex w) {
    return x*w.x + y*w.y;
  }
  
  public static double innerProduct (Complex z, Complex w) {
    return z.x*w.x + z.y*w.y;
  }

  public double normSquared() {
    return x*x + y*y;
  }

  public double norm() {
    return Math.sqrt(x*x + y*y); 
  }
  
  public Complex conjugate() {
    return new Complex (x, -y);
  }

  public Complex negation () {
    return new Complex (- x, y); 
  }
  public Complex minus (Complex w) {
    return new Complex (x - w.x, y - w.y); 
  }
  
  public Complex minus (double t) {
    return new Complex (x - t, y); 
  }

  public static Complex subtract (Complex z, Complex w) {
    return new Complex (z.x - w.x, z.y - w.y);
   }

  public static Complex subtract (double t, Complex w) {
    return new Complex (t - w.x, -w.y);
   }
  
  public static Complex subtract (Complex z, double t) {
    return new Complex (z.x - t, z.y);
   }

  public Complex plus (Complex w) {
    return new Complex (x + w.x, y + w.y); 
  }
  
  public Complex plus (double t) {
    return new Complex (x + t, y); 
  }

  public static Complex add (Complex z, Complex w) {
    return new Complex (z.x + w.x, z.y + w.y);
   }

  public static Complex add (double t, Complex w) {
    return new Complex (t + w.x, w.y);
   }
  
  public static Complex add (Complex z, double t) {
    return new Complex (z.x + t, z.y);
   }

  public Complex times (Complex w) {
    return new Complex (x*w.x - y*w.y, y*w.x + x*w.y); 
  }
  
  public Complex times (double t) {
    return new Complex (t*x, t*y); 
  }

  public static Complex multiply (Complex z, Complex w) {
    return new Complex (z.x*w.x - z.y*w.y, z.y*w.x + z.x*w.y);
   }

  public static Complex multiply (double t, Complex w) {
    return new Complex (t*w.x, t*w.y);
   }
  
  public static Complex multiply (Complex z, double t) {
    return new Complex (t*z.x, t*z.y);
   }

  public Complex reciprocal() {
    double den = normSquared();
    return new Complex (x/den, -y/den);
   }
  public Complex over (Complex w) {
    double den = w.normSquared();
    return new Complex ((x*w.x + y*w.y)/den, (y*w.x - x*w.y)/den);
  }
  
  public Complex over (double t) {
    return new Complex (x/t, y/t); 
  }

  public static Complex divide (Complex z, Complex w) {
    double den = w.normSquared();
    return new Complex ((z.x*w.x + z.y*w.y)/den, (z.y*w.x - z.x*w.y)/den);
   }

  public static Complex divide (double t, Complex w) {
    double den = w.normSquared();
    return new Complex (t*w.x/den, -t*w.y/den);
   }
  
  public static Complex divide (Complex z, double t) {
    return new Complex (z.x/t, z.y/t);
   }
} // Complex