#include <iostream>
#include <cmath>

class Complex
{
public:
  Complex(double = 0., double = 0.);
  double real() const;
  double imag() const  {
    return m_imag;
  }
  double mag() const ;
  double mag2() const ;
  Complex coniugate() const ;
  Complex inverse() const ;
  Complex& operator+=(Complex const& rhs);

 private:
  double m_real;
  double m_imag;
};

inline double Complex::real() const{
  return m_real;
}

Complex operator+(Complex const& lhs, Complex const& rhs);
Complex operator-(Complex const& lhs, Complex const& rhs);
Complex operator*(Complex const& lhs, Complex const& rhs);
Complex operator/(Complex const& lhs, Complex const& rhs);
Complex operator-(Complex const& lhs);
double abs(Complex const& c);

std::ostream& operator<<(std::ostream&, Complex const&);

Complex::Complex(double re, double im)  : m_real(re), m_imag(im){
}

Complex& Complex::operator+=(Complex const& rhs) {
  m_real += rhs.m_real;
  m_imag += rhs.m_imag;
  
  return *this;
}

Complex operator-(Complex const& rhs){
  return Complex(-rhs.real(), -rhs.imag());
}

Complex operator+(Complex const& lhs, Complex const& rhs){
  return Complex(lhs.real() + rhs.real(), lhs.imag() + rhs.imag());
}

Complex operator-(Complex const& lhs, Complex const& rhs){
  return Complex(lhs.real() - rhs.real(), lhs.imag() - rhs.imag());
}

std::ostream& operator<<(std::ostream& os, Complex const& c){
  //(re,im)
  os << c.real() << "+i" << c.imag() ;
  return os;
}

Complex operator*(Complex const& lhs, Complex const& rhs){
  return Complex(lhs.real()*rhs.real()-lhs.imag()*rhs.imag(), lhs.imag()*rhs.real()+ rhs.imag()*lhs.real());
}

Complex operator/(Complex const& lhs, Complex const& rhs){
  return lhs*rhs.inverse();
}


Complex Complex::coniugate() const {
  return Complex(real(),-imag());
}
Complex Complex::inverse() const {
  return 1./mag2()*coniugate();
}
double Complex::mag2() const {
  return (real()*real()+imag()*imag());
}
double Complex::mag() const {
  return sqrt(real()*real()+imag()*imag());
}