Fraction implement by C++

fraction.cpp

#include <cstdio>
#include <cstring>
#include <iostream>
#include <stdexcept>

using namespace std;

class Fraction {
    int numerator;
    int denominator;
public:
    Fraction(int n, int d);
    ~Fraction();
    Fraction operator+(const Fraction& f);
    Fraction operator-(const Fraction& f);
    Fraction operator*(const Fraction& f);
    Fraction operator/(const Fraction& f);
    Fraction operator+=(const Fraction& f);
    Fraction operator++();
    Fraction operator++(int);
    bool operator>(const Fraction& f);
    bool operator==(const Fraction& f);
    operator double();
    friend ostream &operator<<(ostream &output, const Fraction& f);

private:
    void simplify() {
        reduction();
        fix_sign();
    }
    void fix_sign() {
        if (denominator < 0) {
            denominator = -denominator;
            numerator = -numerator;
        }
    }
    void reduction() {
        int common = gcd(numerator, denominator);
        numerator /= common;
        denominator /= common;
    }
    int gcd(int x, int y) {
        return y == 0 ? x : gcd(y, x % y);
    }
};

Fraction::Fraction(int n, int d) {
    if (d == 0) throw invalid_argument("d");
    numerator = n;
    denominator = d;
    simplify();
}

Fraction::~Fraction() {}

Fraction Fraction::operator+(const Fraction& f) {
    int n = numerator * f.denominator + f.numerator * denominator;
    int d = denominator * f.denominator;

    Fraction ff(n, d);
    return ff;
}

Fraction Fraction::operator-(const Fraction& f) {
    int n = numerator * f.denominator - f.numerator * denominator;
    int d = denominator * f.denominator;

    Fraction ff(n, d);
    return ff;
}

Fraction Fraction::operator*(const Fraction& f) {
    int n = numerator * f.numerator;
    int d = denominator * f.denominator;

    Fraction ff(n, d);
    return ff;
}

Fraction Fraction::operator/(const Fraction& f) {
    int n = numerator * f.denominator;
    int d = denominator * f.numerator;

    Fraction ff(n, d);
    return ff;
}

Fraction Fraction::operator+=(const Fraction& f) {
    numerator = numerator * f.denominator + f.numerator * denominator;
    denominator = denominator * f.denominator;
    simplify();

    Fraction ff(numerator, denominator);
    return ff;
}

Fraction Fraction::operator++() {
    Fraction f(1, 1);
    numerator = numerator * f.denominator + f.numerator * denominator;
    denominator = denominator * f.denominator;
    simplify();

    Fraction ff(numerator, denominator);
    return ff;
}

Fraction Fraction::operator++(int) {
    Fraction ff(numerator, denominator);
    Fraction f(1, 1);
    numerator = numerator * f.denominator + f.numerator * denominator;
    denominator = denominator * f.denominator;
    simplify();

    return ff;
}

bool Fraction::operator>(const Fraction& f) {
    int n = numerator * f.denominator - f.numerator * denominator;
    int d = denominator * f.denominator;

    Fraction ff(n, d);
    return ff.numerator > 0;
}

bool Fraction::operator==(const Fraction& f) {
    int n = numerator * f.denominator - f.numerator * denominator;
    int d = denominator * f.denominator;

    Fraction ff(n, d);
    return ff.denominator == 0;
}

Fraction::operator double() {
    return (double)numerator / denominator;
}

ostream &operator<<(ostream &output, const Fraction& f) {
    output << "(" << f.numerator << "/" << f.denominator << ")";
    return output;
}

int main()
{
    Fraction f0(1, 2);
    Fraction f1(1, 3);
    cout << "f0: " << f0 << endl;
    cout << "f1: " << f1 << endl;
    cout << "f0 + f1: " << f0 + f1 << endl;
    cout << "f0 - f1: " << f0 - f1 << endl;
    cout << "f0 * f1: " << f0 * f1 << endl;
    cout << "f0 / f1: " << f0 / f1 << endl;
    f0 += f1;
    cout << "after f0 += f1, f0: " << f0 << endl;
    ++f1;
    cout << "after ++f1, f1: " << f1 << endl;
    cout << "f0++, f0: " << f0++ << endl;
    cout << "after f0++, f0: " << f0 << endl;
    cout << "f0 > f1: " << (f0 > f1) << endl;
    cout << "f0 == f1: " << (f0 == f1) << endl;
    cout << "(double)f1: " << (double)f1 << endl;

    return 0;
}

Mark functions operator>, operator== and operator double() as const for usability in case of declaring const Fraction foo and still works for non const variable