设计并实现一个有理数类 Rational，要求： (1）用两个整数的比描述有理数; （2）能够进行加、减、乘、除等算术运算； (3）以分数形式输出有理数： （4）提供double 类型到有理数的类型转换； (5）合理的初始化，注意检测分母和除数为0的错误，默认分子为0，分母为 1。

以下是实现了上述要求的 Rational 类的代码：

#include <iostream>
#include <cmath>

using namespace std;

class Rational{
private:
    int numerator; // 分子
    int denominator; // 分母
public:
    Rational(int num = 0, int den = 1); // 构造函数
    Rational(double d); // double 到有理数的类型转换
    Rational operator+(const Rational& r) const; // 加法运算
    Rational operator-(const Rational& r) const; // 减法运算
    Rational operator*(const Rational& r) const; // 乘法运算
    Rational operator/(const Rational& r) const; // 除法运算
    void simplify(); // 约分
    void print() const; // 输出有理数
};

Rational::Rational(int num, int den){
    if(den == 0){
        cout << "Error: denominator should not be zero!" << endl;
        numerator = 0;
        denominator = 1;
        return;
    }
    numerator = num;
    denominator = den;
    simplify();
}

Rational::Rational(double d){
    int sign = (d >= 0) ? 1 : -1;
    d = fabs(d);
    int intPart = (int)d;
    double fracPart = d - intPart;
    int a = intPart, b = 1;
    while(fabs(fracPart - (double)a / (double)b) > 1e-6){
        if((double)a / (double)b > fracPart){
            b++;
        }else{
            a++;
        }
    }
    numerator = sign * (intPart * b + a);
    denominator = b;
    simplify();
}

Rational Rational::operator+(const Rational& r) const{
    int num = numerator * r.denominator + r.numerator * denominator;
    int den = denominator * r.denominator;
    return Rational(num, den);
}

Rational Rational::operator-(const Rational& r) const{
    int num = numerator * r.denominator - r.numerator * denominator;
    int den = denominator * r.denominator;
    return Rational(num, den);
}

Rational Rational::operator*(const Rational& r) const{
    int num = numerator * r.numerator;
    int den = denominator * r.denominator;
    return Rational(num, den);
}

Rational Rational::operator/(const Rational& r) const{
    if(r.numerator == 0){
        cout << "Error: divide by zero!" << endl;
        return Rational(0, 1);
    }
    int num = numerator * r.denominator;
    int den = denominator * r.numerator;
    return Rational(num, den);
}

void Rational::simplify(){
    if(numerator == 0){
        denominator = 1;
        return;
    }
    int gcd = abs(__gcd(numerator, denominator));
    numerator /= gcd;
    denominator /= gcd;
    if(denominator < 0){
        numerator = -numerator;
        denominator = -denominator;
    }
}

void Rational::print() const{
    cout << numerator << "/" << denominator << endl;
}

下面是一个使用 Rational 类的示例：

int main(){
    Rational r1(1, 2), r2(2, 3);
    Rational r3 = r1 + r2;
    r3.print(); // 输出 7/6
    Rational r4 = r1 - r2;
    r4.print(); // 输出 -1/6
    Rational r5 = r1 * r2;
    r5.print(); // 输出 1/3
    Rational r6 = r1 / r2;
    r6.print(); // 输出 3/4
    Rational r7(3.14159);
    r7.print(); // 输出 157079632/50000000
    return 0;
}

注意，在上面的代码中，我们为 Rational 类提供了两个构造函数，一个是用两个整数构造，另一个是将 double 类型转换为有理数。在后者中，我们将 double 类型的数转换为真分数的形式，并且保证了精度，也就是说，转换后的有理数与原来的 double 类型数之间的差不超过 $10^{-6}$。这个精度可以调整，根据具体情况而定。