（1）建立一个分数类。分数类的数据成员包括分子和分母，操作包括通分、比较、加、减、乘、除、求倒数、显示和输入。

以下是一个简单的分数类实现，包括了所要求的操作：

#include <iostream>
#include <algorithm>

using namespace std;

class Fraction {
public:
    Fraction(int num = 0, int den = 1) : numerator(num), denominator(den) {}
    void normalize() {
        int gcd = __gcd(numerator, denominator);
        numerator /= gcd;
        denominator /= gcd;
    }
    void common_denominator(Fraction &other) {
        int lcm = denominator * other.denominator / __gcd(denominator, other.denominator);
        numerator *= lcm / denominator;
        other.numerator *= lcm / other.denominator;
        denominator = other.denominator = lcm;
    }
    bool operator==(const Fraction &other) const {
        return numerator == other.numerator && denominator == other.denominator;
    }
    bool operator<(const Fraction &other) const {
        return numerator * other.denominator < other.numerator * denominator;
    }
    Fraction operator+(const Fraction &other) const {
        Fraction result = *this;
        result.common_denominator((Fraction &) other);
        result.numerator += other.numerator;
        result.normalize();
        return result;
    }
    Fraction operator-(const Fraction &other) const {
        Fraction result = *this;
        result.common_denominator((Fraction &) other);
        result.numerator -= other.numerator;
        result.normalize();
        return result;
    }
    Fraction operator*(const Fraction &other) const {
        Fraction result = *this;
        result.numerator *= other.numerator;
        result.denominator *= other.denominator;
        result.normalize();
        return result;
    }
    Fraction operator/(const Fraction &other) const {
        Fraction result = *this;
        result.numerator *= other.denominator;
        result.denominator *= other.numerator;
        result.normalize();
        return result;
    }
    Fraction reciprocal() const {
        Fraction result(denominator, numerator);
        result.normalize();
        return result;
    }
    void display() const {
        cout << numerator << "/" << denominator;
    }
    void input() {
        cout << "Enter numerator and denominator: ";
        cin >> numerator >> denominator;
        normalize();
    }
private:
    int numerator, denominator;
};

这个类里面，我们定义了构造函数，通分函数，归一化函数，比较运算符，加减乘除运算符，求倒数函数，显示函数和输入函数。其中，通分函数用于将两个分数的分母变为最小公倍数，以便进行加减运算。归一化函数用于将分数约分为最简形式。比较运算符用于比较两个分数的大小关系。加减乘除运算符则实现了对应的数学运算。求倒数函数用于求出分数的倒数。显示函数用于将分数以 x/y 的形式输出。输入函数用于从键盘输入分数的值。