用string类型完成大整数运算

使用 `string` 类型可以完成大整数运算。具体实现方法是将大整数存储在 `string` 类型中，然后通过模拟手算的方法进行加减乘除等运算。

以下是使用 `string` 类型进行大整数加法、减法、乘法和除法的示例代码：

#include <iostream>
#include <string>
#include <algorithm>

using namespace std;

// 大整数加法
string add(string num1, string num2) {
    string res;
    int carry = 0;
    int i = num1.size() - 1, j = num2.size() - 1;
    while (i >= 0 || j >= 0 || carry) {
        int x = i >= 0 ? num1[i--] - '0' : 0;
        int y = j >= 0 ? num2[j--] - '0' : 0;
        int sum = x + y + carry;
        res += to_string(sum % 10);
        carry = sum / 10;
    }
    reverse(res.begin(), res.end());
    return res;
}

// 大整数减法
string sub(string num1, string num2) {
    string res;
    int borrow = 0;
    int i = num1.size() - 1, j = num2.size() - 1;
    while (i >= 0 || j >= 0) {
        int x = i >= 0 ? num1[i--] - '0' : 0;
        int y = j >= 0 ? num2[j--] - '0' : 0;
        int diff = x - y - borrow;
        if (diff < 0) {
            diff += 10;
            borrow = 1;
        } else {
            borrow = 0;
        }
        res += to_string(diff);
    }
    reverse(res.begin(), res.end());
    while (res.size() > 1 && res[0] == '0') {
        res.erase(0, 1);
    }
    return res;
}

// 大整数乘法
string multiply(string num1, string num2) {
    string res(num1.size() + num2.size(), '0');
    for (int i = num1.size() - 1; i >= 0; --i) {
        int carry = 0;
        for (int j = num2.size() - 1; j >= 0; --j) {
            int sum = (num1[i] - '0') * (num2[j] - '0') + (res[i + j + 1] - '0') + carry;
            res[i + j + 1] = sum % 10 + '0';
            carry = sum / 10;
        }
        res[i] += carry;
    }
    while (res.size() > 1 && res[0] == '0') {
        res.erase(0, 1);
    }
    return res;
}

// 大整数除法
string divide(string num1, string num2) {
    if (num2 == "0") {
        return "NaN";
    }
    if (num1 == "0") {
        return "0";
    }
    string res;
    string remainder = num1.substr(0, num2.size() - 1);
    for (int i = num2.size() - 1; i < num1.size(); ++i) {
        remainder += num1[i];
        int quotient = 0;
        while (remainder >= num2) {
            remainder = sub(remainder, num2);
            quotient++;
        }
        res += to_string(quotient);
    }
    while (res.size() > 1 && res[0] == '0') {
        res.erase(0, 1);
    }
    return res;
}

int main() {
    string num1 = "12345678901234567890";
    string num2 = "98765432109876543210";

    // 加法
    string sum = add(num1, num2);
    cout << "num1 + num2 = " << sum << endl;

    // 减法
    string diff = sub(num1, num2);
    cout << "num1 - num2 = " << diff << endl;

    // 乘法
    string product = multiply(num1, num2);
    cout << "num1 * num2 = " << product << endl;

    // 除法
    string quotient = divide(num1, num2);
    cout << "num1 / num2 = " << quotient << endl;

    return 0;
}

上述代码中，`add`、`sub`、`multiply`、`divide` 分别对应大整数加法、减法、乘法和除法。这些函数的实现原理都是通过模拟手算的方法进行计算。