加法

#include <iostream>
#include <vector>

using namespace std;

vector<int> add(vector<int> &A, vector<int> &B)
{
    if (A.size() < B.size()) return add(B, A);

    vector<int> C;
    int t = 0;
    for (int i = 0; i < A.size(); i ++ )
    {
        t += A[i];
        if (i < B.size()) t += B[i];
        C.push_back(t % 10);
        t /= 10;
    }

    if (t) C.push_back(t);
    return C;
}

int main()
{
    string a, b;
    vector<int> A, B;
    cin >> a >> b;
    for (int i = a.size() - 1; i >= 0; i -- ) A.push_back(a[i] - '0');
    for (int i = b.size() - 1; i >= 0; i -- ) B.push_back(b[i] - '0');

    auto C = add(A, B);

    for (int i = C.size() - 1; i >= 0; i -- ) cout << C[i];
    cout << endl;

    return 0;
}

减法

主要注意几点：

• 数据处理前，去0操作；
• 计算结果之后，去0操作。

#include <iostream>
#include <vector>

using namespace std;

bool cmp(vector<int> &A, vector<int> &B)
{
    if (A.size() != B.size()) return A.size() > B.size();

    for (int i = A.size() - 1; i >= 0; i -- )
        if (A[i] != B[i])
            return A[i] > B[i];

    return true;//注意两数相等的情况应返回true
}
void trimZero(vector<int>& A)
{
    while (A.back() == 0 && A.size() > 1) A.pop_back();
}
vector<int> sub(vector<int> &A, vector<int> &B)
{
    vector<int> C;
    for (int i = 0, t = 0; i < A.size(); i ++ )
    {
        t = A[i] - t;
        if (i < B.size()) t -= B[i];
        C.push_back((t + 10) % 10);
        if (t < 0) t = 1;
        else t = 0;
    }
    //trimZero(C);
    while (C.size() > 1 && C.back() == 0) C.pop_back();//注意删去前导零，例如：100 - 10，结果存在前导零
    return C;
}

int main()
{
    string a, b;
    vector<int> A, B;
    cin >> a >> b;
    for (int i = a.size() - 1; i >= 0; i -- ) A.push_back(a[i] - '0');
    for (int i = b.size() - 1; i >= 0; i -- ) B.push_back(b[i] - '0');
    trimZero(A);   trimZero(B);

    vector<int> C;

    if (cmp(A, B)) C = sub(A, B);
    else C = sub(B, A), cout << '-';

    for (int i = C.size() - 1; i >= 0; i -- ) cout << C[i];
    cout << endl;

    return 0;
}

乘法

高精度 × 低精度

#include <iostream>
#include <vector>

using namespace std;


vector<int> mul(vector<int> &A, int b)
{
    vector<int> C;

    int t = 0;
    for (int i = 0; i < A.size() || t; i ++ )
    {
        if (i < A.size()) t += A[i] * b;
        C.push_back(t % 10);
        t /= 10;
    }

    while (C.size() > 1 && C.back() == 0) C.pop_back();

    return C;
}


int main()
{
    string a;
    int b;

    cin >> a >> b;

    vector<int> A;
    for (int i = a.size() - 1; i >= 0; i -- ) A.push_back(a[i] - '0');

    auto C = mul(A, b);

    for (int i = C.size() - 1; i >= 0; i -- ) printf("%d", C[i]);

    return 0;
}

高精度×高精度

#include <iostream>
#include <vector>

using namespace std;

void trimzero(vector<int>& A)
{
    while (A.size() > 1 && A.back() == 0) A.pop_back();
}
vector<int> mul(vector<int>& A, vector<int>& B) {
    vector<int> C(A.size() + B.size(), 0); // 初始化为 0，且999*99最多 5 位

    for (int i = 0; i < A.size(); i++)
        for (int j = 0; j < B.size(); j++)
            C[i + j] += A[i] * B[j];

    int t = 0;
    for (int i = 0; i < C.size(); i++) { // i = C.size() - 1时 t 一定小于 10
        t += C[i];
        C[i] = t % 10;
        t /= 10;
    }

    trimzero(C); // 必须要去前导 0，因为最高位很可能是 0
    return C;
}

int main() {
    string a, b;
    cin >> a >> b; // a = "189", b = "92", C = "17388"

    vector<int> A, B;
    for (int i = a.size() - 1; i >= 0; i--)
        A.push_back(a[i] - '0');
    for (int i = b.size() - 1; i >= 0; i--)
        B.push_back(b[i] - '0');
    trimzero(A); trimzero(B);

    auto C = mul(A, B);

    for (int i = C.size() - 1; i >= 0; i--)
        cout << C[i];

    return 0;
}

除法（高精度/低精度）

#include<bits/stdc++.h>
#include <iostream>
#include <vector>
#include <algorithm>

using namespace std;
void trimzero(vector<int>& A)
{
    while (A.size() > 1 && A.back() == 0) A.pop_back();
}
vector<int> div(vector<int>& A, int b, int& r)
{
    vector<int> C;
    r = 0;
    for (int i = 0; i < A.size(); i++)
    {
        r = r * 10 + A[i];
        C.push_back(r / b);
        r %= b;
    }

    reverse(C.begin(), C.end());//注意区分 reverse 和 reserve
    trimzero(C);
    return C;
}

int main()
{
    string a;
    vector<int> A;

    int B;
    cin >> a >> B;
    for (int i = 0; i < a.size(); i++) A.push_back(a[i] - '0');
    trimzero(A);

    int r;
    auto C = div(A, B, r);

    for (int i = C.size() - 1; i >= 0; i--) cout << C[i];

    cout << endl << r << endl;

    return 0;
}