> For the complete documentation index, see [llms.txt](https://docs.1l.lol/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://docs.1l.lol/number-theory/big-number.md).

# 高精度运算

### 为什么需要高精度

C++ 的 `long long` 最大约 9.2×10¹⁸，超过就溢出。当数值上千位时，用**数组模拟**大整数运算，即高精度。

### 存储方式

把数字的每一位存进数组，通常**倒序存放**（个位在 a\[0]），便于进位。

```cpp
string s = "12345";
int a[100] = {0}, len = s.size();
for (int i = 0; i < len; i++)
    a[i] = s[len - 1 - i] - '0';  // 逆序存入：a[0]=5, a[1]=4...
```

### 高精度加法

```cpp
int c[105] = {0};
for (int i = 0; i < max(len1, len2); i++) {
    c[i] += a[i] + b[i];
    c[i + 1] += c[i] / 10;   // 进位
    c[i] %= 10;
}
```

### 高精度减法

处理借位，并去除前导零，注意比较大小确定符号。

### 高精度乘法

```cpp
for (int i = 0; i < len1; i++)
    for (int j = 0; j < len2; j++) {
        c[i + j] += a[i] * b[j];
        c[i + j + 1] += c[i + j] / 10;
        c[i + j] %= 10;
    }
```

### 高精度除法

从高位到低位试商，与手算除法一致（较复杂，可结合二分试商）。

> 考纲要求：用数组模拟高精度加、减、乘、除。乘法结果位数最多为 len1+len2。


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