> 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/basics/complement.md).

# 数学与补码/反码

### 幂的运算

* 任何非零数的 0 次方等于 1：`a^0 = 1 (a ≠ 0)`
* `0^0` 无意义
* 同底数幂相乘：`a^m × a^n = a^(m+n)`

### 原码、反码、补码

计算机中**整数用补码存储**。以 8 位为例：

| 编码 | 正数         | 负数           |
| -- | ---------- | ------------ |
| 原码 | 符号位 0 + 数值 | 符号位 1 + 数值   |
| 反码 | 与原码相同      | 符号位不变，其余各位取反 |
| 补码 | 与原码相同      | 反码 + 1       |

```
+5 原码/反码/补码：0000 0101
-5 原码：1000 0101
-5 反码：1111 1010
-5 补码：1111 1011
```

### 8 位有符号整数的范围

**-128 \~ 127**（不是 -127）。

* 最大值 `0111 1111 = 127`
* 最小值 `1000 0000 = -128`（补码中多出一个数用于表示 -128）

### 为什么用补码

* 把减法变成加法：`a - b = a + (-b)`，硬件只需加法器
* 0 只有一种表示，避免 +0 / -0 的歧义

### 相关

更详细的编码与进制见「计算机基础」分组的《数据编码（原码、反码、补码）》。


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