负数取模怎么算, -10 % 3 = ?

柔光的暖阳◎ 2023-10-09 05:31 31阅读 0赞

# 负数取模怎么算, -10 % 3 = ? #

**简介：**  
今天遇到了一个很奇怪的问题：**-10 % 3 = ？和 10 % -3 = ？**，相信很多人都跟我一样脱口而出-10 % 3 = -1，10 % -3 = 1，然后我用计算器算了一下，结果让我很惊讶，-10 % 3 = 2，10 % -3 = -2。于是我用Java执行了两个式子，算出来的结果跟我一开始想的一样的，这怎么回事呢，经过查资料发现原来每一种语言算出来的结果可能都不一样。

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3RyYWN5ZHJhZ29ubHh5_size_16_color_FFFFFF_t_70]  
关于取模操作，WIKI里是这么定义的：[https://en.wikipedia.org/wiki/Modulo\_operation][https_en.wikipedia.org_wiki_Modulo_operation]  
In mathematics, the result of the modulo operation is an equivalence class, and any member of the class may be chosen as representative; however, the usual representative is the least positive residue, the smallest nonnegative integer that belongs to that class, i.e. the remainder of the Euclidean division. However, other conventions are possible. Computers and calculators have various ways of storing and representing numbers; thus their definition of the modulo operation depends on the programming language or the underlying hardware.

In nearly all computing systems, the quotient q and the remainder r of a divided by n satisfy：  
![在这里插入图片描述][20190925114114847.jpg]  
However, this still leaves a sign ambiguity if the remainder is nonzero: two possible choices for the remainder occur, one negative and the other positive, and two possible choices for the quotient occur. Usually, in number theory, the positive remainder is always chosen, but programming languages choose depending on the language and the signs of a or n.

r = a - nq,而q的计算主要有：

1.  **truncated division**，定义如下：

q = t r u n c ( a n ) q=trunc(\\frac\{a\}\{n\}) q=trunc(na)  
相当于截去小数部分。3 / 2 = 1，-3 / 2 = -1

1.  Donald Knuth提出的 **floored division**：

q = ⌊ a n ⌋ q=⌊\\frac\{a\}\{n\}⌋ q=⌊na⌋  
向下取整。3 / 2 = 1，-3 / 2 = -2

1.  **Euclidean division** ：  
     q = ⌊ a n ⌋ ， n > 0 q=⌊\\frac\{a\}\{n\}⌋，n > 0 q=⌊na⌋，n>0  
     q = ⌈ a n ⌉ ， n < 0 q = ⌈\\frac\{a\}\{n\}⌉，n < 0 q=⌈na⌉，n<0

不同程序设计语言实现的方式可能不同。

**1. C/Java的处理方式：**  
大多数语言的处理方式都与 C/Java 一致，采用了 truncate 除法。

> 10 % （-3）= 1 的计算结果如下：r = 10 - (10 / -3) \* -3 = 10 - (-3 x -3) = 1  
> \-10 % 3 = -1 的计算结果如下：r = -10 - (-10 / 3) \* 3 = -10 - (-3 \* 3) = -1。

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3RyYWN5ZHJhZ29ubHh5_size_16_color_FFFFFF_t_70 1]

**2. Python的处理方式：**  
Python 语言除法采用的是 floor 除法。

> 10 % （-3）= -2 的计算结果如下：r = 10 - (10 / -3) \* -3 = 10 - (-4 x -3) = -2  
> \-10 % 3 = 2 的计算结果如下：r = -10 - (-10 / 3) \* 3 = -10 - (-4 \* 3) = 2。

![在这里插入图片描述][20190925144321190.jpg]

[watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3RyYWN5ZHJhZ29ubHh5_size_16_color_FFFFFF_t_70]: https://img-blog.csdnimg.cn/20190925112358332.png?x-oss-process=image/watermark,type_ZmFuZ3poZW5naGVpdGk,shadow_10,text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3RyYWN5ZHJhZ29ubHh5,size_16,color_FFFFFF,t_70
[https_en.wikipedia.org_wiki_Modulo_operation]: https://en.wikipedia.org/wiki/Modulo_operation
[20190925114114847.jpg]: https://img-blog.csdnimg.cn/20190925114114847.jpg
[watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3RyYWN5ZHJhZ29ubHh5_size_16_color_FFFFFF_t_70 1]: https://img-blog.csdnimg.cn/20190925143809818.jpg?x-oss-process=image/watermark,type_ZmFuZ3poZW5naGVpdGk,shadow_10,text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3RyYWN5ZHJhZ29ubHh5,size_16,color_FFFFFF,t_70
[20190925144321190.jpg]: https://img-blog.csdnimg.cn/20190925144321190.jpg