模除

Template:Colorbox 商數 ( $q$ ) 和 Template:Colorbox 餘數 ( $r$ ) 作為被除數 ( $a$ ) 的函數時的圖像。左侧是除数为正的情况，右侧除数为负。从上至下分别使用了：向零取整、向下取整和欧几里得除法。

模除（Template:Lang-en 有时也称作 modulus），又稱模数、取模、取模運算等，它得出一个数除以另一个数的余数。给定两个正数：被除数 $a$ 和除数 $n$ ， $a modulo n$ （通常缩写为 $a mod n$ ），得到的是使用欧几里得除法时的余数。 $a mod 1$ 所得之数被称为 $a$ 的小数部份。

举个例子：计算表达式 $5 mod 2$ 得到 $1$ ，因为 $5 \div 2$ 的商为 $2$ 而余数为 $1$ ；而 $9 mod 3$ 得到 $0$ ，因为 $9 \div 3$ 的商为 $3$ 而余数为 $0$ ；做除法不能整除时得到是有理数，平常使用的计算器采用了有限个数位的十进制表示法，它以小数点分隔整数部份和小数部份，比如 $5 \div 2 = 2.5$ 。

虽然通常情况下模除的被除数 $a$ 和除数 $n$ 都是整数，但许多计算系统允许其他类型的数值运算，比如对浮点数取模。在所有计算系统中当被除数 $a$ 和除数 $n$ 都是正数时，结果的余数 $r$ 的范围为 $0 \leq r < n$ ；而 $a mod 0$ 经常是未定义的，在编程语言里可能会导致除以零错误。当被除数 $a$ 或除数 $n$ 为负数时，不同的编程语言对结果有不同的处理。

各种定义

在数学中，取模运算的结果就是欧几里德除法的余数。当然也有许多其他的定义方式。计算机和计算器有许多种表示和储存数字的方法，因此在不同的硬件环境下、不同的编程语言中，取模运算有着不同的定义。

几乎所有的计算系统中， $a$ 除以 $n$ 得到商 $q$ 和余数 $r$ 均满足以下式子： Template:NumBlk

即使如此，当余数非 $0$ 时，余数的符号仍然是有歧义的：余数非 $0$ 时，它的符号有两种选择，一个是正号而另一个是负号Template:Efn。通常情况下，在数论中总是选择正余数。但在编程中，选择余数的符号依赖于编程语言和被除数 $a$ 或除数 $n$ 的符号。在编程语言所定义的整数模除中，ISO/IEC标准Pascal^[1]和ALGOL 68^[2]，在计算出的余数r为负数时，返回正数 $r + | n |$ 作为结果Template:Efn；另一些编程语言如ISO/IEC C90，当被除数 $a$ 或除数 $n$ 是负数时，C90标准并没有做具体的规定，而是留给编译器去定义并实现^[3]。在大多数系统中 $a mod 0$ 是未定义的，虽然有些系统定义它就等于 $a$ 。更多详情参见后续章节表格。

Template:Bulleted list

记号

一些计算器有取模Template:Math按钮，很多编程语言里也有类似的函数，通常像mod(a, n)这样。有些语言也支持在表达式内使用%、mod或Mod作为取模或取余操作符，比如a % n或a mod n。

在一些没有Template:Math函数的环境中或许可使用等价的：a - (n * int(a/n))。这里的int()函数事实上等价于截断函数。

性质及恒等式

一些取模操作，经过分解和展开可以等同于其他数学运算。这在密码学的证明中十分有用，例如：迪菲-赫爾曼密鑰交換。

恒等式：
- $(a mod n) mod n = a mod n$
- 对所有的正数 $x$ 有： $n^{x} mod n = 0$
- 如果 $p$ 是一个质数，且不为 $b$ 的因数，此时由费马小定理有： $a b^{p - 1} mod p = a mod p$
逆运算：
- $[(- a mod n) + (a mod n)] mod n = 0$ .
- $b^{- 1} mod n$ 表示模反元素。当且仅当 $b$ 与 $n$ 互质时，等式左侧有定义： $[(b^{- 1} mod n) (b mod n)] mod n = 1$ 。
分配律：
- $(a + b) mod n = [(a mod n) + (b mod n)] mod n$
- $a b mod n = [(a mod n) (b mod n)] mod n$
除法定义：仅当式子右侧有定义时，即 $b$ 、 $n$ 互质时有： $\frac{a}{b} mod n = [(a mod n) (b^{- 1} mod n)] mod n$ ，其他情况为未定义的。
乘法逆元： $[(a b mod n) (b^{- 1} mod n)] mod n = a mod n$ .

编程语言实现

在各种编程语言中的模除运算
语言	算符	整数	浮点数	定义
ABAP	Template:Code	Template:Yes	Template:Yes	欧几里得式
ActionScript	Template:Code	Template:Yes	Template:No	截断
Ada	Template:Code	Template:Yes	Template:No	下取整^[4]
Ada	Template:Code	Template:Yes	Template:No	截断^[4]
ALGOL 68	Template:Code, Template:Code	Template:Yes	Template:No	类似欧几里得式Template:Efn
AMPL	Template:Code	Template:Yes	Template:No	截断
APL	`\|`Template:Efn	Template:Yes	Template:Yes	下取整
AppleScript	Template:Code	Template:Yes	Template:No	截断
AutoLISP	Template:Code	Template:Yes	Template:No	截断
AWK	Template:Code	Template:Yes	Template:No	截断
bash	Template:Code	Template:Yes	Template:No	截断
BASIC	Template:Code	Template:Yes	Template:No	实现各异
bc	Template:Code	Template:Yes	Template:No	截断
C Template:Break C++	Template:Code, Template:Code	Template:Yes	Template:No	截断Template:Efn
	Template:Code (C)Template:Break Template:Code (C++)	Template:No	Template:Yes	截断^[5]
	Template:Code (C)Template:Break Template:Code (C++)	Template:No	Template:Yes	修约
C#	Template:Code	Template:Yes	Template:Yes	截断
C#	Template:Code	Template:No	Template:Yes	修约^[6]
Template:En-link	Template:Code	Template:Yes	Template:No	截断
Clean	Template:Code	Template:Yes	Template:No	截断
Clojure	Template:Code	Template:Yes	Template:No	下取整^[7]
Clojure	Template:Code	Template:Yes	Template:No	截断^[8]
COBOL	Template:Code	Template:Yes	Template:No	下取整^[9]
COBOL	Template:Code	Template:Yes	Template:Yes	截断^[9]
CoffeeScript	Template:Code	Template:Yes	Template:No	截断
CoffeeScript	Template:Code	Template:Yes	Template:No	下取整^[10]
ColdFusion	Template:Code, Template:Code	Template:Yes	Template:No	截断
Common Intermediate Language	Template:Code (有符号)	Template:Yes	Template:Yes	截断^[11]
Common Intermediate Language	Template:Code (无符号)	Template:Yes	Template:No	Template:N/A
Common Lisp	Template:Code	Template:Yes	Template:Yes	下取整
Common Lisp	Template:Code	Template:Yes	Template:Yes	截断
Template:En-link	Template:Code, Template:Code	Template:Yes	Template:Yes	下取整
Template:En-link	Template:Code	Template:Yes	Template:Yes	截断
D	Template:Code	Template:Yes	Template:Yes	截断^[12]
Dart	Template:Code	Template:Yes	Template:Yes	欧几里得式^[13]
Dart	Template:Code	Template:Yes	Template:Yes	截断^[14]
Eiffel	Template:Code	Template:Yes	Template:No	截断
Elixir	Template:Code	Template:Yes	Template:No	截断^[15]
Elixir	Template:Code	Template:Yes	Template:No	下取整^[16]
Elm	Template:Code	Template:Yes	Template:No	下取整^[17]
Elm	Template:Code	Template:Yes	Template:No	截断^[18]
Erlang	Template:Code	Template:Yes	Template:No	截断
Erlang	Template:Code	Template:No	Template:Yes	截断 (同于C)^[19]
Euphoria	Template:Code	Template:Yes	Template:No	下取整
Euphoria	Template:Code	Template:Yes	Template:No	截断
F#	Template:Code	Template:Yes	Template:Yes	截断
F#	Template:Code	Template:No	Template:Yes	修约^[6]
Factor	Template:Code	Template:Yes	Template:No	截断
FileMaker	Template:Code	Template:Yes	Template:No	下取整
Forth	Template:Code	Template:Yes	Template:No	实现定义
	Template:Code	Template:Yes	Template:No	下取整
	Template:Code	Template:Yes	Template:No	截断
Fortran	Template:Code	Template:Yes	Template:Yes	截断
Fortran	Template:Code	Template:Yes	Template:Yes	下取整
Template:En-link	Template:Code	Template:Yes	Template:No	下取整
Template:En-link	Template:Code	Template:Yes	Template:Yes	下取整^[20]
Template:En-link	Template:Code	Template:Yes	Template:Yes	截断^[21]
GLSL	Template:Code	Template:Yes	Template:No	未定义^[22]
GLSL	Template:Code	Template:No	Template:Yes	下取整^[23]
GameMaker Studio (GML)	Template:Code, Template:Code	Template:Yes	Template:No	截断
GDScript (Godot)	Template:Code	Template:Yes	Template:No	截断
	Template:Code	Template:No	Template:Yes	截断
	Template:Code	Template:Yes	Template:No	欧几里得式
	Template:Code	Template:No	Template:Yes	欧几里得式
Go	Template:Code	Template:Yes	Template:No	截断^[24]
	Template:Code	Template:No	Template:Yes	截断^[25]
	Template:Code	Template:Yes	Template:No	欧几里得式^[26]
	Template:Code	Template:Yes	Template:No	截断^[27]
Groovy	Template:Code	Template:Yes	Template:No	截断
Haskell	Template:Code	Template:Yes	Template:No	下取整^[28]
	Template:Code	Template:Yes	Template:No	截断^[28]
	Template:Code (GHC)	Template:No	Template:Yes	下取整
Haxe	Template:Code	Template:Yes	Template:No	截断
HLSL	Template:Code	Template:Yes	Template:Yes	未定义^[29]
J	`\|`Template:Efn	Template:Yes	Template:No	下取整
Java	Template:Code	Template:Yes	Template:Yes	截断
Java	Template:Code	Template:Yes	Template:No	下取整
JavaScript Template:Break TypeScript	Template:Code	Template:Yes	Template:Yes	截断
Julia	Template:Code	Template:Yes	Template:Yes	下取整^[30]
Julia	Template:Code, Template:Code	Template:Yes	Template:Yes	截断^[31]
Kotlin	Template:Code, Template:Code	Template:Yes	Template:Yes	截断^[32]
Kotlin	Template:Code	Template:Yes	Template:Yes	下取整^[33]
ksh	Template:Code	Template:Yes	Template:No	截断 (同于POSIX sh)
ksh	Template:Code	Template:No	Template:Yes	截断
LabVIEW	Template:Code	Template:Yes	Template:Yes	截断
LibreOffice	Template:Code	Template:Yes	Template:No	下取整
Logo	Template:Code	Template:Yes	Template:No	下取整
Logo	Template:Code	Template:Yes	Template:No	截断
Lua 5	Template:Code	Template:Yes	Template:Yes	下取整
Lua 4	Template:Code	Template:Yes	Template:Yes	截断
Template:En-link	Template:Code	Template:Yes	Template:No	截断
Mathcad	Template:Code	Template:Yes	Template:No	下取整
Maple	Template:Code (默认), Template:Code	Template:Yes	Template:No	欧几里得式
	Template:Code	Template:Yes	Template:No	修约
	Template:Code	Template:Yes	Template:Yes	修约
Mathematica	Template:Code	Template:Yes	Template:No	下取整
MATLAB	Template:Code	Template:Yes	Template:No	下取整
MATLAB	Template:Code	Template:Yes	Template:No	截断
Maxima	Template:Code	Template:Yes	Template:No	下取整
Maxima	Template:Code	Template:Yes	Template:No	截断
Template:En-link	Template:Code	Template:Yes	Template:No	截断
Microsoft Excel	Template:Code	Template:Yes	Template:Yes	下取整
Minitab	Template:Code	Template:Yes	Template:No	下取整
Modula-2	Template:Code	Template:Yes	Template:No	下取整
Modula-2	Template:Code	Template:Yes	Template:No	截断
Template:En-link	Template:Code	Template:Yes	Template:No	下取整
Netwide Assembler (NASM, NASMX)	Template:Code, Template:Code (无符号)	Template:Yes	Template:No	Template:N/A
Netwide Assembler (NASM, NASMX)	Template:Code (有符号)	Template:Yes	Template:No	实现定义^[34]
Nim	Template:Code	Template:Yes	Template:No	截断
Oberon	Template:Code	Template:Yes	Template:No	类似下取整Template:Efn
Objective-C	Template:Code	Template:Yes	Template:No	截断 (同于C99)
Object Pascal, Delphi	Template:Code	Template:Yes	Template:No	截断
OCaml	Template:Code	Template:Yes	Template:No	截断^[35]
OCaml	Template:Code	Template:No	Template:Yes	截断^[36]
Occam	Template:Code	Template:Yes	Template:No	截断
Pascal (ISO-7185和ISO-10206)	Template:Code	Template:Yes	Template:No	类似欧几里得式Template:Efn
Perl	Template:Code	Template:Yes	Template:No	下取整Template:Efn
Perl	Template:Code	Template:No	Template:Yes	截断
PHP	Template:Code	Template:Yes	Template:No	截断^[37]
PHP	Template:Code	Template:No	Template:Yes	截断^[38]
PIC BASIC Pro	Template:Code	Template:Yes	Template:No	截断
PL/I	Template:Code	Template:Yes	Template:No	下取整 (ANSI PL/I)
PowerShell	Template:Code	Template:Yes	Template:No	截断
Programming Code (Template:En-link)	Template:Code	Template:Yes	Template:No	未定义
Template:En-link	Template:Code	Template:Yes	Template:No	截断
Prolog (ISO 1995^[39])	Template:Code	Template:Yes	Template:No	下取整
Prolog (ISO 1995^[39])	Template:Code	Template:Yes	Template:No	截断
PureBasic	Template:Code, Template:Code	Template:Yes	Template:No	截断
PureScript	Template:Code	Template:Yes	Template:No	欧几里得式^[40]
Pure Data	Template:Code	Template:Yes	Template:No	截断 (同于C)
Pure Data	Template:Code	Template:Yes	Template:No	下取整
Python	Template:Code	Template:Yes	Template:Yes	下取整
	Template:Code	Template:No	Template:Yes	截断
	Template:Code	Template:No	Template:Yes	修约
Q#	Template:Code	Template:Yes	Template:No	截断^[41]
R	Template:Code	Template:Yes	Template:Yes	下取整^[42]
Racket	Template:Code	Template:Yes	Template:No	下取整
Racket	Template:Code	Template:Yes	Template:No	截断
Raku	Template:Code	Template:No	Template:Yes	下取整
Template:En-link	Template:Code	Template:Yes	Template:No	截断
Reason	Template:Code	Template:Yes	Template:No	截断
Rexx	Template:Code	Template:Yes	Template:Yes	截断
Template:En-link	Template:Code	Template:Yes	Template:No	截断
Ruby	Template:Code, Template:Code	Template:Yes	Template:Yes	下取整
Ruby	Template:Code	Template:Yes	Template:Yes	截断
Rust	Template:Code	Template:Yes	Template:Yes	截断
Rust	Template:Code	Template:Yes	Template:Yes	欧几里得式^[43]
SAS	Template:Code	Template:Yes	Template:No	截断
Scala	Template:Code	Template:Yes	Template:Yes	截断
Scheme	Template:Code	Template:Yes	Template:No	下取整
Scheme	Template:Code	Template:Yes	Template:No	截断
Scheme R⁶RS	Template:Code	Template:Yes	Template:No	欧几里得式^[44]
	Template:Code	Template:Yes	Template:No	修约^[44]
	Template:Code	Template:No	Template:Yes	欧几里得式
	Template:Code	Template:No	Template:Yes	修约
Scratch	Template:Code	Template:Yes	Template:Yes	下取整
Template:En-link	Template:Code	Template:Yes	Template:Yes	下取整
Template:En-link	Template:Code	Template:Yes	Template:Yes	截断
Template:En-link	Template:Code	Template:Yes	Template:No	下取整
Template:En-link	Template:Code	Template:Yes	Template:No	截断
Unix shell (包括bash、ash等。)	Template:Code	Template:Yes	Template:No	截断 (同于C)^[45]
Smalltalk	Template:Code	Template:Yes	Template:No	下取整
Smalltalk	Template:Code	Template:Yes	Template:No	截断
Snap!	Template:Code	Template:Yes	Template:No	下取整
Template:En-link	Template:Code	Template:Yes	Template:No	下取整
Solidity	Template:Code	Template:Yes	Template:No	下取整
SQL (Template:En-link)	Template:Code	Template:Yes	Template:No	截断
SQL (Template:En-link)	Template:Code	Template:Yes	Template:No	截断
Standard ML	Template:Code	Template:Yes	Template:No	下取整
	Template:Code	Template:Yes	Template:No	截断
	Template:Code	Template:No	Template:Yes	截断
Stata	Template:Code	Template:Yes	Template:No	欧几里得式
Swift	Template:Code	Template:Yes	Template:No	截断^[46]
	Template:Code	Template:No	Template:Yes	修约^[47]
	Template:Code	Template:No	Template:Yes	截断^[48]
Tcl	Template:Code	Template:Yes	Template:No	下取整
Tcl	Template:Code	Template:No	Template:Yes	截断 (同于C)
tcsh	Template:Code	Template:Yes	Template:No	截断
Template:En-link	Template:Code	Template:Yes	Template:No	截断
Template:En-link	Template:Code	Template:Yes	Template:No	下取整
Verilog (2001)	Template:Code	Template:Yes	Template:No	截断
VHDL	Template:Code	Template:Yes	Template:No	下取整
VHDL	Template:Code	Template:Yes	Template:No	截断
VimL	Template:Code	Template:Yes	Template:No	截断
Visual Basic	Template:Code	Template:Yes	Template:No	截断
WebAssembly	Template:Code, Template:Code (无符号)	Template:Yes	Template:No	Template:N/A
WebAssembly	Template:Code, Template:Code (有符号)	Template:Yes	Template:No	截断^[49]
Template:En-link	Template:Code(无符号)	Template:Yes	Template:No	Template:N/A
Template:En-link	Template:Code(有符号)	Template:Yes	Template:No	截断
Template:En-link	Template:Code	Template:Yes	Template:Yes	截断
Template:En-link	Template:Code	Template:Yes	Template:Yes	下取整
Zig	Template:Code, Template:Code, Template:Code	Template:Yes	Template:Yes	截断^[50]
Template:En-link	Template:Code, Template:Code	Template:Yes	Template:No	欧几里得式

此外，很多计算机系统提供Template:Code功能，它同时产生商和余数。例子x86架构的Template:Code指令，C编程语言的Template:Code函数，和Python的Template:Code函数。

常见错误

当取模的结果与被除数符号相同时，可能会导致意想不到的错误。

举个例子：如果需要判断一个整数是否为奇数，有人可能会测试这个数除 2 的余数是否为 1：

bool is_odd(int n) {
    return n % 2 == 1;
}

但在一个取模结果与被除数符号相同的编程语言里，这样做是错的。因为当被除数 Template:Math 是奇数且为负数时， $n mod 2$ 得到 −1，此时函数返回“假”。

一种正确的实现是测试取模结果是否为 0，因为余数为 0 时没有符号的问题：

bool is_odd(int n) {
    return n % 2 != 0;
}

或者考虑余数的符号，有两种情况：余数可能为 1 或 -1。

bool is_odd(int n) {
    return n % 2 == 1 || n % 2 == -1;
}

性能问题

可以通过依次计算带余数的除法实现取模操作。特殊情况下，如某些硬件上，存在更快的实现。例如：2 的 n 次幂的模，可以通过逐位与运算实现：

x % 2ⁿ == x & (2ⁿ - 1)

例子，假定 Template:Math 为正数：

x % 2 == x & 1

x % 4 == x & 3

x % 8 == x & 7

在进行位操作比取模操作效率更高的设备或软件环境中，以上形式的取模运算速度更快。^[51]

编译器可以自动识别出对 2 的 n 次幂取模的表达式，自动将其优化为 expression & (constant-1)。这样可以在兼顾效率的情况下写出更整洁的代码。这个优化在取模结果与被除数符号一致的语言中（包括 C 语言）不能使用，除非被除数是无符号整数。这是因为如果被除数是负数，则结果也是负数，但 expression & (constant-1) 总是正数，进行这样的优化就会导致错误，无符号整数则没有这个问题。

用途

取模运算可用于判断一个数是否能被另一个数整除。对 2 取模即可判断整数的奇偶性；从 2 到 $n - 1$ 取模则可判断一个数是否为质数。
進制之間的轉換。
用于求取最大公约数的輾轉相除法使用取模运算。
密码学中的应用：从古老的凯撒密码到现代常用的RSA、椭圆曲线密码，它们的实现过程均使用了取模运算。

參見

模 (消歧义)和Template:Tsl —— “模数（Modulo）”这个词的许多用法，都是 1801 年卡爾·弗里德里希·高斯引入模算數时产生的。
模幂运算
同餘

脚注

Template:Reflist

参考文献

Template:Reflist

↑ Template:Cite web
A term of the form i div j shall be an error if j is zero; otherwise, the value of i div j shall be such that

abs(i) - abs(j) < abs((i div j) * j) <= abs(i)
where the value shall be zero if abs(i) < abs(j); otherwise, the sign of the value shall be positive if i and j have the same sign and negative if i and j have different signs.
A term of the form i mod j shall be an error if j is zero or negative; otherwise, the value of i mod j shall be that value of (i - (k * j)) for integral k such that 0 <= i mod j < j.
NOTE 2 — Only for i >= 0 and j > 0 does the relation (i div j) * j + i mod j = i hold.
Template:Cite web

div   divide and truncate (i.e., value is not rounded)

mod   modulus: let Remainder = A - (A div B) * B;
  if Remainder < 0 then A mod B = Remainder + B

  otherwise A mod B = Remainder

↑ Template:Cite web
10.2.3.3. Operations on integral operands
……

m) OP «÷, %, OVER» = (L INT a, b) L INT:
      IF b ≠ L 0
      THEN L INT q := L 0, r := ABS a;
         WHILE (r := r - ABS b) ≥ L 0 DO q := q + L 1 OD;
         (a < L 0 AND b > L 0 OR a ≥ L 0 AND b < L 0 | -q | q)
      FI;
n) OP «÷×, ÷*, %*, MOD» = (L INT a, b) L INT:
      (L INT r = a - a ÷ b × b; r < 0 | r + ABS b | r);

↑
Template:Cite book
C99 reflects almost universal processor behavior (as does the Fortran Standard). This definition truncates toward zero and the expression (-(a/b) == (-a)/b) && (-(a/b) == a/(-b)) is always true. It also means that the absolute value of the result does not depend on the signs of the operands; for example:

+10 / +3 == +3 +10 % +3 == +1 -10 / +3 == -3 -10 % +3 == -1
+10 / -3 == -3 +10 % -3 == +1 -10 / -3 == +3 -10 % -3 == -1
C90

When integers are divided and the division is inexact, if both operands are positive the result of the / operator is the largest integer less than the algebraic quotient and the result of the % operator is positive. If either operand is negative, whether the result of the / operator is the largest integer less than or equal to the algebraic quotient or the smallest integer greater than or equal to the algebraic quotient is implementation-defined, as is the sign of the result of the % operator.
If either operand is negative, the behavior may differ between C90 and C99, depending on the implementation-defined behavior of the C90 implementation.
#include <stdio.h> int main(void) { int x = -1, y = +3; if ((x%y > 0) || ((x+y)%y == x%y)) printf("This is a C90 translator behaving differently than C99\n"); }
Quoting from the C9X Revision Proposal, WG14/N613, that proposed this change:

The origin of this practice seems to have been a desire to map C’s division directly to the “natural” behavior of the target instruction set, whatever it may be, without requiring extra code overhead that might be necessary to check for special cases and enforce a particular behavior. However, the argument that Fortran programmers are unpleasantly surprised by this aspect of C and that there would be negligible impact on code efficiency was accepted by WG14, who agreed to require Fortran-like behavior in C99.
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[1] Template:Cite web
A term of the form i div j shall be an error if j is zero; otherwise, the value of i div j shall be such that

abs(i) - abs(j) < abs((i div j) * j) <= abs(i)
where the value shall be zero if abs(i) < abs(j); otherwise, the sign of the value shall be positive if i and j have the same sign and negative if i and j have different signs.
A term of the form i mod j shall be an error if j is zero or negative; otherwise, the value of i mod j shall be that value of (i - (k * j)) for integral k such that 0 <= i mod j < j.
NOTE 2 — Only for i >= 0 and j > 0 does the relation (i div j) * j + i mod j = i hold.
Template:Cite web

div divide and truncate (i.e., value is not rounded)

mod modulus: let Remainder = A - (A div B) * B;
if Remainder < 0 then A mod B = Remainder + B

otherwise A mod B = Remainder

[2] Template:Cite web
10.2.3.3. Operations on integral operands
……

m) OP «÷, %, OVER» = (L INT a, b) L INT: IF b ≠ L 0 THEN L INT q := L 0, r := ABS a; WHILE (r := r - ABS b) ≥ L 0 DO q := q + L 1 OD; (a < L 0 AND b > L 0 OR a ≥ L 0 AND b < L 0 | -q | q) FI; n) OP «÷×, ÷*, %*, MOD» = (L INT a, b) L INT: (L INT r = a - a ÷ b × b; r < 0 | r + ABS b | r);

[C-CPP-Std-3] Template:Cite book
C99 reflects almost universal processor behavior (as does the Fortran Standard). This definition truncates toward zero and the expression (-(a/b) == (-a)/b) && (-(a/b) == a/(-b)) is always true. It also means that the absolute value of the result does not depend on the signs of the operands; for example:

+10 / +3 == +3 +10 % +3 == +1 -10 / +3 == -3 -10 % +3 == -1
+10 / -3 == -3 +10 % -3 == +1 -10 / -3 == +3 -10 % -3 == -1
C90

When integers are divided and the division is inexact, if both operands are positive the result of the / operator is the largest integer less than the algebraic quotient and the result of the % operator is positive. If either operand is negative, whether the result of the / operator is the largest integer less than or equal to the algebraic quotient or the smallest integer greater than or equal to the algebraic quotient is implementation-defined, as is the sign of the result of the % operator.
If either operand is negative, the behavior may differ between C90 and C99, depending on the implementation-defined behavior of the C90 implementation.

#include <stdio.h> int main(void) { int x = -1, y = +3; if ((x%y > 0) || ((x+y)%y == x%y)) printf("This is a C90 translator behaving differently than C99\n"); }
Quoting from the C9X Revision Proposal, WG14/N613, that proposed this change:

The origin of this practice seems to have been a desire to map C’s division directly to the “natural” behavior of the target instruction set, whatever it may be, without requiring extra code overhead that might be necessary to check for special cases and enforce a particular behavior. However, the argument that Fortran programmers are unpleasantly surprised by this aspect of C and that there would be negligible impact on code efficiency was accepted by WG14, who agreed to require Fortran-like behavior in C99.

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[CoffeeScript-10] CoffeeScript operators

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[39] ISO 1995

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[r6rs-44] 44.0 ^44.1 r6rs.org

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模除

目录

各种定义

记号

性质及恒等式

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参考文献

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`div`	divide and truncate (i.e., value is not rounded)
`mod`	modulus: let `Remainder = A - (A div B) * B`; if `Remainder < 0` then `A mod B = Remainder + B` otherwise `A mod B = Remainder`

模除

各种定义

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