Modulo

"fmod" redirects here. For the computer audio library, see FMOD.

In computing, the modulo operation finds the remainder of division of one number by another.

Quotient (red) and remainder (green) functions using different algorithms.

Given two numbers, a (the dividend) and n (the divisor), a modulo n (abbreviated as a mod n) is the remainder, on division of a by n. For instance, the expression "7 mod 3" would evaluate to 1, while "9 mod 3" would evaluate to 0. Although typically performed with a and n both being integers, many computing systems allow other types of numeric operands.

See modular arithmetic for an older and related convention applied in number theory.

Remainder calculation for the modulo operation

Modulo operators in various programming languages

Language	Operator	Result has the same sign as
ActionScript	%	Dividend
Ada	mod	Divisor
	rem	Dividend
ASP	Mod	Not defined
C (ISO 1990)	%	Not defined
C (ISO 1999)	%	Dividend
C++	%	Not defined
C#	%	Dividend
ColdFusion	MOD	Dividend
Common Lisp	mod	Divisor
	rem	Dividend
Eiffel	\\	Dividend
Microsoft Excel	=MOD()	Divisor
Euphoria	remainder	Dividend
FileMaker	Mod	Divisor
Fortran	mod	Dividend
	modulo	Divisor
GML (Game Maker)	mod	Dividend
Haskell	mod	Divisor
	rem	Dividend
J	|~	Divisor
Java	%	Dividend
JavaScript	%	Dividend
Lua	%	Divisor
MathCad	mod(x,y)	Divisor
Mathematica	Mod	Divisor
MATLAB	mod	Divisor
	rem	Dividend
MySQL	MOD %	Dividend
Objective Caml	mod	Not defined
Occam	\	Dividend
Pascal (Delphi)	mod	Dividend
Perl	%	Divisor[1]
PHP	%	Dividend
PL/I	mod	Divisor (ANSI PL/I)
Prolog (ISO 1995)	mod	Divisor
	rem	Dividend
Python	%	Divisor
QBasic	MOD	Dividend
R	%%	Divisor
RPG	%REM	Dividend
Ruby	%	Divisor
Scheme	modulo	Divisor
SenseTalk	modulo	Divisor
	rem	Dividend
Smalltalk	\\	Divisor
Tcl	%	Divisor
Verilog (2001)	%	Dividend
VHDL	mod	Divisor
	rem	Dividend
Visual Basic	Mod	Dividend

There are various ways of defining a remainder, and computers and calculators have various ways of storing and representing numbers, so what exactly constitutes the result of a modulo operation depends on the programming language and/or the underlying hardware.

In nearly all computing systems, the quotient q and the remainder r satisfy

• $a=q+nr\,$
• $q\,$ is an integer
• $0\leq |r|<|n|$

Pascal and Algol68 do not satisfy these for negative divisors. Some programming languages, such as C89, don't even define a result if either of n or a is negative. See the table for details. a modulo 0 is undefined in the majority of systems, although some do define it to be a.

Many implementations use truncated division where the quotient is defined by truncation q = trunc(a/n) and the remainder by r=a-n q. With this definition the quotient is rounded towards zero and the remainder has the same sign as the dividend.

Knuth described floored division where the quotient is defined by the floor function q=floor(a/n) and the remainder r is

$r=a-n\left\lfloor {a \over n}\right\rfloor .$

Here the quotient rounds towards negative infinity and the remainder has the same sign as the divisor.

Raymond T. Boute introduces the Euclidean definition which is consistent with the division algorithm. Let q be the integer quotient of a and n, then:

$q\in \mathbb {Z}$

$a=n\times q+r\,$

$0\leq r<|n|.$

Two corollaries are that

$n>0\to q=\left\lfloor a\div n\right\rfloor$

$n<0\to q=\left\lceil a\div n\right\rceil .$

As described by Leijen,

Boute argues that Euclidean division is superior to the other ones in terms of regularity and useful mathematical properties, although floored division, promoted by Knuth, is also a good definition. Despite its widespread use, truncated division is shown to be inferior to the other definitions.

Common Lisp also defines round- and ceiling-division where the quotient is given by q=round(a/n), q=ceil(a/n). IEEE 754 defines a remainder function where the quotient is a/n rounded according to the round to nearest convention.

Modulo operation expression

Some calculators have a mod() function button, and many programming languages have a mod() function or similar, expressed as mod(a, n), for example. Some also support expressions that use "%", "mod", or "Mod" as a modulo or remainder operator, such as

a % n

or

a mod n.

Performance issues

Modulo operations might be implemented such that division with remainder is calculated each time. For special cases, there are faster alternatives on some hardware. For example, the modulo of powers of 2 can alternatively be expressed as a bitwise AND operation:

x % 2n == x & (2n - 1).

Examples (assuming x is an integer):

x % 2 == x & 1

x % 4 == x & 3

x % 8 == x & 7.

In devices and software that implement bitwise operations more efficiently than modulo, these alternative forms can result in faster calculations.

In the C programming language, compiling with heavy speed optimizations will typically (depending on compiler and hardware) automatically convert modulo operations to bitwise AND in the assembly file.

In some compilers, the modulo operation is implemented as mod(a, n) = a - n * floor(a / n). When performing both modulo and division on the same numbers, one can get the same result somewhat more efficiently by avoiding the actual modulo operator, and using the formula above on the result, avoiding an additional division operation.

See also

• Modulo — many uses of the word "modulo", all of which grew out of Carl F. Gauss's introduction of modular arithmetic in 1801.

Notes

• ^ Perl usually uses arithmetic modulo operator that is machine-independent. See the Perl documentation for exceptions and examples.
• ^ Mathematically, these two choices are but two of the infinite number of choices available for the inequality satisfied by a remainder.