MODULO OPERATION

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

Contents

Remainder calculation for the modulo operation

Modulo operation expression

Performance issues

x & (2n - 1) Examples (assuming x is an integer): :x % 2

x & 3 :x % 8

See also

Notes

References

Remainder calculation for the modulo operation

>>>

Modulo operators in various programming languages

Language	Operator	Result has the same sign as
Ada	mod	Divisor
	rem	Dividend
ASP	Mod	Not defined
C (1989)	%	Not defined
C (1999)	%	Dividend
C#	%	Dividend
ColdFusion	MOD	Dividend
Common Lisp	mod	Divisor
Eiffel	\	Dividend
Fortran	mod	Dividend
	modulo	Divisor
J	>~	Divisor
Java	%	Dividend
JavaScript	%	Dividend
MATLAB	mod	Divisor
	rem	Dividend
MySQL	MOD %	Dividend
Objective Caml	mod	Not defined
Pascal (Delphi)	mod	Dividend
Perl	%	Not defined
PHP	%	Dividend
PL/I	mod	Divisor (ANSI PL/I)
Prolog (ISO 1995)	mod	Divisor
Python	%	Divisor
RPG	%REM	Dividend
Ruby	%	Divisor
SenseTalk	modulo	Divisor
	rem	Dividend
Verilog (2001)	%	Dividend
VHDL	mod	Divisor
	rem	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 resulting from the division is constrained to the set of integers, and the remainder ''r'' is typically constrained by either or . The choice between the two possible remainders depends on the signs of ''a'' and ''n'' and the programming language being used. 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''. If the definition is consistent with the division algorithm, then ''n'' = 0 implies , which is a contradiction (i.e., the usual remainder does not exist in this case).
The remainder can be calculated by using equations, in terms of other functions. Differences may arise according to the scope of the variables, which in common implementations is broader than in the definition just given. One useful equation for calculating the remainder ''r'' is
:$r\; =\; a\; -\; n\; leftlfloor\; \{a\; over\; n\}$
ight
floor
where $lfloor\; x$
floor is the floor function of ''x''. See e.g. [1], [2], [3].
Raymond T. Boute^[1] analyzed several definitions of integer division and modulo, and he introduces the “Euclidean” definition. Let ''q'' be the integer quotient of ''a'' and ''n'', then:
: $q\; in\; mathbb\{Z\}$
: $a\; =\; n\; imes\; q\; +\; r$
:
Two corollaries are that
: $n\; >\; 0\; o\; q\; =\; leftlfloor\; a\; div\; n$
ight
floor
:
ight
ceil
As described by Leijen,^[2]
: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.

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 modulus 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.

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.

★ Modular arithmetic

Notes

★ The semantics of the modulo operator in Perl are defined to be those of the modulo operator of the C compiler that was used to compile the Perl interpreter itself.

★ Mathematically, these two choices are but two of the infinite number of choices available for the inequality satisfied by a remainder.

References

1. The Euclidean definition of the functions div and mod, , Raymond T., Boute, ACM Transactions on Programming Languages and Systems (TOPLAS),
2. Division and Modulus for Computer Scientists