Jump to content

Modulo

From Wikipedia, the free encyclopedia

This is an old revision of this page, as edited by 164.67.235.98 (talk) at 23:59, 24 September 2009 (Remainder calculation for the modulo operation). The present address (URL) is a permanent link to this revision, which may differ significantly from the current revision.

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

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.

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
BASIC Mod Not defined
C (ISO 1990) % Implementation defined
C (ISO 1999) % Dividend
C++ % Implementation defined[1]
C# % Dividend
CLARION % Dividend
Clojure rem Dividend
ColdFusion MOD Dividend
Common Lisp mod Divisor
rem Dividend
Eiffel \\ Dividend
Erlang rem Dividend
Microsoft Excel =MOD() Divisor
Euphoria mod Divisor
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
Oberon MOD Divisor
Objective Caml mod Dividend
Occam \ Dividend
Pascal (Delphi) mod Dividend
Perl % Divisor[1]
PHP % Dividend
PL/I mod Divisor (ANSI PL/I)
PowerBuilder mod(x,y) ?
Prolog (ISO 1995) mod Divisor
rem Dividend
Python % Divisor
QBasic MOD Dividend
R %% Divisor
RPG %REM Dividend
Ruby % Divisor
Scheme modulo Divisor
remainder Dividend
Scheme R6RS[2] mod Always Nonnegative (Euclidean)
mod0 Closest to zero
SenseTalk modulo Divisor
rem Dividend
Smalltalk \\ Divisor
SQL (SQL:1999) mod(x,y) Dividend
Standard ML mod Divisor
Int.rem Dividend
Tcl % Divisor
Torque Game Engine % Dividend
TI-BASIC fPart(x/y)*y Dividend
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

This means there are two possible choices for the remainder, one negative and the other positive, and there are also two possible choices for the quotient. Usually, in number theory, the positive remainder is always chosen, but programming languages choose depending on the language and the signs of a and n.[3] However, Pascal and Algol68 do not satisfy these conditions for negative divisors, and 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[2] described floored division where the quotient is defined by the floor function q=floor(a/n) and the remainder r is

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

Raymond T. Boute[3] introduces the Euclidean definition which is consistent with the division algorithm. Let q be the integer quotient of a and n, then:

Two corollaries are that

As described by Leijen,[4]

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.

Common pitfalls

When the result of a modulo operation has the sign of the dividend, it can sometimes lead to surprising mistakes:

For example, to test whether an integer is odd, one might be inclined to test whether the remainder by 2 is equal to 1:

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

But in a language where modulo has the sign of the dividend, that is incorrect, because when n (the dividend) is negative and odd, n % 2 returns -1, and the function returns false.

One correct alternative is to test that it is not 0 (because remainder 0 is the same regardless of the signs):

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

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

or equivalent, for environments lacking a mod() function

a - n * int(a/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.

Optimizing C compilers generally recognize expressions of the form expression % constant where constant is a power of two and automatically implement them as expression & (constant-1). This can allow the programmer to write clearer code without compromising performance.

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

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.

References

  1. ^ ISO/IEC 14882:2003 : Programming languages -- C++, 5.6.4: ISO, IEC, 2003{{citation}}: CS1 maint: location (link). "the binary % operator yields the remainder from the division of the first expression by the second. .... If both operands are nonnegative then the remainder is nonnegative; if not, the sign of the remainder is implementation-defined".
  2. ^ Knuth, Donald. E. (1972). The Art of Computer Algorithms. Addison-Wesley.
  3. ^ Boute, Raymond T. (April 1992). "The Euclidean definition of the functions div and mod". ACM Transactions on Programming Languages and Systems (TOPLAS). 14 (2). ACM Press (New York, NY, USA): 127–144. doi:10.1145/128861.128862.
  4. ^ Leijen, Daan (December 3, 2001). "Division and Modulus for Computer Scientists" (PDF). Retrieved 2006-08-27. {{cite web}}: Check date values in: |date= (help)