Modulo
Template:About Template:Short description In computing, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another, called the modulus of the operation.
Given two positive numbers Template:Math and Template:Math, Template:Math modulo Template:Math (often abbreviated as Template:Math) is the remainder of the Euclidean division of Template:Math by Template:Math, where Template:Math is the dividend and Template:Math is the divisor.[1]
For example, the expression "5 mod 2" evaluates to 1, because 5 divided by 2 has a quotient of 2 and a remainder of 1, while "9 mod 3" would evaluate to 0, because 9 divided by 3 has a quotient of 3 and a remainder of 0.
Although typically performed with Template:Math and Template:Math both being integers, many computing systems now allow other types of numeric operands. The range of values for an integer modulo operation of Template:Math is 0 to Template:Math. Template:Math mod 1 is always 0.
When exactly one of Template:Math or Template:Math is negative, the basic definition breaks down, and programming languages differ in how these values are defined.
Variants of the definition
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 non-negative integer that belongs to that class (i.e., the remainder of the Euclidean division).[2] 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 Template:Math and the remainder Template:Math of Template:Math divided by Template:Math satisfy the following conditions: Template:NumBlk
This still leaves a sign ambiguity if the remainder is non-zero: two possible choices for the remainder occur, one negative and the other positive; that choice determines which of the two consecutive quotients must be used to satisfy equation (1). In number theory, the positive remainder is always chosen, but in computing, programming languages choose depending on the language and the signs of Template:Math or Template:Math.Template:Efn Standard Pascal and ALGOL 68, for example, give a positive remainder (or 0) even for negative divisors, and some programming languages, such as C90, leave it to the implementation when either of Template:Math or Template:Math is negative (see the table under Template:Section link for details). Some systems leave Template:Math modulo 0 undefined, though others define it as Template:Math.
If both the dividend and divisor are positive, then the truncated, floored, and Euclidean definitions agree. If the dividend is positive and the divisor is negative, then the truncated and Euclidean definitions agree. If the dividend is negative and the divisor is positive, then the floored and Euclidean definitions agree. If both the dividend and divisor are negative, then the truncated and floored definitions agree.
As described by Leijen, Template:Quote
However, truncated division satisfies the identity .[3]
Notation
Some calculators have a Template:Math function button, and many programming languages have a similar function, expressed as Template:Math, for example. Some also support expressions that use "%", "mod", or "Mod" as a modulo or remainder operator, such as Template:Code or Template:Code.
For environments lacking a similar function, any of the three definitions above can be used.
Common pitfalls
When the result of a modulo operation has the sign of the dividend (truncated definition), it can lead to surprising mistakes.
For example, to test if an integer is odd, one might be inclined to test if 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 Template:Math (the dividend) is negative and odd, Template:Math mod 2 returns −1, and the function returns false.
One correct alternative is to test that the remainder is not 0 (because remainder 0 is the same regardless of the signs):
bool is_odd(int n) {
return n % 2 != 0;
}
Or with the binary arithmetic:
bool is_odd(int n) {
return n & 1;
}
Performance issues
Modulo operations might be implemented such that a division with a remainder is calculated each time. For special cases, on some hardware, faster alternatives exist. For example, the modulo of powers of 2 can alternatively be expressed as a bitwise AND operation (assuming Template:Math is a positive integer, or using a non-truncating definition):
x % 2n == x & (2n - 1)
Examples:
In devices and software that implement bitwise operations more efficiently than modulo, these alternative forms can result in faster calculations.[4]
Compiler optimizations may recognize expressions of the form Template:Code where Template:Code is a power of two and automatically implement them as Template:Code, allowing the programmer to write clearer code without compromising performance. This simple optimization is not possible for languages in which the result of the modulo operation has the sign of the dividend (including C), unless the dividend is of an unsigned integer type. This is because, if the dividend is negative, the modulo will be negative, whereas Template:Code will always be positive. For these languages, the equivalence x % 2n == x < 0 ? x | ~(2n - 1) : x & (2n - 1) has to be used instead, expressed using bitwise OR, NOT and AND operations.
Optimizations for general constant-modulus operations also exist by calculating the division first using the constant-divisor optimization.
Properties (identities)
Template:See also Some modulo operations can be factored or expanded similarly to other mathematical operations. This may be useful in cryptography proofs, such as the Diffie–Hellman key exchange. The properties involving multiplication, division, and exponentiation generally require that Template:Math and Template:Math are integers.
- Identity:
- Template:Math.
- Template:Math for all positive integer values of Template:Math.
- If Template:Math is a prime number which is not a divisor of Template:Math, then Template:Math, due to Fermat's little theorem.
- Inverse:
- Template:Math.
- Template:Math denotes the modular multiplicative inverse, which is defined if and only if Template:Math and Template:Math are relatively prime, which is the case when the left hand side is defined: Template:Math.
- Distributive:
- Division (definition): Template:Math, when the right hand side is defined (that is when Template:Math and Template:Math are coprime), and undefined otherwise.
- Inverse multiplication: Template:Math.
In programming languages
In addition, many computer systems provide a Template:Code functionality, which produces the quotient and the remainder at the same time. Examples include the x86 architecture's Template:Code instruction, the C programming language's Template:Code function, and Python's Template:Code function.
Generalizations
Modulo with offset
Sometimes it is useful for the result of Template:Mvar modulo Template:Mvar to lie not between 0 and Template:Math, but between some number Template:Mvar and Template:Math. In that case, Template:Mvar is called an offset and Template:Math is particularly common.
There does not seem to be a standard notation for this operation, so let us tentatively use Template:Math. We thus have the following definition:[54] Template:Math just in case Template:Math and Template:Math. Clearly, the usual modulo operation corresponds to zero offset: Template:Math.
The operation of modulo with offset is related to the floor function as follows:
To see this, let . We first show that Template:Math. It is in general true that Template:Math for all integers Template:Mvar; thus, this is true also in the particular case when ; but that means that , which is what we wanted to prove. It remains to be shown that Template:Math. Let Template:Mvar and Template:Mvar be the integers such that Template:Math with Template:Math (see Euclidean division). Then , thus . Now take Template:Math and add Template:Mvar to both sides, obtaining Template:Math. But we've seen that Template:Math, so we are done.
The modulo with offset Template:Math is implemented in Mathematica as Template:Code .[54]
Implementing other modulo definitions using truncation
Despite the mathematical elegance of Knuth's floored division and Euclidean division, it is generally much more common to find a truncated division-based modulo in programming languages. Leijen provides the following algorithms for calculating the two divisions given a truncated integer division:[55]
/* Euclidean and Floored divmod, in the style of C's ldiv() */
typedef struct {
/* This structure is part of the C stdlib.h, but is reproduced here for clarity */
long int quot;
long int rem;
} ldiv_t;
/* Euclidean division */
inline ldiv_t ldivE(long numer, long denom) {
/* The C99 and C++11 languages define both of these as truncating. */
long q = numer / denom;
long r = numer % denom;
if (r < 0) {
if (denom > 0) {
q = q - 1;
r = r + denom;
} else {
q = q + 1;
r = r - denom;
}
}
return (ldiv_t){.quot = q, .rem = r};
}
/* Floored division */
inline ldiv_t ldivF(long numer, long denom) {
long q = numer / denom;
long r = numer % denom;
if ((r > 0 && denom < 0) || (r < 0 && denom > 0)) {
q = q - 1;
r = r + denom;
}
return (ldiv_t){.quot = q, .rem = r};
}
For both cases, the remainder can be calculated independently of the quotient, but not vice versa. The operations are combined here to save screen space, as the logical branches are the same.
See also
- Modulo (disambiguation) – many uses of the word modulo, all of which grew out of Carl F. Gauss' approach to modular arithmetic in 1801.
- Modulo (mathematics), general use of the term in mathematics
- Modular exponentiation
- Turn (angle)
Notes
References
External links
- Different kinds of integer division
- Modulorama, animation of a cyclic representation of multiplication tables (explanation in French)
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