What is the modulo of 4?

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 a and n, a modulo n (abbreviated as a mod n) is the remainder of the Euclidean division of a by n, where a is the dividend and n is the divisor. The modulo operation is to be distinguished from the symbol mod, which refers to the modulus[1] (or divisor) one is operating from.

For example, the expression "5 mod 2" would evaluate 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 the division of 9 by 3 has a quotient of 3 and a remainder of 0; there is nothing to subtract from 9 after multiplying 3 times 3.

Although typically performed with a and n both being integers, many computing systems now allow other types of numeric operands. The range of values for an integer modulo operation of n is 0 to n 1 inclusive (a mod 1 is always 0; a mod 0 is undefined, possibly resulting in a division by zero error in some programming languages). See Modular arithmetic for an older and related convention applied in number theory.

When exactly one of a or n is negative, the naive definition breaks down, and programming languages differ in how these values are defined.

Variants of the definitionEdit

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 q and the remainder r of a divided by n satisfy the following conditions:

qZa=nq+r|r|<|n|{\displaystyle {\begin{aligned}q\,&\in \mathbb {Z} \\a\,&=nq+r\\|r|&<|n|\end{aligned}}}

(1)

However, 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, and two possible choices for the quotient occur. In number theory, the positive remainder is always chosen, but in computing, programming languages choose depending on the language and the signs of a or n.[a] 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 n or a is negative (see the table under §In programming languages for details). a modulo 0 is undefined in most systems, although some do define it as a.

  • Quotient (q) and remainder (r) as functions of dividend (a), using truncated division

    Many implementations use truncated division, where the quotient is defined by truncation (integer part) q=trunc(an){\textstyle q=\operatorname {trunc} \left({\frac {a}{n}}\right)} and thus according to equation (1) the remainder would have the same sign as the dividend. The quotient is rounded towards zero: equal to the first integer in the direction of zero from the exact rational quotient.r=antrunc(an){\displaystyle r=a-n\operatorname {trunc} \left({\frac {a}{n}}\right)}
  • Quotient and remainder using floored division

    Donald Knuth[3] described floored division where the quotient is defined by the floor function q=an{\textstyle q=\left\lfloor {\frac {a}{n}}\right\rfloor } and thus according to equation (1) the remainder would have the same sign as the divisor. Due to the floor function, the quotient is always rounded downwards, even if it is already negative.r=anan{\displaystyle r=a-n\left\lfloor {\frac {a}{n}}\right\rfloor }
  • Quotient and remainder using Euclidean division

    Raymond T. Boute[4] describes the Euclidean definition in which the remainder is non-negative always, 0 r, and is thus consistent with the Euclidean division algorithm. In this case,n>0⟹q=an{\displaystyle n>0\implies q=\left\lfloor {\frac {a}{n}}\right\rfloor }n<0⟹q=an{\displaystyle n<0\implies q=\left\lceil {\frac {a}{n}}\right\rceil }

    or equivalently

    q=sgn(n)a|n|{\displaystyle q=\operatorname {sgn}(n)\left\lfloor {\frac {a}{\left|n\right|}}\right\rfloor }

    where sgn is the sign function, and thus

    r=a|n|a|n|{\displaystyle r=a-|n|\left\lfloor {\frac {a}{\left|n\right|}}\right\rfloor }
  • Quotient and remainder using rounding division

    A round-division is where the quotient is q=round(an){\textstyle q=\operatorname {round} \left({\frac {a}{n}}\right)}, i.e. rounded to the nearest integer. It is found in Common Lisp and IEEE 754 (see the round to nearest convention in IEEE-754). Thus, the sign of the remainder is chosen to be nearest to zero.
  • Quotient and remainder using ceiling division

    Common Lisp also defines ceiling-division (remainder different sign from divisor) where the quotient is given by q=an{\textstyle q=\left\lceil {\frac {a}{n}}\right\rceil }. Thus, the sign of the remainder is chosen to be different from that of the divisor.

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.

 Daan Leijen, Division and Modulus for Computer Scientists[5]

However, truncated division satisfies the identity (a)/b=(a/b)=a/(b){\displaystyle (-a)/b=-(a/b)=a/(-b)}.[6]

Some calculators have a mod() function button, and many programming languages have a similar function, 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.

For environments lacking a similar function, any of the three definitions above can be used.

Common pitfallsEdit

When the result of a modulo operation has the sign of the dividend (truncating 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 n (the dividend) is negative and odd, n 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; }

Another alternative is to use the fact that for any odd number, the remainder may be either 1 or 1:

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

Performance issuesEdit

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 x is a positive integer, or using a non-truncating definition):

x% 2n == x & (2n - 1)

Examples:

x% 2 == x & 1x% 4 == x & 3x% 8 == x & 7

In devices and software that implement bitwise operations more efficiently than modulo, these alternative forms can result in faster calculations.[7]

Compiler optimizations may recognize expressions of the form expression% constant where constant is a power of two and automatically implement them as expression & (constant-1), 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 expression & (constant-1) 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)Edit

Some modulo operations can be factored or expanded similarly to other mathematical operations. This may be useful in cryptography proofs, such as the DiffieHellman key exchange.

  • Identity:
  • Inverse:
  • Distributive:
    • (a + b) mod n = [(a mod n) + (b mod n)] mod n.
    • ab mod n = [(a mod n)(b mod n)] mod n.
  • Division (definition): a/b mod n = [(a mod n)(b1 mod n)] mod n, when the right hand side is defined (that is when b and n are coprime), and undefined otherwise.
  • Inverse multiplication: [(ab mod n)(b1 mod n)] mod n = a mod n.

In programming languagesEdit

In addition, many computer systems provide a divmod functionality, which produces the quotient and the remainder at the same time. Examples include the x86 architecture's IDIV instruction, the C programming language's div() function, and Python's divmod() function.

GeneralizationsEdit

Modulo with offsetEdit

Sometimes it is useful for the result of a modulo n to lie not between 0 and n 1, but between some number d and d + n 1. In that case, d is called an offset. There does not seem to be a standard notation for this operation, so let us tentatively use a modd n. We thus have the following definition:[38] x = a modd n just in case d x d + n 1 and x mod n = a mod n. Clearly, the usual modulo operation corresponds to zero offset: a mod n = a mod0 n. The operation of modulo with offset is related to the floor function as follows:

amoddn=anadn.{\displaystyle a\operatorname {mod} _{d}n=a-n\left\lfloor {\frac {a-d}{n}}\right\rfloor .}

(To see this, let x=anadn{\textstyle x=a-n\left\lfloor {\frac {a-d}{n}}\right\rfloor }. We first show that x mod n = a mod n. It is in general true that (a + bn) mod n = a mod n for all integers b; thus, this is true also in the particular case when b=adn{\textstyle b=-\!\left\lfloor {\frac {a-d}{n}}\right\rfloor }; but that means that xmodn=(anadn)modn=amodn{\textstyle x{\bmod {n}}=\left(a-n\left\lfloor {\frac {a-d}{n}}\right\rfloor \right)\!{\bmod {n}}=a{\bmod {n}}}, which is what we wanted to prove. It remains to be shown that d x d + n 1. Let k and r be the integers such that a d = kn + r with 0 r n 1 (see Euclidean division). Then adn=k{\textstyle \left\lfloor {\frac {a-d}{n}}\right\rfloor =k}, thus x=anadn=ank=d+r{\textstyle x=a-n\left\lfloor {\frac {a-d}{n}}\right\rfloor =a-nk=d+r}. Now take 0 r n 1 and add d to both sides, obtaining d d + r d + n 1. But we've seen that x = d + r, so we are done. )

The modulo with offset a modd n is implemented in Mathematica as Mod[a, n, d] .[38]

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:[5]

/* 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}; }

Note that 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.

  1. ^ Mathematically, these two choices are but two of the infinite number of choices available for the inequality satisfied by a remainder.
  2. ^ a b Argument order reverses, i.e., α|ω computes ωmodα{\displaystyle \omega {\bmod {\alpha }}}, the remainder when dividing ω by α.
  3. ^ C99 and C++11 define the behavior of % to be truncated.[8] The standards before then leave the behavior implementation-defined.[9]
  4. ^ As implemented in ACUCOBOL, Micro Focus COBOL, and possible others.
  5. ^ Divisor must be positive, otherwise undefined.
  6. ^ As discussed by Boute, ISO Pascal's definitions of div and mod do not obey the Division Identity of D = d · (D / d) + D% d, and are thus fundamentally broken.
  7. ^ Perl usually uses arithmetic modulo operator that is machine-independent. For examples and exceptions, see the Perl documentation on multiplicative operators.[27]

ReferencesEdit

  • Modulorama, animation of a cyclic representation of multiplication tables (explanation in French)