Computing is usually defined as the activity of using and improving computer hardware and software. It is the computer-specific part of information technology...
right|thumb|200px|20 \div 4=5In mathematics, especially in elementary arithmetic, division is an arithmetic operation.Specifically, if c times b equals a, written:c \times b = a\,...
Dividends are payments made by a corporation to its shareholder members. It is the portion of corporate profits paid out to stockholders. When a corporation earns a profit or surplus, that money can be put to two uses: it can either be re-invested in the business , or it can be distributed to...
In mathematics, a divisor of an integer n, also called a factor of n, is an integer which divides n without leaving a remainder.-Explanation:...
), amodulon (abbreviated as a mod n) can be thought of as the remainder, on division of a by n. For instance, the expression "5 mod 4" would evaluate to 1 because 5 divided by 4 leaves a remainder of 1, while "9 mod 3" would evaluate to 0 because the division of 9 by 3 leaves a remainder of 0; there is nothing to subtract from 9 after multiplying 3 times 3. (Notice that doing the division with a calculator won't show you the result referred to here by this operation, the quotient will be expressed as a decimal.) When either or is negative, this naive definition breaks down and programming languages differ in how these values are defined. Although typically performed with a and n both being integers, many computing systems allow other types of numeric operands. The range of numbers for an integer modulo of n is 0 to n - 1. (n mod 1 is always 0; n mod 0 is undefined, possibly resulting in a "Division by zero" error in computer programming languages) See modular arithmetic
Modular arithmetic
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value—the modulus....
for an older and related convention applied in number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
.
Remainder calculation for the modulo operation
Integer modulo operators in various programming languages
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IDIV
Dividend
Floating-point modulo operators in various programming languages
C is a general-purpose computer programming language developed between 1969 and 1973 by Dennis Ritchie at the Bell Telephone Laboratories for use with the Unix operating system....
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C++ is a statically typed, free-form, multi-paradigm, compiled, general-purpose programming language. It is regarded as an intermediate-level language, as it comprises a combination of both high-level and low-level language features. It was developed by Bjarne Stroustrup starting in 1979 at Bell...
Common Lisp, commonly abbreviated CL, is a dialect of the Lisp programming language, published in ANSI standard document ANSI INCITS 226-1994 , . From the ANSI Common Lisp standard the Common Lisp HyperSpec has been derived for use with web browsers...
The D programming language is an object-oriented, imperative, multi-paradigm, system programming language created by Walter Bright of Digital Mars. It originated as a re-engineering of C++, but even though it is mainly influenced by that language, it is not a variant of C++...
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Real.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
Programming language
A programming language is an artificial language designed to communicate instructions to a machine, particularly a computer. Programming languages can be used to create programs that control the behavior of a machine and/or to express algorithms precisely....
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In mathematics, a quotient is the result of division. For example, when dividing 6 by 3, the quotient is 2, while 6 is called the dividend, and 3 the divisor. The quotient further is expressed as the number of times the divisor divides into the dividend e.g. The quotient of 6 and 2 is also 3.A...
and the remainder satisfy
This means, that if the remainder is nonzero, 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. However, Pascal and Algol68 give a positive remainder (or 0) even 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
Truncation
In mathematics and computer science, truncation is the term for limiting the number of digits right of the decimal point, by discarding the least significant ones.For example, consider the real numbersThe result would be:- Truncation and floor function :...
q = trunc(a/n), in other words it is the first integer in the direction of 0 from the exact rational quotient, and the remainder by r=a − nq. Informally speaking the quotient is "rounded towards zero", and the remainder therefore has the same sign as the dividend.
Donald Ervin Knuth is a computer scientist and Professor Emeritus at Stanford University.He is the author of the seminal multi-volume work The Art of Computer Programming. Knuth has been called the "father" of the analysis of algorithms...
described floored division where the quotient is defined by the floor function
Floor function
In mathematics and computer science, the floor and ceiling functions map a real number to the largest previous or the smallest following integer, respectively...
q=floor(a/n) and the remainder r is
Here the quotient is always rounded downwards (even if it is already negative) and the remainder has the same sign as the divisor.
Raymond T. Boute introduces the Euclidean definition, which is the one in which the remainder is always positive or 0, and is therefore consistent with the division algorithm
Division algorithm
In mathematics, and more particularly in arithmetic, the usual process of division of integers producing a quotient and a remainder can be specified precisely by a theorem stating that these exist uniquely with given properties. An integer division algorithm is any effective method for producing...
. This definition is marked as "Always positive" in the table. Let q be the integer quotient of a and n, then:
Two corollaries are that
or, equivalently,
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 , .
IEEE 754 defines a remainder function where the quotient is 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
Operator (programming)
Programming languages typically support a set of operators: operations which differ from the language's functions in calling syntax and/or argument passing mode. Common examples that differ by syntax are mathematical arithmetic operations, e.g...
, such as
a % n
or
a mod n
or equivalent, for environments lacking a mod function
a - (n * int(a/n)).
In most cases means modulo function and not remainder function. For example
Performance issues
Modulo operations might be implemented such that a division with a 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
Bitwise operation
A bitwise operation operates on one or more bit patterns or binary numerals at the level of their individual bits. This is used directly at the digital hardware level as well as in microcode, machine code and certain kinds of high level languages...
operation:
x % 2n x & (2n - 1).
Examples (assuming x is a positive 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.
Compiler optimization is the process of tuning the output of a compiler to minimize or maximize some attributes of an executable computer program. The most common requirement is to minimize the time taken to execute a program; a less common one is to minimize the amount of memory occupied...
C is a general-purpose computer programming language developed between 1969 and 1973 by Dennis Ritchie at the Bell Telephone Laboratories for use with the Unix operating system....
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. (Note: This will not work for the languages whose modulo have the sign of the dividend (including C), because if the dividend is negative, the modulo will be negative; however, expression & (constant-1) will always produce a positive result. So special treatment has to be made when the dividend is negative.)
In some compilers, the modulo operation is implemented as mod(a, n) = a - n * floor(a / n). For example, mod(7, 3) = 7 - 3 * floor(7 / 3) = 7 - 3 * floor(2.33) = 7 - 3 * 2 = 7 - 6 = 1.
See also
In the mathematical community, the word modulo is often used informally. Generally, to say "A is the same as B modulo C" means, more-or-less, "A and B are the same except for differences accounted for or explained by C"....
The word modulo is the Latin ablative of modulus which itself means "a small measure."It was introduced into mathematics in the book Disquisitiones Arithmeticae by Carl Friedrich Gauss in 1801...
– many uses of the word "modulo", all of which grew out of Carl F. Gauss's introduction of modular arithmetic
Modular arithmetic
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value—the modulus....
in 1801.
The source of this article is wikipedia, the free encyclopedia. The text of this article is licensed under the GFDL.
means modulo function and not remainder function. For example
