Remainder

in remainderlaverremainder for natural numbersremaindersresidues
In mathematics, the remainder is the amount "left over" after performing some computation.wikipedia
114 Related Articles

Modulo operation

modulomodmodulus
The modulo operation is the operation that produces such a remainder when given a dividend and divisor. Extending the definition of remainder for floating-point numbers as described above is not of theoretical importance in mathematics; however, many programming languages implement this definition, see modulo operation.
In computing, the modulo operation finds the remainder after division of one number by another (called the modulus of the operation).

Division (mathematics)

divisiondividingdivided
In arithmetic, the remainder is the integer "left over" after dividing one integer by another to produce an integer quotient (integer division).
Unlike the other basic operations, when dividing natural numbers there is sometimes a remainder that will not go evenly into the dividend; for example,

Quotient

In arithmetic, the remainder is the integer "left over" after dividing one integer by another to produce an integer quotient (integer division).
The quotient is also less commonly defined as the greatest whole number of times a divisor may be subtracted from a dividend without the remainder becoming negative.

Division algorithm

Newton–Raphson divisionSRT divisiondivision by a constant
See Euclidean division for a proof of this result and division algorithm for algorithms describing how to calculate the remainder.
A division algorithm is an algorithm which, given two integers N and D, computes their quotient and/or remainder, the result of Euclidean division.

Euclidean division

Division theoremdividedivisible by two without remainder
See Euclidean division for a proof of this result and division algorithm for algorithms describing how to calculate the remainder.
In arithmetic, Euclidean division — or division with remainder — is the process of dividing one integer (the dividend) by another (the divisor), in such a way that produces a quotient and a remainder smaller than the divisor.

Integer

integersintegralZ
In arithmetic, the remainder is the integer "left over" after dividing one integer by another to produce an integer quotient (integer division).
is called the remainder of the division of

Euclidean algorithm

Euclid's algorithmEuclideanEuclid
In mathematics, the Euclidean algorithm, or Euclid's algorithm, is an efficient method for computing the greatest common divisor (GCD) of two numbers, the largest number that divides both of them without leaving a remainder.

Long division

(long)Division tableauSchoolbook long division
The quotient (rounded down to an integer) becomes the first digit of the result, and the remainder is calculated (this step is notated as a subtraction).

Divisibility rule

divisibility testcan be divideddivisibility rule for 11

Ancient Egyptian multiplication

Egyptian multiplicationPeasant multiplicationEgyptian multiplication and division
Quotient and remainder answers were multiplied by divisor inverses, 1/3, 1/7, 1/10, 1/11 and 1/13, exactly returning the beginning rational number (64/64).

Arithmetic

arithmetic operationsarithmeticsarithmetic operation
In arithmetic, the remainder is the integer "left over" after dividing one integer by another to produce an integer quotient (integer division).

Algebra

algebraicAlgebra IAlgebra 1
In algebra, the remainder is the polynomial "left over" after dividing one polynomial by another.

Subtraction

differencesubtrahendminuend
Formally it is also true that a remainder is what is left after subtracting one number from another, although this is more precisely called the difference.

Function (mathematics)

functionfunctionsmathematical function
However, the term "remainder" is still used in this sense when a function is approximated by a series expansion and the error expression ("the rest") is referred to as the remainder term.

Series expansion

However, the term "remainder" is still used in this sense when a function is approximated by a series expansion and the error expression ("the rest") is referred to as the remainder term.

Series (mathematics)

infinite seriesseriespartial sum
However, the term "remainder" is still used in this sense when a function is approximated by a series expansion and the error expression ("the rest") is referred to as the remainder term.

Floating-point arithmetic

floating pointfloating-pointfloating-point number
When a and d are floating-point numbers, with d non-zero, a can be divided by d without remainder, with the quotient being another floating-point number.

Programming language

programming languageslanguagedialect
Extending the definition of remainder for floating-point numbers as described above is not of theoretical importance in mathematics; however, many programming languages implement this definition, see modulo operation.

Pascal (programming language)

PascalPascal programming languageISO 7185
* Pascal chooses the result of the mod operation positive, but does not allow d to be negative or zero (so, a = (a div d ) × d + a mod d is not always valid).

C99

C99 programming language1999 standardC
* C99 chooses the remainder with the same sign as the dividend a.

Perl

Perl 5Perl programming languagePerl Script
* Perl, Python (only modern versions), and Common Lisp choose the remainder with the same sign as the divisor d.

Python (programming language)

PythonPython programming languagePython 2
* Perl, Python (only modern versions), and Common Lisp choose the remainder with the same sign as the divisor d.

Common Lisp

QuicklispANSI Common LispArmed Bear Common Lisp
* Perl, Python (only modern versions), and Common Lisp choose the remainder with the same sign as the divisor d.

Haskell (programming language)

HaskellHaskell programming languageHackage
* Haskell and Scheme offer two functions, remainder and modulo – PL/I has mod and rem, while Fortran has mod and modulo; in each case, the former agrees in sign with the dividend, and the latter with the divisor.

Scheme (programming language)

SchemeScheme programming languageR6RS
* Haskell and Scheme offer two functions, remainder and modulo – PL/I has mod and rem, while Fortran has mod and modulo; in each case, the former agrees in sign with the dividend, and the latter with the divisor.