Dictionary Definition
remainder
Noun
1 something left after other parts have been
taken away; "there was no remainder"; "he threw away the rest"; "he
took what he wanted and I got the balance" [syn: balance, residual, residue, residuum, rest]
2 the part of the dividend that is left over when
the dividend is not evenly divisible by the divisor
3 the number that remains after subtraction; the
number that when added to the subtrahend gives the minuend [syn:
difference]
4 a piece of cloth that is left over after the
rest has been used or sold [syn: end, remnant, oddment] v : sell cheaply as
remainders; "The publisher remaindered the books"
User Contributed Dictionary
English
Pronunciation

 Rhymes: eɪndə(r)
Noun
 A part or parts remaining after some has/have
been removed.
 My son ate part of his cake and I ate the remainder.
 You can have the remainder of my clothes.
 In the context of "commerce": Items left unsold and subject to
reduction in price.
 I got a really good price on this shirt because it was a remainder.
 The amount left over after subtracting the divisor as many times as
possible from the dividend without producing a
negative result. If
(dividend) and d (divisor) are integers, then can always be
expressed in the form n = dq + r, where q (quotation) and r (remainder)
are also integers, and 0 ≤ r < d.
 17 leaves a remainder of 2 when divided by 3.
 11 divided by 2 is 5 remainder 1.
Synonyms
 sense a part or parts remaining remnant, residue, rest
Translations
what remains after some has been removed
items left unsold and subject to reduction in
price
 Czech: zbytek
 Finnish: jäännöserä
 German: Restposten
 Japanese: 売れ残り
 Russian: остаток (ostátok) , остатки (ostátki) p
mathematics: amount left over after repeatedly
subtracting the divisor
 Czech: zbytek
 Finnish: jakojäännös
 Japanese: 剰余
 Russian: остаток (ostátok)
Translations to be checked
 ttbc Arabic:
 ttbc Chinese: 剩馀 (shèng yú)
 ttbc Dutch: rest
 ttbc French: reste , invendu (2)
 ttbc German: Rest
 ttbc Hungarian: maradék p
 ttbc Italian: resto (1), avanzo (2,3,4)
 ttbc Korean: 나머지 (nameoji)
 ttbc Portuguese: resto
 ttbc Spanish: resto
 ttbc Swedish: rest (1,2,3)
See also
Extensive Definition
In arithmetic, when the result
of the division
of two integers cannot
be expressed with an integer quotient, the remainder is the
amount "left over."
The remainder for natural numbers
If a and d are natural numbers, with d nonzero, it can be proved that there exist unique integers q and r, such that a = qd + r and 0 ≤ r < d. The number q is called the quotient, while r is called the remainder. The division algorithm provides a proof of this result and also an algorithm describing how to calculate the remainder.Examples
 When dividing 13 by 10, 1 is the quotient and 3 is the remainder, because 13=1×10+3.
 When dividing 26 by 4, 6 is the quotient and 2 is the remainder, because 26=6×4+2.
 When dividing 56 by 7, 8 is the quotient and 0 is the remainder, because 56=7×8+0.
 When dividing 3 by 10, 3 is the remainder as we always take the front number as the remainder when the second number is of higher value.
The case of general integers
If a and d are integers, with d nonzero, then a remainder is an integer r such that a = qd + r for some integer q, and with 0 ≤ r < d.When defined this way, there are two possible
remainders. For example, the division of −42 by
−5 can be expressed as either
 −42 = 9×(−5) + 3
as is usual for mathematicians, or
 −42 = 8×(−5) + (−2).
So the remainder is then either 3 or
−2.
This ambiguity in the value of the remainder can
be quite serious computationally; for mission critical computing
systems, the wrong choice can lead to dangerous consequences. In
the case above, the negative remainder is obtained from the
positive one just by subtracting 5, which is d. This holds in
general. When dividing by d, if the positive remainder is r1, and
the negative one is r2, then
 r1 = r2 + d.
The remainder for real numbers
When a and d are real numbers, with d nonzero, a can be divided by d without remainder, with the quotient being another real number. If the quotient is constrained to being an integer however, the concept of remainder is still necessary. It can be proved that there exists a unique integer quotient q and a unique real remainder r such that a=qd+r with 0≤r < d. As in the case of division of integers, the remainder could be required to be negative, that is, d < r ≤ 0.Extending the definition of remainder for real
numbers as described above is not of theoretical importance in
mathematics; however, many programming
languages implement this definition — see modulo
operation.
The inequality satisfied by the remainder
The way remainder was defined, in addition to the equality a=qd+r an inequality was also imposed, which was either 0≤ r < d or d < r ≤ 0. Such an inequality is necessary in order for the remainder to be unique — that is, for it to be welldefined. The choice of such an inequality is somewhat arbitrary. Any condition of the form x < r ≤ x+d (or x ≤ r < x+d), where x is a constant, is enough to guarantee the uniqueness of the remainder.Quotient and remainder in programming languages
With two choices for the inequality, there are two possible choices for the remainder, one is negative and the other is positive. This means that there are also two possible choices for the quotient. Usually, in number theory, we always choose the positive remainder. But programming languages do not. C99 and Pascal choose the remainder with the same sign as the dividend a. (Before C99, the C language allowed either choice.) Perl and Python choose the remainder with the same sign as the divisor d.See also
References
 The higher arithmetic: an introduction to the theory of numbers
 Arithmetic: A Straightforward Approach
remainder in Catalan: Residu
remainder in Spanish: Resto
remainder in French: Reste
remainder in Italian: Resto
remainder in Dutch: Rest
remainder in Polish: Reszta
remainder in Portuguese: Resto da divisão
inteira
remainder in Sicilian: Rimasugghiu
remainder in Simple English: Remainder
remainder in Finnish: Jakojäännös
remainder in Chinese: 余数
Synonyms, Antonyms and Related Words
adjunct, afterlife, balance, bonus, component, contingent, copyhold, credit, cross section, deficit, detachment, detail, difference, discrepancy, dividend, division, dole, epact, equitable estate, estate at
sufferance, estate for life, estate for years, estate in
expectancy, estate in fee, estate in possession, estate tail,
excess, extra, fee, fee simple, fee tail, feod, feodum, feud, feudal estate, fief, following, fraction, future time, gratuity, hangover, heel, installment, item, lagniappe, lateness, lease, leasehold, leavings, leftover, leftovers, legal estate,
margin, net, next life, overage, overmeasure, overplus, overrun, overset, overstock, oversupply, paramount estate,
parcel, part, particular, particular
estate, percentage,
plus, portion, postdate, postdating, posteriority, pourboire, provenience, quadrant, quarter, quota, random sample, remains, remnant, residual, residue, residuum, rest, reversion, sample, sampling, section, sector, segment, sequence, share, something extra, spare, subdivision, subgroup, subsequence, subspecies, succession, supervenience, supervention, surplus, surplusage, tip, vested estate