AskDefine | Define remainder

Dictionary Definition



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




remain + d + -er


  1. 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.
  2. 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.
  3. 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.


what remains after some has been removed
items left unsold and subject to reduction in price
mathematics: amount left over after repeatedly subtracting the divisor
  • Czech: zbytek
  • Finnish: jakojäännös
  • Japanese: 剰余
  • Russian: остаток (ostátok)
Translations to be checked

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 non-zero, 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.


  • 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 non-zero, 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 non-zero, 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 well-defined. 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.


  • 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
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