Remainder


Remainder

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 proven 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&times;10+3.
* When dividing 26 by 4, 6 is the quotient and 2 is the remainder, because 26=6&times;4+2.
* When dividing 56 by 7, 8 is the quotient and 0 is the remainder, because 56=7&times;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&times;(−5) + 3

as is usual for mathematicians, or

:−42 = 8&times;(−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 "r"1, and the negative one is "r"2, then

:r1 = "r"2 + "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 &mdash; 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 &mdash; 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".

See also

*Chinese remainder theorem
*Division algorithm
*Euclidean algorithm
*Modulo
*Modular arithmetic
*Modulo operation

References

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  • remainder — re·main·der n [Anglo French, from Old French remaindre to remain] 1: an estate in property in favor of one other than the grantor that follows upon the natural termination of a prior intervening possessory estate (as a life estate) created at the …   Law dictionary

  • remainder — remainder, residue, residuum, remains, leavings, rest, balance, remnant can all mean what is left after the subtraction or removal of a part. Remainder is the technical term for the result in the arithmetical process of subtraction {subtract 8… …   New Dictionary of Synonyms

  • Remainder — Re*main der (r? m?n d?r), n. [OF. remaindre, inf. See {Remain}.] 1. Anything that remains, or is left, after the separation and removal of a part; residue; remnant. The last remainders of unhappy Troy. Dryden. [1913 Webster] If these decoctions… …   The Collaborative International Dictionary of English

  • remainder — [ri mān′dər] n. [ME remaindre < Anglo Fr substantive use of OFr inf.: see REMAIN] 1. those remaining 2. what is left when a part is taken away; the rest 3. a copy or number of copies of a book still held by a publisher when the sale has fallen …   English World dictionary

  • Remainder — Re*main der, a. Remaining; left; left over; refuse. [1913 Webster] Which is as dry as the remainder biscuit After a voyage. Shak. [1913 Webster] …   The Collaborative International Dictionary of English

  • remainder — early 15c., from Anglo Fr. remainder (O.Fr. remaindre), variant of O.Fr. remanoir (see REMAIN (Cf. remain)) …   Etymology dictionary

  • remainder — /ri meində/, it. /re mɛinder/ s. ingl. (propr. resto ), usato in ital. al masch. (comm.) [libro rimasto invenduto, messo in vendita a prezzo ridotto] ▶◀ ⇑ giacenza, rimanenza …   Enciclopedia Italiana

  • remainder — /reˈmɛnder, ingl. rɪˈmeɪndə(r)/ [vc. ingl., propriamente «resto», dall ant. fr. remaindre «rimanere»] s. m. inv. (di libro) giacenza di magazzino, fondo di magazzino …   Sinonimi e Contrari. Terza edizione

  • remainder — [n] balance, residue bottom of barrel*, butt, carry over, detritus, dregs, excess, fragment, garbage, hangover*, heel, junk, leavings, leftover, obverse, oddment, odds and ends*, overplus, refuse, relic, remains, remnant, residuum, rest, ruins,… …   New thesaurus

  • remainder — ► NOUN 1) a part, number, or quantity that is left over. 2) a part that is still to come. 3) the number which is left over in a division in which one quantity does not exactly divide another. 4) a copy of a book left unsold when demand has fallen …   English terms dictionary


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