- 関
- rest

- sell cheaply as remainders; "The publisher remaindered the books"
- 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" (同)balance, residual, residue, residuum, rest
- the number that remains after subtraction; the number that when added to the subtrahend gives the minuend (同)difference
- the part of the dividend that is left over when the dividend is not evenly divisible by the divisor
- put something in a resting position, as for support or steadying; "Rest your head on my shoulder"
- euphemisms for death (based on an analogy between lying in a bed and in a tomb); "she was laid to rest beside her husband"; "they had to put their family pet to sleep" (同)eternal rest, sleep, eternal sleep, quietus
- a state of inaction; "a body will continue in a state of rest until acted upon"
- be at rest
- freedom from activity (work or strain or responsibility); "took his repose by the swimming pool" (同)ease, repose, relaxation
- a support on which things can be put; "the gun was steadied on a special rest"
- a musical notation indicating a silence of a specified duration
- take a short break from one
*s activities in order to relax (同)breathe, catch one''s breath, take a breather* - be inherent or innate in; (同)reside, repose
- be inactive, refrain from acting; "The committee is resting over the summer"
- give a rest to; "He rested his bad leg"; "Rest the dogs for a moment"
- not move; be in a resting position

- 〈U〉《集合的に;複数扱い》残り,残りの人(物) / 〈C〉(引き算の)残り;(割り算の)余り / 〈特に残本〉‘を'安売りする
- 〈U〉〈C〉『休息』,休憩,休養;休息の時間(期間);(…から)解放されて休むこと《+『from』+『名』》 / 〈U〉〈C〉睡眠(sleep) / 動かないこと,停止,静止 / 〈C〉《複合語に用いて》「(物を載せる)台,支え」の意を表す / 〈U〉《詩》死,永眠 / 〈C〉(音楽で)休止;休止符 / 『休む』,『休息する』;(仕事などを)やめて休む《+『from』+『名』》 / 『安心する』,落ち着く,くつろぐ / (…に)『載っている』,『支えられている』《+『on』(『upon』)+『名』》;(…に)もたれている(lean)《+『against』+『名』》 / 〈光・視線などが〉(…に)注がれる,じっと留まる《+『on』(『upon』)+『名』》 / (変化しないで)そのままでいる,休止(静止)する / 永眠する / (訴訟事件で)証拠提出を自発的に中止する / 《しばしば受動態で》…‘を'『休息(休養)させる』,休ませる / (…に)…‘を'置く,載せる,基づかせる《+『名』+『on』(『upon』)+『名』(do『ing』)》;(…に)…‘を'よりかからせる》+『名』+『against』+『名』》 / 〈訴訟事件〉‘の'証拠提出を自発的に中止する
- (…の)『残り』,残与《+『of』+『名』》 / 〈U〉《the~》《複数扱い》『そのほかの物(人)』 / 《rest+形(名)〈補〉》(ある状態の)ままである(remain)

出典(authority):フリー百科事典『ウィキペディア（Wikipedia）』「2013/10/09 10:14:35」(JST)

For other uses, see Remainder (disambiguation).

This article includes a list of references, related reading or external links, but its sources remain unclear because it lacks inline citations. Please improve this article by introducing more precise citations. (April 2010) |

In arithmetic, the **remainder** (or **modulus**) is the amount "left over" after performing the division of two integers which do not divide evenly, that is, where the result of the division cannot be expressed as an integer.

- 1 For natural numbers
- 2 General case for integers
- 3 For floating-point numbers
- 4 Satisfaction of an inequality
- 5 In programming languages
- 6 See also
- 7 References
- 8 Further reading

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*. See Euclidean division for a proof of this result and division algorithm for algorithms describing how to calculate the remainder.

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 |*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,^{[citation needed]} 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 *r*_{1}, and the negative one is *r*_{2}, then

- r
_{1}=*r*_{2}+*d*.

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

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.

Main article: Modulo operation

With two choices for the inequality, there are two choices for the remainder, one negative and the other positive. This means that there are also two possible choices for the quotient. In number theory the positive remainder is chosen by convention. But programming languages need not, and different languages have adopted different conventions: Pascal goes with number theory in choosing the result of the *mod* operation positive (while not always *a* = (*a div d* )**d* + *a mod d* ).^{[1]} C99 chooses the remainder with the same sign as the dividend *a*. (Before C99, the C language allowed either choice.) Perl, Python (only modern versions), and Common Lisp choose the remainder with the same sign as the divisor *d*. 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.

- Chinese remainder theorem
- Divisibility rule
- Egyptian multiplication and division
- Euclidean algorithm
- Long division
- Modular arithmetic
- Polynomial long division
- Polynomial remainder theorem
- Taylor's theorem

**^**Pascal ISO 7185:1990 6.7.2.2

- Davenport, Harold (1 x999).
*The higher arithmetic: an introduction to the theory of numbers*. Cambridge, UK: Cambridge University Press. p. 25. ISBN 0-521-63446-6. - Katz, Victor, ed. (2007).
*The mathematics of Egypt, Mesopotamia, China, India, and Islam : a sourcebook*. Princeton: Princeton University Press. ISBN 9780691114859. - Schwartzman, Steven (1994). "remainder (noun)".
*The words of mathematics : an etymological dictionary of mathematical terms used in english*. Washington: Mathematical Association of America. ISBN 9780883855119. - Smith, David Eugene (1958).
*History of Mathematics, Volume 2*(1st ed.). New York: Courier Dover. ISBN 9780486204307. - Zuckerman, Martin M.
*Arithmetic: A Straightforward Approach*. Lanham, Md: Rowman & Littlefield Publishers, Inc. ISBN 0-912675-07-1.

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- In arithmetic, the remainder (or residue) is the amount "left over" after the division of two integers which cannot be expressed with an integer quotient. The general form of a linear equation can be expressed as. a = q × d + r. In this equation, q ...

- remainder. 出典: フリー多機能辞典『ウィクショナリー日本語版（Wiktionary）』. 移動: 案内、 検索 ... [編集] 動詞. remainder (三単現: remainders, 現在分詞: remaindering, 過去形: remaindered, 過去分詞: remaindered ). 安く売る。

- [名]1 残り, 残り物, 残留品；((通例〜s))遺物, 遺跡the remainder of the property財産 の残り.2 ((the 〜))((単数・複数扱い))残部, 残りの人［物］drink up the remainder of the water残りの水を飲み干すThe r... - goo辞書は国語、英和、和英、中国語、百科 事典等 ...

リンク元 | 「rest」「残り」「残部」 |

- n.

- 急速、休養、西洋、睡眠。休憩。永眠
- 安息所、宿泊所
- 安静、安楽、安堵、安心、平温
- 休止、停止、静止
- 台

- vi.

- 休む
- 安心している、落ち着いている
- 休止する、静止する
- ある、位置を占める、かかる。(目・視線が)とまる、向けられる、注がれる
- たよる。希望を賭ける、信ずる
- (～に)基づく、依存している。

- vt.