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In mathematics, the remainder is the amount "left over" after performing some computation. In arithmetic, the remainder is the integer "left over" after dividing one integer by another to produce an integer quotient (integer division). In algebra, the remainder is the polynomial "left over" after dividing one polynomial by another. The ''modulo operation'' is the operation that produces such a remainder when given a dividend and divisor. Formally it is also true that a remainder is what is left after subtracting one number from another, although this is more properly called the ''difference''. This usage can be found in some elementary textbooks; colloquially it is replaced by the expression "the rest" as in "Give me two dollars back and keep the rest." 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. ==Integer division== If ''a'' and ''d'' are integers, 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. The remainder, as defined above, is called the ''least positive remainder'' or simply the ''remainder''.〔. But if the remainder is 0, it is not positive, even though it is called a "positive remainder". 〕 The integer ''a'' is either a multiple of ''d'' or lies in the interval between consecutive multiples of ''d'', namely, ''q⋅d'' and (''q'' + 1)''d'' (for positive ''q''). At times it is convenient to carry out the division so that ''a'' is as close as possible to an integral multiple of ''d'', that is, we can write :''a'' = ''k⋅d'' + ''s'', with |''s''| ≤ |''d''/2| for some integer ''k''. In this case, ''s'' is called the ''least absolute remainder''. As with the quotient and remainder, ''k'' and ''s'' are uniquely determined except in the case where ''d'' = 2''n'' and ''s'' = ± ''n''. For this exception we have, : ''a'' = ''k⋅d'' + ''n'' = (''k'' + 1)''d'' - ''n''. A unique remainder can be obtained in this case by some convention such as always taking the positive value of ''s''. ==Examples== In the division of 43 by 5 we have: : 43 = 8 × 5 + 3, so 3 is the least positive remainder. We also have, : 43 = 9 × 5 - 2, and −2 is the least absolute remainder. These definitions are also valid if ''d'' is negative, for example, in the division of 43 by −5, :43 = (−8)×(−5) + 3, and 3 is the least positive remainder, while, :43 = (−9)×(−5) + (−2) and −2 is the least absolute remainder. In the division of 42 by 5 we have: :42 = 8 × 5 + 2, and since 2 < 5/2, 2 is both the least positive remainder and the least absolute remainder. In these examples, the (negative) least absolute remainder is obtained from the least positive remainder by subtracting 5, which is ''d''. This holds in general. When dividing by ''d'', either both remainders are positive and therefore equal, or they have opposite signs. If the positive remainder is ''r''1, and the negative one is ''r''2, then :r1 = ''r''2 + ''d''. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「remainder」の詳細全文を読む スポンサード リンク
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