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In mathematics, a multiplicative inverse or reciprocal for a number ''x'', denoted by 1/''x'' or ''x''−1, is a number which when multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a fraction ''a''/''b'' is ''b''/''a''. For the multiplicative inverse of a real number, divide 1 by the number. For example, the reciprocal of 5 is one fifth (1/5 or 0.2), and the reciprocal of 0.25 is 1 divided by 0.25, or 4. The reciprocal function, the function ''f''(''x'') that maps ''x'' to 1/''x'', is one of the simplest examples of a function which is its own inverse (an involution). The term ''reciprocal'' was in common use at least as far back as the third edition of ''Encyclopædia Britannica'' (1797) to describe two numbers whose product is 1; geometrical quantities in inverse proportion are described as ''reciprocall'' in a 1570 translation of Euclid's ''Elements''.〔" In equall Parallelipipedons the bases are reciprokall to their altitudes". ''OED'' "Reciprocal" §3a. Sir Henry Billingsley translation of Elements XI, 34.〕 In the phrase ''multiplicative inverse'', the qualifier ''multiplicative'' is often omitted and then tacitly understood (in contrast to the additive inverse). Multiplicative inverses can be defined over many mathematical domains as well as numbers. In these cases it can happen that ; then "inverse" typically implies that an element is both a left and right inverse. The notation ''f'' −1 is sometimes also used for the inverse function of the function ''f'', which is not in general equal to the multiplicative inverse. For example, the multiplicative inverse is different from the inverse sin of ''x'', denoted or . Only for linear maps are they strongly related (see below). The terminology difference ''reciprocal'' versus ''inverse'' is not sufficient to make this distinction, since many authors prefer the opposite naming convention, probably for historical reasons (for example in French, the inverse function is preferably called application réciproque). ==Examples and counterexamples== In the real numbers, zero does not have a reciprocal because no real number multiplied by 0 produces 1 (the product of any number with zero is zero). With the exception of zero, reciprocals of every real number are real, reciprocals of every rational number are rational, and reciprocals of every complex number are complex. The property that every element other than zero has a multiplicative inverse is part of the definition of a field, of which these are all examples. On the other hand, no integer other than 1 and -1 has an integer reciprocal, and so the integers are not a field. In modular arithmetic, the modular multiplicative inverse of ''a'' is also defined: it is the number ''x'' such that ''ax'' ≡ 1 (mod ''n''). This multiplicative inverse exists if and only if ''a'' and ''n'' are coprime. For example, the inverse of 3 modulo 11 is 4 because 4 · 3 ≡ 1 (mod 11). The extended Euclidean algorithm may be used to compute it. The sedenions are an algebra in which every nonzero element has a multiplicative inverse, but which nonetheless has divisors of zero, i.e. nonzero elements ''x'', ''y'' such that ''xy'' = 0. A square matrix has an inverse if and only if its determinant has an inverse in the coefficient ring. The linear map that has the matrix ''A''−1 with respect to some base is then the reciprocal function of the map having ''A'' as matrix in the same base. Thus, the two distinct notions of the inverse of a function are strongly related in this case, while they must be carefully distinguished in the general case (as noted above). The trigonometric functions are related by the reciprocal identity: the cotangent is the reciprocal of the tangent; the secant is the reciprocal of the cosine; the cosecant is the reciprocal of the sine. A ring in which every nonzero element has a multiplicative inverse is a division ring; likewise an algebra in which this holds is a division algebra. 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「multiplicative inverse」の詳細全文を読む スポンサード リンク
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