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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Irreducible fraction ： ウィキペディア英語版 Irreducible fraction An irreducible fraction (or fraction in lowest terms or reduced fraction) is a fraction in which the numerator and denominator are integers that have no other common divisors than 1 (and -1, when negative numbers are considered). In other words, a fraction a⁄b is irreducible if and only if ''a'' and ''b'' are coprime, that is, if ''a'' and ''b'' have a greatest common divisor of 1. In higher mathematics, "irreducible fraction" may also refer to rational fractions such that the numerator and the denominator are coprime polynomials.〔E.g., see 〕 Every positive rational number can be represented as an irreducible fraction in exactly one way.〔.〕 An equivalent definition is sometimes useful: if ''a'', ''b'' are integers, then the fraction ''a''⁄''b'' is irreducible if and only if there is no other equal fraction ''c''⁄''d'' such that |''c''| < |''a''| or |''d''| < |''b''|, where |''a''| means the absolute value of ''a''. (Two fractions ''a''⁄''b'' and ''c''⁄''d'' are ''equal'' or ''equivalent'' if and only if ''ad'' = ''bc''.) For example, 1⁄4, 5⁄6, and −101⁄100 are all irreducible fractions. On the other hand, 2⁄4 is reducible since it is equal in value to 1⁄2, and the numerator of 1⁄2 is less than the numerator of 2⁄4. A fraction that is reducible can be reduced by dividing both the numerator and denominator by a common factor. It can be fully reduced to lowest terms if both are divided by their greatest common divisor.〔.〕 In order to find the greatest common divisor, the Euclidean algorithm or prime factorization may be used. The Euclidean algorithm is commonly preferred because it allows one to reduce fractions with numerators and denominators too large to be easily factored.〔.〕 ==Examples== :$\backslash frac=\backslash frac=\backslash frac\; \backslash ,.$ In the first step both numbers were divided by 10, which is a factor common to both 120 and 90. In the second step, they were divided by 3. The final result, 4/3, is an irreducible fraction because 4 and 3 have no common factors other than 1. The original fraction could have also been reduced in a single step by using the greatest common divisor of 90 and 120, which is 30 (i.e., gcd(90,120)=30). :$\backslash frac=\backslash frac\; \backslash ,.$ Which method is faster "by hand" depends on the fraction and the ease with which common factors are spotted. In case a denominator and numerator remain that are too large to ensure they are coprime by inspection, a greatest common divisor computation is needed anyway to ensure the fraction is actually irreducible. 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「Irreducible fraction」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.