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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 Mediant (mathematics) ： ウィキペディア英語版 Mediant (mathematics) :''For mediant in music, see mediant. "Mediant" should not be confused with median.'' In mathematics, the mediant of two fractions :$\backslash frac\; \backslash text\; \backslash frac$ is :$\backslash frac\; .$ that is to say, the numerator and denominator of the mediant are the sums of the numerators and denominators of the given fractions, respectively. It is sometimes called the freshman sum, as it is a common mistake in the usual addition of fractions. In general, this is an operation on fractions rather than on rational numbers. That is to say, for two rational numbers ''q''1, ''q''2, the value of the mediant depends on how the rational numbers are expressed using integer pairs. For example, the mediant of 1/1 and 1/2 is 2/3, but the mediant of 2/2 and 1/2 is 3/4. A way around this, where required, is to specify that both rationals are to be represented as fractions in their lowest terms (with ''c'' > 0, ''d'' > 0). With such a restriction, mediant becomes a well-defined binary operation on rationals. The Stern-Brocot tree provides an enumeration of all positive rational numbers, in lowest terms, obtained purely by iterative computation of the mediant according to a simple algorithm. ==Properties== * The mediant inequality: An important property (also explaining its name) of the mediant is that it lies strictly between the two fractions of which it is the mediant: If and $a,b,c,d\backslash geq\; 0$, then :: :This property follows from the two relations ::$\backslash frac-\backslash frac\; a\; c=\; =-\backslash frac\; a\; c\; \backslash right)$ :and ::$\backslash frac\; b\; d-\backslash frac=\; =-\backslash frac\; a\; c\; \backslash right).$ * Assume that the pair of fractions ''a''/''c'' and ''b''/''d'' satisfies the determinant relation $bc-ad=1$. Then the mediant has the property that it is the ''simplest'' fraction in the interval (''a''/''c'', ''b''/''d''), in the sense of being the fraction with the smallest denominator. More precisely, if the fraction $a\text{'}/c\text{'}$ with positive denominator c' lies (strictly) between ''a''/''c'' and ''b''/''d'', then its numerator resp. denominator can be written as $\backslash ,a\text{'}=\backslash lambda\_1\; a+\backslash lambda\_2\; b$ and $\backslash ,c\text{'}=\backslash lambda\_1\; c+\backslash lambda\_2\; d$ with two ''positive'' real (in fact rational) numbers $\backslash lambda\_1,\backslash ,\backslash lambda\_2$. To see why the $\backslash lambda\_i$ must be positive note that ::$\backslash frac-\backslash frac\; a\; c=\backslash lambda\_2$ :and ::$\backslash frac\; b\; d-\backslash frac=\backslash lambda\_1$ :must be positive. The determinant relation ::$bc-ad=1\; \backslash ,$ :then implies that both $\backslash lambda\_1,\backslash ,\backslash lambda\_2$ must be integers, solving the system of linear equations ::$\backslash ,\; a\text{'}=\backslash lambda\_1\; a+\backslash lambda\_2\; b$ ::$\backslash ,\; c\text{'}=\backslash lambda\_1\; c+\backslash lambda\_2\; d$ :for $\backslash lambda\_1,\backslash lambda\_2$. Therefore $c\text{'}\backslash ge\; c+d.$ * The converse is also true: assume that the pair of reduced fractions ''a''/''c'' < ''b''/''d'' has the property that the ''reduced'' fraction with smallest denominator lying in the interval (''a''/''c'', ''b''/''d'') is equal to the mediant of the two fractions. Then the determinant relation ''bc'' − ''ad'' = 1 holds. This fact may be deduced e.g. with the help of Pick's theorem which expresses the area of a plane triangle whose vertices have integer coordinates in terms of the number vinterior of lattice points (strictly) inside the triangle and the number vboundary of lattice points on the boundary of the triangle. Consider the triangle $\backslash Delta(v\_1,v\_2,v\_3)$ with the three vertices ''v''1 = (0, 0), ''v''2 = (''a'', ''c''), ''v''3 = (''b'', ''d''). Its area is equal to ::$\backslash text(\backslash Delta)=\backslash over\; 2\}\; \backslash \; .$ :A point $p=(p\_1,p\_2)$ inside the triangle can be parametrized as ::$p\_1=\backslash lambda\_1\; a+\backslash lambda\_2\; b,\backslash ;\; p\_2=\backslash lambda\_1\; c+\backslash lambda\_2\; d,$ :where ::$\backslash lambda\_1\backslash ge\; 0,\backslash ,\backslash lambda\_2\; \backslash ge\; 0,\; \backslash ,\backslash lambda\_1+\backslash lambda\_2\; \backslash le\; 1.\; \backslash ,$ :The Pick formula ::$\backslash text(\backslash Delta)=v\_\backslash mathrm+-1$ :now implies that there must be a lattice point ''q'' = (''q''1, ''q''2) lying inside the triangle different from the three vertices if ''bc'' − ''ad'' >1 (then the area of the triangle is $\backslash ge\; 1$). The corresponding fraction ''q''1/''q''2 lies (strictly) between the given (by assumption reduced) fractions and has denominator :: :as ::$\backslash lambda\_1+\backslash lambda\_2\; \backslash le\; 1.\; \backslash ,$ *Relatedly, if ''p''/''q'' and ''r''/''s'' are reduced fractions on the unit interval such that |''ps'' − ''rq''| = 1 (so that they are adjacent elements of a row of the Farey sequence) then :$?\backslash left(\backslash frac\backslash right)\; =\; \backslash frac12\; \backslash left(?\backslash bigg(\backslash frac\; pq\backslash bigg)\; +\; \{\}?\backslash bigg(\backslash frac\; rs\backslash bigg)\backslash right)$ :where ? is Minkowski's question mark function. :In fact, mediants commonly occur in the study of continued fractions and in particular, Farey fractions. The ''n''th Farey sequence ''F''''n'' is defined as the (ordered with respect to magnitude) sequence of reduced fractions ''a''/''b'' (with coprime ''a'', ''b'') such that ''b'' ≤ ''n''. If two fractions ''a''/''c'' < ''b''/''d'' are adjacent (neighbouring) fractions in a segment of Fn then the determinant relation $bc-ad=1$ mentioned above is generally valid and therefore the mediant is the ''simplest'' fraction in the interval (''a''/''c'', ''b''/''d''), in the sense of being the fraction with the smallest denominator. Thus the mediant will then (first) appear in the (''c'' + ''d'')th Farey sequence and is the "next" fraction which is inserted in any Farey sequence between ''a''/''c'' and ''b''/''d''. This gives the rule how the Farey sequences ''F''''n'' are successively built up with increasing ''n''. 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「Mediant (mathematics)」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.