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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 Difference of two squares ： ウィキペディア英語版 Difference of two squares In mathematics, the difference of two squares is a squared (multiplied by itself) number subtracted from another squared number. Every difference of squares may be factored according to the identity :$a^2-b^2\; =\; (a+b)(a-b)\backslash ,\backslash !$ in elementary algebra. ==Proof== The proof of the factorization identity is straightforward. Starting from the left-hand side, apply the distributive law to get :$(a+b)(a-b)\; =\; a^2+ba-ab-b^2\backslash ,\backslash !$, and set :$ba\; -\; ab\; =\; 0\backslash ,\backslash !$ as an application of the commutative law. The resulting identity is one of the most commonly used in mathematics. Among many uses, it gives a simple proof of the AM–GM inequality in two variables. The proof just given indicates the scope of the identity in abstract algebra: it will hold in any commutative ring ''R''. Conversely, if this identity holds in a ring ''R'' for all pairs of elements ''a'' and ''b'' of the ring, then ''R'' is commutative. To see this, apply the distributive law to the right-hand side of the original equation and get :$a^2\; +\; ba\; -\; ab\; -\; b^2\backslash ,\backslash !$ and for this to be equal to $a^2\; -\; b^2$, we must have :$ba\; -\; ab\; =\; 0\backslash ,\backslash !$ for all pairs ''a'', ''b'' of elements of ''R'', so the ring ''R'' is commutative. ==Geometrical demonstrations== The difference of two squares can also be illustrated geometrically as the difference of two square areas in a plane. In the diagram, the shaded part represents the difference between the areas of the two squares, i.e. $a^2\; -\; b^2$. The area of the shaded part can be found by adding the areas of the two rectangles; $a(a-b)\; +\; b(a-b)$, which can be factorized to $(a+b)(a-b)$. Therefore $a^2\; -\; b^2\; =\; (a+b)(a-b)$ Another geometric proof proceeds as follows: We start with the figure shown in the first diagram below, a large square with a smaller square removed from it. The side of the entire square is a, and the side of the small removed square is b. The area of the shaded region is $a^2-b^2$. A cut is made, splitting the region into two rectangular pieces, as shown in the second diagram. The larger piece, at the top, has width a and height a-b. The smaller piece, at the bottom, has width a-b and height b. Now the smaller piece can be detached, rotated, and placed to the right of the larger piece. In this new arrangement, shown in the last diagram below, the two pieces together form a rectangle, whose width is $a+b$ and whose height is $a-b$. This rectangle's area is $(a+b)(a-b)$. Since this rectangle came from rearranging the original figure, it must have the same area as the original figure. Therefore, $a^2-b^2\; =\; (a+b)(a-b)$. 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「Difference of two squares」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.