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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 Pythagorean addition ： ウィキペディア英語版 Pythagorean addition In mathematics, Pythagorean addition is the following binary operation on the real numbers: :$a\; \backslash oplus\; b\; =\; \backslash sqrt.$ The name recalls the Pythagorean theorem, which states that the length of the hypotenuse of a right triangle is where ''a'' and ''b'' are the lengths of the other sides. This operation provides a simple notation and terminology when the summands are complicated; for example, the energy-momentum relation in physics becomes :$E\; =\; mc^2\; \backslash oplus\; pc.$ ==Properties== The operation ⊕ is associative and commutative, and :$\backslash sqrt\; =\; x\_1\; \backslash oplus\; x\_2\; \backslash oplus\; \backslash cdots\; \backslash oplus\; x\_n$. This is enough to form the real numbers into a commutative semigroup. However, ⊕ is not a group operation for the following reasons. The only element which could potentially act as an identity element is 0, since an identity ''e'' must satisfy ''e''⊕''e'' = ''e''. This yields the equation $\backslash sqrte=e$, but if ''e'' is nonzero that implies $\backslash sqrt=1$, so ''e'' could only be zero. Unfortunately 0 does not work as an identity element after all, since 0⊕(−1) = 1. This does indicate, however, that if the operation ⊕ is restricted to nonnegative real numbers, then 0 ''does'' act as an identity. Consequently the operation ⊕ acting on the nonnegative real numbers forms a commutative monoid. 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「Pythagorean addition」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.