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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 Split-complex number ： ウィキペディア英語版 Split-complex number In abstract algebra, the split-complex numbers (or hyperbolic numbers, also perplex numbers, and double numbers) are a two-dimensional commutative algebra over the real numbers different from the complex numbers. Every split-complex number has the form : ''x'' + ''y'' ''j'', where ''x'' and ''y'' are real numbers. The number ''j'' is similar to the imaginary unit i, except that : ''j'' 2 = +1. As an algebra over the reals, the split-complex numbers are the same as the direct sum of algebras (under the isomorphism sending to ). The name ''split'' comes from this characterization: as a real algebra, the split-complex numbers ''split'' into the direct sum . It arises, for example, as the real subalgebra generated by an involutory matrix. Geometrically, split-complex numbers are related to the modulus in the same way that complex numbers are related to the square of the Euclidean norm . Unlike the complex numbers, the split-complex numbers contain nontrivial idempotents (other than 0 and 1), as well as zero divisors, and therefore they do not form a field. In interval analysis, a split complex number represents an interval with midpoint ''x'' and radius ''y''. Another application involves using split-complex numbers, dual numbers, and ordinary complex numbers, to interpret a real matrix as a complex number. Split-complex numbers have many other names; see the synonyms section below. See the article Motor variable for functions of a split-complex number. ==Definition== A split-complex number is an ordered pair of real numbers, written in the form :$z\; =\; x\; +\; jy$ where ''x'' and ''y'' are real numbers and the quantity ''j'' satisfies :$j^2\; =\; +1$ Choosing $j^2\; =\; -1$ results in the complex numbers. It is this sign change which distinguishes the split-complex numbers from the ordinary complex ones. The quantity ''j'' here is not a real number but an independent quantity; that is, it is not equal to ±1. The collection of all such ''z'' is called the split-complex plane. Addition and multiplication of split-complex numbers are defined by :(''x'' + ''j'' ''y'') + (''u'' + ''j'' ''v'') = (''x'' + ''u'') + ''j'' (''y'' + ''v'') :(''x'' + ''j'' ''y'')(''u'' + ''j'' ''v'') = (''xu'' + ''yv'') + ''j'' (''xv'' + ''yu''). This multiplication is commutative, associative and distributes over addition. 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「Split-complex number」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.