翻訳と辞書 Words near each other ・ Cayley process ・ Cayley surface ・ Cayley table ・ Cayley transform ・ Cayley's formula ・ Cayley's mousetrap ・ Cayley's nodal cubic surface ・ Cayley's ruled cubic surface ・ Cayley's sextic ・ Cayley's theorem ・ Cayley's Ω process ・ Cayley, Alberta ・ Cayley/A. J. Flying Ranch Airport ・ Cayleyan ・ Cayley–Bacharach theorem ・ Cayley–Dickson construction ・ Cayley–Galt Tariff ・ Cayley–Hamilton theorem ・ Cayley–Klein metric ・ Cayley–Purser algorithm ・ Cayli ・ Caylib Oosthuizen ・ Caylin Hauptmann ・ Caylloma ・ Caylloma District ・ Caylloma Province ・ Caylor ・ Caylor, Virginia ・ Caylus ・ Caylus (disambiguation) Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Cayley–Dickson construction ： ウィキペディア英語版 Cayley–Dickson construction In mathematics, the Cayley–Dickson construction, named after Arthur Cayley and Leonard Eugene Dickson, produces a sequence of algebras over the field of real numbers, each with twice the dimension of the previous one. The algebras produced by this process are known as Cayley–Dickson algebras. They are useful composition algebras frequently applied in mathematical physics. The Cayley–Dickson construction defines a new algebra based on the direct sum of an algebra with itself, with multiplication defined in a specific way and an involution known as conjugation. The product of an element and its conjugate (or sometimes the square root of this) is called the norm. The symmetries of the real field disappear as the Cayley–Dickson construction is repeatedly applied: first losing order, then commutativity of multiplication, and next associativity of multiplication. More generally, the Cayley–Dickson construction takes any algebra with involution to another algebra with involution of twice the dimension.〔Schafer (1995) p.45〕 == Complex numbers as ordered pairs == (詳細はcomplex numbers can be written as ordered pairs (''a'', ''b'') of real numbers ''a'' and ''b'', with the addition operator being component-by-component and with multiplication defined by : $(a,\; b)\; (c,\; d)\; =\; (a\; c\; -\; b\; d,\; a\; d\; +\; b\; c).\backslash ,$ A complex number whose second component is zero is associated with a real number: the complex number (''a'', 0) is the real number ''a''. Another important operation on complex numbers is conjugation. The conjugate (''a'', ''b'') * of (''a'', ''b'') is given by : $(a,\; b)^$ * = (a, -b).\, The conjugate has the property that : $(a,\; b)^$ * (a, b) = (a a + b b, a b - b a) = (a^2 + b^2, 0),\, which is a non-negative real number. In this way, conjugation defines a ''norm'', making the complex numbers a normed vector space over the real numbers: the norm of a complex number ''z'' is : $|z|\; =\; (z^$ * z)^.\, Furthermore, for any nonzero complex number ''z'', conjugation gives a multiplicative inverse, : $z^\; =\; .\backslash ,$ In as much as complex numbers consist of two independent real numbers, they form a 2-dimensional vector space over the real numbers. Besides being of higher dimension, the complex numbers can be said to lack one algebraic property of the real numbers: a real number is its own conjugate. 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「Cayley–Dickson construction」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.