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In linear algebra, the dual numbers extend the real numbers by adjoining one new element ε with the property ε2 = 0 (ε is nilpotent). The collection of dual numbers forms a particular two-dimensional commutative unital associative algebra over the real numbers. Every dual number has the form ''z'' = ''a'' + ''b''ε with ''a'' and ''b'' uniquely determined real numbers. Dual numbers can also be thought of as the exterior algebra of a one dimensional vector space. The algebra of dual numbers is a ring that is a local ring since the principal ideal generated by ε is its only maximal ideal. Dual numbers form the coefficients of dual quaternions. ==Linear representation== Using matrices, dual numbers can be represented as :. The sum and product of dual numbers are then calculated with ordinary matrix addition and matrix multiplication; both operations are commutative and associative within the algebra of dual numbers. This correspondence is analogous to the usual matrix representation of complex numbers. However, it is ''not'' the only representation with 2 × 2 real matrices, as is shown in the profile of 2 × 2 real matrices. Like the complex plane and split-complex number plane, the dual numbers are one of the realizations of planar algebra. 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「dual number」の詳細全文を読む スポンサード リンク
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