dualnumber (pluraldual numbers)

1. (grammar) A grammaticalnumber denoting a quantity of exactly two.

   Coordinate terms:singular number,trial number,plural number

   • 1840,George Grey,A Vocabulary of the Dialects of South Western Australia, 2nd edition, T. & W. Boone,page xi:

     This is particularly the case with regard to the pronouns of thedual number, which form one of the peculiarities of the language.

   • 1998, William Blake Tyrrell, Larry J. Bennett,Recapturing Sophocles' Antigone,Rowman & Littlefield,page43:

     Eteocles and Polyneices are enemy brothers, that is, they are alike in spite of the difference erected by being on opposite sides of the Theban battlements.¹ By uniting them with thedual number in the description of their deaths, Sophocles has the elders speak of them as indistinguishable in their hatred, spear work, and deaths.

   • 2006, Andrea Albrette,Teach Yourself Slovene, McGraw-Hill,page19:

     In most languages, singular and plural are used to indicate number. Slovene, however, has an additional‘dual’ number, which is used when referring to two of anything. Thisdual number existed in many languages at one stage in their development, but has been lost; in Slovene, though,[…]

   • 2009, Giuliano Lancioni, “Formulaic models and formulaicity in Classical and Modern Standard Arabic”, in Roberta Corrigan, Edith A. Moravcsik, Hamid Ouali, Kathleen M. Wheatley, editors,Formulaic Language, Volume 1: Distribution and historical change,John Benjamins Publishing Company,page225:

     Classical Arabic has a fully functionaldual number, which requires dual agreement in verbs, adjectives and pronouns.

2. (algebra) An element of analgebra (thealgebra of dual numbers) which includes thereal numbers and an element ε which satisfies ε ≠ 0 and ε² = 0.

   Hypernym:hypercomplex number

   Iff(x) is a polynomial or a power series then its derivative can be obtained “automatically” usingdual numbers as follows:${\displaystyle f^{\prime }(x)=\frac{f(x+ϵ)-f(x)}{ϵ}}$ where${\displaystyle f(x+ϵ)}$ should be expanded using the algebra ofdual numbers. The role of theε is therein similar to that of an infinitesimal.

   • 1993, Ivan Kolář, Peter W. Michor, Jan Slovák,Natural Operations in Differential Geometry, Springer,page336:

     In the special case of the algebra${\displaystyle \mathbf{D}}$ ofdual numbers we get the vertical tangent bundle${\displaystyle V}$.

   • 1996, Andrei N. Kolmogorov, Adolf-Andrei P. Yushkevich, translated by Roger Cooke,Mathematics of the 19th Century: Geometry, Analytic Function Theory, Birkhäuser,page87:

     These numbers are now calleddual numbers (a term due toStudy) and can be defined as expressions of the form${\displaystyle a+bϵ}$,${\displaystyle {ϵ}^2=0}$. When this is done asKotel'nikov and Study showed, the dual distance between two points of this sphere representing two lines, forming an angle${\displaystyle \phi }$ and having shortest distance${\displaystyle d}$, is equal to thedual number${\displaystyle \phi +ϵd}$, while the motions of Euclidean space are represented by rotations of the dual sphere, and consequently depend on three dual parameters.

   • 2002, Jiří Tomáš, “Some Classification Problems on Natural Bundles Related to Weil Bundles”, in Olga Gil-Medrano, Vicente Miquel, editors,Differential Geometry, Valencia 2001: Proceedings of the International Conference, World Scientific,page297:

     Thus we investigate natural${\displaystyle T}$-functions defined on${\displaystyle T^{\ast }{T}^{\mathbf{D}\otimes A}M}$ to determine all natural operators${\displaystyle T\to T{T}^{\ast }{T}^A}$ for${\displaystyle \mathbf{D}}$ denoting the algebra ofdual numbers.

• nilpotent

• Dual (grammatical number) on Wikipedia.Wikipedia
• Dual number on Wikipedia.Wikipedia

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