# dual number
## English
### Noun

dual number (plural dual numbers)
* (grammar) A grammatical number 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 the dual number, which form one of the peculiarities of the language.
** 1998, William Blake Tyrrell, Larry J. Bennett, Recapturing Sophocles' Antigone, Rowman & Littlefield, page 43: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 the dual 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, page 19: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. This dual 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, page 225:Classical Arabic has a fully functional dual number, which requires dual agreement in verbs, adjectives and pronouns.
* (algebra) An element of an algebra (the algebra of dual numbers) which includes the real numbers and an element ε which satisfies ε ≠ 0 and ε² = 0.
    Hypernym: hypercomplex number
    If f(x) is a polynomial or a power series then its derivative can be obtained “automatically” using dual numbers as follows: $f'(x)={f(x+\epsilon )-f(x) \over \epsilon }$ where $f(x+\epsilon )$ should be expanded using the algebra of dual numbers. The role of the ε is therein similar to that of an infinitesimal.
** 1993, Ivan Kolář, Peter W. Michor, Jan Slovák, Natural Operations in Differential Geometry, Springer, page 336:In the special case of the algebra $\mathbf {D}$ of dual numbers we get the vertical tangent bundle $V$.
** 1996, Andrei N. Kolmogorov, Adolf-Andrei P. Yushkevich, translated by Roger Cooke, Mathematics of the 19th Century: Geometry, Analytic Function Theory, Birkhäuser, page 87:These numbers are now called dual numbers (a term due to Study) and can be defined as expressions of the form $a+b\epsilon$, $\epsilon ^{2}=0$. When this is done as Kotel'nikov and Study showed, the dual distance between two points of this sphere representing two lines, forming an angle $\varphi$ and having shortest distance $d$, is equal to the dual number $\varphi +\epsilon d$, 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, page 297:Thus we investigate natural $T$-functions defined on $T^{*}T^{\mathbf {D} \otimes A}M$ to determine all natural operators $T\to TT^{*}T^{A}$ for $\mathbf {D}$ denoting the algebra of dual numbers.

#### Translations

    The translations below need to be checked and inserted above into the appropriate translation tables. See instructions at Wiktionary:Entry layout § Translations.

### See also

* nilpotent

### Further reading

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