A dual number is given by an expression of the form $a + \epsilon b$, where $a$ and $b$ are real numbers and $\epsilon^2 = 0$ (but $\epsilon \ne 0$). The set of dual numbers is a topological vector space and a commutative algebra over the real numbers.
We can generalise (at least the algebraic aspects) from $\mathbb{R}$ to any commutative ring $R$.
This can be thought of as:
We think of $\mathbb{R}$ as a subset of $\mathbb{D}$ by identifying $a$ with $a + 0 \epsilon$.
$\mathbb{D}$ is equipped with an involution that maps $\epsilon$ to $\bar{\epsilon} = -\epsilon$:
$\mathbb{D}$ also has an absolute value:
notice that the absolute value of a dual number is a non-negative real number, with
But this absolute value is degenerate, in that ${|z|} = 0$ need not imply that $z = 0$.
Some concepts in analysis can be extended from $\mathbb{R}$ to $\mathbb{D}$, but not as many as work for the complex numbers. Even algebraically, the dual numbers are not as nice as the real or complex numbers, as they do not form a field.
The original articles:
William Clifford: pp. 385 in: Preliminary Sketch of Biquaternions, Proceedings of the London Mathematical Society (1871) [doi:10.1112/plms/s1-4.1.381]
Josef Grünwald: Über duale Zahlen und ihre Anwendung in der Geometrie, Monatsh. f. Mathematik und Physik 17 (1906) 81–136 [doi:10.1007/BF01697639]
(relating to projective geometry)
In monographs on algebraic geometry or synthetic differential geometry:
Anders Kock, p xi & 4 in: Synthetic Differential Geometry, Cambridge University Press (1981, 2006) [pdf, doi:10.1017/CBO9780511550812]
David Mumford, pp. 218 of: Red book of varieties and schemes, Lecture Notes in Mathematics 1358, Springer (1988, 1999) [doi:10.1007/b62130]
(not using the term “dual numbers”)
Ieke Moerdijk, Gonzalo Reyes, p. 19 of: Models for Smooth Infinitesimal Analysis (1991) [doi:10.1007/978-1-4757-4143-8]
See also:
Last revised on August 15, 2024 at 12:35:57. See the history of this page for a list of all contributions to it.