ncatlab.org

dual number in nLab

Contents

Idea

A dual number is given by an expression of the form a+ϵba + \epsilon b, where aa and bb are real numbers and ϵ 2=0\epsilon^2 = 0 (but ϵ≠0\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 ℝ\mathbb{R} to any commutative ring RR.

Interpretations

This can be thought of as:

the vector space ℝ 2\mathbb{R}^2 made into an algebra by the rule
(a,b)⋅(c,d)=(ac,ad+bc); (a, b) \cdot (c, d) = (a c, a d + b c) ;
the subalgebra of those 22-by-22 real matrices of the form
(a b 0 a); \left(\array { a & b \\ 0 & a } \right);
the polynomial ring ℝ[x]\mathbb{R}[\mathrm{x}] modulo x 2\mathrm{x}^2;
the parabolic 22-dimensional algebra of hypercomplex numbers;
the algebra of functions on the infinitesimal interval (the smallest of the infinitesimally thickened points) in synthetic differential geometry.
if ϵ\epsilon is regarded as being of degree 1 and ℝ⊕ϵℝ\mathbb{R} \oplus \epsilon \mathbb{R} is regarded accordingly as a superalgebra then this is the algebra of functions on the odd line ℝ 0|1\mathbb{R}^{0|1}.
the square-0-extension corresponding to the ℝ\mathbb{R}-module (see there) given by ℝ\mathbb{R} itself.

We think of ℝ\mathbb{R} as a subset of 𝔻\mathbb{D} by identifying aa with a+0ϵa + 0 \epsilon.

Properties

𝔻\mathbb{D} is equipped with an involution that maps ϵ\epsilon to ϵ¯=−ϵ\bar{\epsilon} = -\epsilon:

a+ϵb¯=a−ϵb. \overline{a + \epsilon b} = a - \epsilon b .

𝔻\mathbb{D} also has an absolute value:

|a+ϵb|=|a|; {|a + \epsilon b|} = {|a|} ;

notice that the absolute value of a dual number is a non-negative real number, with

|z| 2=zz¯. {|z|^2} = z \bar{z}.

But this absolute value is degenerate, in that |z|=0{|z|} = 0 need not imply that z=0z = 0.

Some concepts in analysis can be extended from ℝ\mathbb{R} to 𝔻\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.

References

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]