nLab dual number

Idea

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 $R$.

Interpretations

This can be thought of as:

• the vector space $\mathbb{R}^2$ made into an algebra by the rule
  $$(a, b) \cdot (c, d) = (a c, a d + b c) ;$$
• the subalgebra of those $2$-by-$2$ real matrices of the form
  $$\left(\array { a & b \\ 0 & a } \right);$$
• the polynomial ring $\mathbb{R}[\mathrm{x}]$ modulo $\mathrm{x}^2$;
• the parabolic $2$-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 $\epsilon$ is regarded as being of degree 1 and $\mathbb{R} \oplus \epsilon \mathbb{R}$ is regarded accordingly as a superalgebra then this is the algebra of functions on the odd line $\mathbb{R}^{0|1}$.
• the square-0-extension corresponding to the $\mathbb{R}$-module (see there) given by $\mathbb{R}$ itself.

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

Properties

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

$$\overline{a + \epsilon b} = a - \epsilon b .$$

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

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

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

$${|z|^2} = z \bar{z}.$$

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.

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]

See also:

• Wikipedia, Dual number