nLab
dual number

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 ϵ0\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 RR.

Interpretations

This can be thought of as:

  • the vector space 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 [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 ϵ\mathbb{R} \oplus \epsilon \mathbb{R} is regarded accordingly as a superalgebra then this is the algebra of functions on the odd line 0|1\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 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¯=aϵb. \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 \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.

Revised on November 14, 2011 06:02:25 by Toby Bartels (139.55.238.24)