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dual number

Idea

A dual number is given by an expression of the form $a+ϵb$, where $a$ and $b$ are real numbers and ${ϵ}^2=0$ (but $ϵ\ne 0$). 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 $ℝ$ to any commutative ring $R$.

Interpretations

This can be thought of as:

• the vector space ${ℝ}^2$ made into an algebra by the rule

  $$(a,b)\cdot (c,d)=(ac,ad+bc);$$(a, b) \cdot (c, d) = (a c, a d + b c) ;

• the subalgebra of those $2$-by-$2$ real matrices of the form

  $$\left(\begin{array}{cc}a& b\\ 0& a\end{array}\right);$$\left(\array { a & b \\ 0 & a } \right);

• the polynomial ring $ℝ[\mathrm{x}]$ 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 $ϵ$ is regarded as being of degree 1 and $ℝ\oplus ϵℝ$ is regarded accordingly as a superalgebra then this is the algebra of functions on the odd line ${ℝ}^{0\mid 1}$.

• the square-0-extension corresponding to the $ℝ$-module (see there) given by $ℝ$ itself.

We think of $ℝ$ as a subset of $𝔻$ by identifying $a$ with $a+0ϵ$.

Properties

$𝔻$ is equipped with an involution that maps $ϵ$ to $\overline{ϵ}=-ϵ$:

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

$𝔻$ also has an absolute value:

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

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

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

But this absolute value is degenerate, in that $\mid z\mid =0$ need not imply that $z=0$.

Some concepts in analysis can be extended from $ℝ$ to $𝔻$, 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.