The Dual Numbers

Definition: Dual Number

A dual number is an expression of the form

$$a+b\epsilon ,$$

where $a,b\in \mathbb{R}$ are real numbers and $\epsilon$ is defined such that

$${\epsilon }^2=0\text{ and }\epsilon \ne 0.$$

Notation

The set of all dual numbers is usually denoted by $\mathbb{R}[\epsilon ]$.