Generalization
This construction can be carried out more generally: for a commutative ring R one can define the dual numbers over R as the quotient of the polynomial ring R by the ideal (X2): the image of X then has square equal to zero and corresponds to the element ε from above.
This ring and its generalisations play an important part in the algebraic theory of derivations and Kähler differentials (purely algebraic differential forms).
Over any ring R, the dual number a + bε is a unit (i.e. multiplicatively invertible) if and only if a is a unit in R. In this case, the inverse of a + bε is a−1 − ba−2ε. As a consequence, we see that the dual numbers over any field (or any commutative local ring) form a local ring, its maximal ideal being the principal ideal generated by ε.
Read more about this topic: Dual Number