**Algebraic Properties**

In abstract algebra terms, the dual numbers can be described as the quotient of the polynomial ring **R** by the ideal generated by the polynomial *X*2,

**R**/(*X*2).

The image of *X* in the quotient is the "imaginary" unit ε. With this description, it is clear that the dual numbers form a commutative ring with characteristic 0. Moreover the inherited multiplication gives the dual numbers the structure of a commutative and associative algebra over the reals of dimension two. The algebra is *not* a division algebra or field since the imaginary elements are not invertible. In fact, all of the nonzero imaginary elements are zero divisors (also see the section "Division"). The algebra of dual numbers is isomorphic to the exterior algebra of .

Read more about this topic: Dual Number

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—Henry David Thoreau (1817–1862)