Are dual numbers a special case of grassmann numbers?

Question

Dual numbers are defined in analogy to complex numbers like $$ z = a + \varepsilon b. $$ But instead of $i^2=-1$ it is defined that $\varepsilon^2=0$.

The multiplication rule for Grassmann numbers $\theta_i$ is $$ \theta_i\theta_j = - \theta_j \theta_i $$ so that $\theta_i\theta_i = 0$.

So it seem that dual numbers are just a special case of a Grassmann algebra with just one generator ($i = j = 1$). Is this correct?

Answer 1

Yes. By the way, both of these are also what mathematicians call an exterior algebra.

Answer 2

Yes, the dual numbers are isomorphic to the Clifford algebra $\mathcal{Cl}_{0,0,1}$ with one imaginary element.