Geometry of the dual numbers

Solution 1:

Dual numbers, unlike complex numbers, illustrate Galileo's invariance principle. The infinitesimal part of a dual number represents the velocity.

In particular, whereas the multiplication of two complex numbers can be understood as a combination of scaling and rotation, the multiplication of dual numbers is actually equivalent to a scaling and shear mapping of plane, since $(1 + p \varepsilon)(1 + q \varepsilon) = 1 + (p+q) \varepsilon$. You can see that the classical velocity addition law emerges.

The "hyperbolic" multiplication law of special relativity (namely the corresponding velocity addition law $v\oplus u=(v+u)/(1+vu)$) requires Lorentz transformations, which can be packaged into different types of numbers like, for example, quaternions (see Generalized complex numbers).

Solution 2:

Dual numbers are numbers with tangents. For any analytical function $f$ one obtains exactly $f(x+ε\dot x)=f(x)+εf'(x)\dot x$.

An old trick in computing derivatives is to turn an implementation of complex numbers into dual numbers by initializing $z=x+i\cdot 10^{-40}\cdot\dot x$, so that the imaginary part has practically no influence on the real part, and the derivative can easily be recovered from the imaginary part.