Geometry of the dual numbers A dual number is a number of the form $a+b\varepsilon$, where $a,b \in \mathbb{R}$ and $\varepsilon$ is a nonreal number with the property $\varepsilon^2=0$. Dual numbers are in some ways similar to the complex numbers $a+bi$, where $i^2=-1$.
Complex numbers have a very elegant geometric interpretation. Specifically, we can treat complex numbers as vectors in the complex plane. Addition and subtraction then follow the regular rules of vector arithmetic, and complex multiplication can be seen as a scaling and rotation of one vector by the magnitude and argument of another, respectively.
Is there a corresponding geometric interpretation for dual number arithmetic in the dual plane?
 A: 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).
A: 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.
A: You can imagine the dual numbers of the form $a + \epsilon \cdot b$ as situated on the orthogonal line centered at $a$ in the dual number "plane". 
You can also treat them as vectors but then the modulus is evaluated as $|| a + \epsilon \cdot b || = abs (a)$ because of the nilpotency. 
