A simple toy implementation of the hypercomplex number system called the Dual numbers.
Dual numbers are of the form $z = a + b\varepsilon$ where $a,b\in\mathbb{R}$, where $\varepsilon^2=0$.
Whereas for standard real $(\mathbb{R})$ algebra:
$$
\begin{align*}
(a + b)(x + y) &= ax +ay + bx + by\\
(a + b)^2 &= a^2 + 2ab + b^2
\end{align*}
$$
And how in complex $(\mathbb{C})$ domain, where $i^2=-1$:
$$
\begin{align*}
(a + bi)(x + yi) &= ax - by + (ay + bx)i\\
(a + bi)^2 &= a^2 + 2abi - b^2
\end{align*}
$$
In the Dual $(\mathbb{D})$ number system:
$$
\begin{align*}
(a + b\varepsilon)(x + y\varepsilon) &= ax + (ay + bx)\varepsilon\\
(a + b\varepsilon)^2 &= a^2 + 2ab\varepsilon
\end{align*}
$$
One property of interest is automatic differentiation. For any analytic function $f:\mathbb{R}\mapsto\mathbb{R}$, it can be shown that the domain can be extended to include the Dual numbers in such a way that, for $z\in\mathbb{D}$, where $x,y\in\mathbb{R}$, $z=(x + y\varepsilon)$:
$$
f(z) = f(a) + bf'(a)\varepsilon
$$
This is done by manipulating the function's Taylor Series, making use of the fact that, $\forall n\geq 2,,\varepsilon^n=0$, which gives the prior result.
Resources:
-
F. Messelmi,
Analysis of Dual Functions,
Annual Review of Chaos Theory, Bifurcations and Dynamical Systems Vol. 4, (2013) 37-54,
DOI:10.13140/2.1.1006.4006
-
Behr, Nicolas, Giuseppe Dattoli, Ambra Lattanzi, and Silvia Licciardi,
Dual Numbers and Operational Umbral Methods, Axioms 8, no. 3: 77 (2019)
DOI:10.3390/axioms8030077