DualNumbers

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