| date |
title |
id |
2020-10-09 13:35:29 -0700 |
Dual Numbers |
2020-10-09t20-35-29z |
We define dual numbers as the set of numbers $a + b\epsilon$ where
$a, b \in
\mathbb{R}$ and $\epsilon^2 = 0$.
Just how we can
represent complex numbers as matrices, an analog
isomorphism is present with dual numbers. In fact the
ring of dual numbers can be
represented by the $2 \times 2$ matrices of the form
$$
\begin{pmatrix}
a & 0 \\
b & a
\end{pmatrix}.
$$
Basic operations are straightforward:
$$
\begin{aligned}
(a+b\epsilon) + (c+d\epsilon) &= (a+c) + (b+d)\epsilon \\
(a+b\epsilon) (c+d\epsilon) &= ac + (ad+bc)\epsilon \\
\frac{a+b\epsilon}{c+d\epsilon} &= \frac ac + \frac{bc - ad}{c^2}\epsilon.
\end{aligned}
$$