Questions tagged [dual-numbers]

Dual numbers of the form $z = x + y \cdot \varepsilon \wedge \left\{ x,\, y \right\} \in \mathbb{R} \wedge \varepsilon \ne 0 = \varepsilon^{2}$. $\Re\left( z \right) = x$ is called the "real part", $\Im\left( z \right) = y$ is called the "imaginary part" and $\varepsilon$ is called "imaginary unit" of the dual numbers. $\overline{z} = z^{\ast} = x - y \cdot \varepsilon$ is called the conjugate of $z$ and its absolute value $\left|z\right|=\sqrt{x^2+y^2}$ is called magnitude.

Dual numbers are commutative, associative, alternative and power-associative $2$-dimensional hypercomplex numbers with zero divisors. They have the square matrix representation $x + y \cdot \varepsilon \mapsto \begin{pmatrix} x & y\\ 0 & x\\ \end{pmatrix}$.

Dual numbers find applications in Algebraic geometry, automatic differentiation, mechanics (e.g. screw theory) and many more more fields. Generalizations such as bicomplex numbers and dual quaternions also have useful applications, e.g. in fluid mechanics.

Important basic relations are: \begin{align*} \left( a + b \cdot \varepsilon \right) \pm \left( c + d \cdot \varepsilon \right) &= a \pm c + \left( b \pm c \right) \cdot \varepsilon\\ \left( a + b \cdot \varepsilon \right) \cdot \left( c + d \cdot \varepsilon \right) &= a \cdot c + \left( a \cdot d + b \cdot c \right) \cdot \varepsilon\\ \frac{a + b \cdot \varepsilon}{c + d \cdot \varepsilon} &= \frac{a}{c} + \frac{b \cdot c - a \cdot d}{c^{2}} \cdot \varepsilon \tag{if $c \ne 0$}\\ f\left( a + b \cdot \varepsilon \right) &= f\left( a \right) + b \cdot f'\left( a \right) \cdot \varepsilon \tag{if $f$ is analytic}\\ \end{align*}

Read more about dual numbers and their properties here.