이원수
Definition
The following form for $\epsilon$ satisfying $\epsilon^{2} = 0 (\epsilon \neq 0)$ is called dual numbers.
$$ a + b\epsilon,\qquad a, b \in \mathbb{R} $$
Explanation
As can be seen from the definition, $\epsilon$ plays a role similar to the $i$ of complex numbers in that it creates the second dimension in the ordered pair. Of course, its properties are entirely different.
$$ x + yi = (x, y) \\[1em] a + b\epsilon = (a, b) $$
Dual numbers are not only of pure mathematical interest and significance but also find important applications in applied mathematics, making them very fascinating objects in my view.
Pure Mathematical Aspect
The $\epsilon$ in the definition is a zero divisor because it is not $0$ but becomes $0$ when squared. Thus, it must be chosen from outside an integral domain. The set of dual numbers formed with such $\epsilon$ constitutes a ring, and for it to be $xy = 0$ for two dual numbers $x, y$, either $x$ or $y$ must be $0 + 0\epsilon$. Thus, the set of dual numbers is an integral domain. In other words, the process of defining dual numbers, assigning operations, and forming a ring is a method of creating an integral domain from a non-integral domain ring.
Applied Mathematical Aspect
The operations of dual numbers have very interesting properties from the viewpoint of a differential coefficient. The differential coefficient becomes the second component, and it is preserved under addition and multiplication. For a differentiable function $f : \mathbb{R} \to \mathbb{R}$, if it is well-defined, it can be verified that the differential coefficient is well preserved in both function composition and evaluation. Thanks to this characteristic, dual numbers, although very theoretical, can be applied in automatic differentiation, a method for calculating differential coefficients used in the backpropagation algorithm for optimizing artificial neural networks in deep learning.
The concept of dual numbers also appears in stochastic differential equations, which seem to have no connection with algebra at all. Specifically, when dealing with the Wiener process ( W_t ), the Ito multiplication table, which involves operations between the infinitesimal increments ( dt ) and ( dW_t ), corresponds to this. Here, although ( dt > 0 ) is clearly positive, it is assumed that from its square onwards, it is so small that it can be neglected. It is very interesting that such an analytically non-rigorous assumption is supported by abstract algebra.
Operations
If the set of dual numbers is endowed with the following two operations, that set constitutes a ring.
Addition
The addition of two dual numbers $a+b\epsilon$ and $c + d\epsilon$ is defined as follows.
$$ (a + b\epsilon) + (c + d\epsilon) = (a + c) + (b + d)\epsilon $$
If a dual number $a + b\epsilon$ is represented as an ordered pair $(a, b)$, then addition is simply taking the component-wise sum.
$$ (a, b) + (c, d) = (a + c, b + d) = (a + c) + (b + d)\epsilon $$
An inverse with respect to addition always exists, and the inverse of $a + b\epsilon$ is as follows.
$$ -(a + b\epsilon) = (-a) + (-b)\epsilon $$
Multiplication
The multiplication of two dual numbers $a+b\epsilon$ and $c + d\epsilon$ is defined as follows.
$$ (a + b\epsilon)(c + d\epsilon) = ac + (bc+ad)\epsilon $$
It is the result of multiplying and organizing as one would apply the distributive law.
$$ (a + b\epsilon)(c + d\epsilon) = ac + ad\epsilon + dc\epsilon + bd\epsilon^{2} = ac + (bc+ad)\epsilon $$
In terms of ordered pairs, it is as follows.
$$ (a, b)(c, d) = (ac, bc+ad) $$
Though it resembles complex number multiplication, it differs in that it lacks $-bd$.
$$ (a+bi)(c+di) = ac - bd + (bc+ad)i $$
For $a + b\epsilon$, which is $a \ne 0$, an inverse with respect to multiplication exists and is as follows.
$$ (a + b\epsilon)^{-1} = \dfrac{1}{a} - \dfrac{b}{a^{2}}\epsilon $$
By actual calculation,
$$ (a + b\epsilon) \left( \dfrac{1}{a} - \dfrac{b}{a^{2}}\epsilon \right) = a\cdot\dfrac{1}{a} + \left(b\cdot\dfrac{1}{a} - a\cdot\dfrac{b}{a^{2}}\right)\epsilon = 1 + \left( \dfrac{b}{a} - \dfrac{b}{a} \right)\epsilon = 1 $$