# 二元数

## 矩陣表示法

$\varepsilon=\begin{pmatrix}0 & 1 \\0 & 0 \end{pmatrix}\quad$$\quad a + b\varepsilon = \begin{pmatrix}a & b \\ 0 & a \end{pmatrix}$.

## 幾何

$\exp(b \varepsilon) = \left(\sum^\infty_{n=0} (b\varepsilon)^n / n!\right) = 1 + b \varepsilon \!$,

a ≠ 0且m = b /a，則z = a(1 + m ε)為二元數z極分解斜率m則與輻角相關。二元數平面中的「旋轉」等價於一個垂直錯切，原因是(1 + p ε)(1 + q ε) = 1 + (p+q) ε。

### 伽利略變換

$(t',x') = (t,x)\begin{pmatrix}1 & v \\0 & 1 \end{pmatrix}$，亦即
$\ \ t'=t,\ \ x' = vt + x \!$

### 循環

$x_1 = x ,\ \ y_1 = vx + y \ \$

$x' = x_1 = v/2a ,\ \ y' = y_1 + v^2/4a \$

## 除法

${a+b\varepsilon \over c+d\varepsilon}$

$= {(a+b\varepsilon)(c-d\varepsilon) \over (c+d\varepsilon)(c-d\varepsilon)} = {ac-ad\varepsilon+bc\varepsilon-bd\varepsilon^2 \over (c^2+cd\varepsilon-cd\varepsilon-d^2\varepsilon^2)} = {ac-ad\varepsilon+bc\varepsilon-0 \over c^2-0}$
$= {ac + \varepsilon(bc - ad) \over c^2}$
$= {a \over c} + {(bc - ad) \over c^2}\varepsilon$

${a+b\varepsilon = (x+y\varepsilon) d\varepsilon} = {xd\varepsilon + 0}$
1. 當a非零時沒有解
2. 當a為零時，以下的二元數都是它的解：
${b \over d} + {y\varepsilon}$.

## 冪

$(a+b\varepsilon)^{c+d\varepsilon}=a^c+\varepsilon(b (c a^{c-1})+d (a^c \ln a))$

## 參考資料

1. ^ V.V. Kisil (2007) "Inventing a Wheel, the Parabolic One" arXiv:0707.4024