# 二元数

${\displaystyle \mathbb {N} \subseteq \mathbb {Z} \subseteq \mathbb {Q} \subseteq \mathbb {R} \subseteq \mathbb {C} }$

## 矩陣表示法

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

## 幾何

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

a≠0且m${\textstyle {b \over a}}$，則za（1＋mε）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＋pε）（1＋qε）＝1＋（pq）ε。

### 伽利略變換

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

### 循環

${\displaystyle x_{1}=x,\ \ y_{1}=vx+y\ \ }$

${\displaystyle x'=x_{1}=v/2a,\ \ y'=y_{1}+v^{2}/4a\ }$

## 除法

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

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

${\displaystyle {a+b\varepsilon =(x+y\varepsilon )d\varepsilon }={xd\varepsilon +0}}$
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：
${\displaystyle {b \over d}+{y\varepsilon }}$

## 冪

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

## 參考資料

1. ^ V.V. Kisil (2007) "Inventing a Wheel, the Parabolic One" arXiv:0707.4024页面存档备份，存于互联网档案馆