# 十六元數

• ${\displaystyle \mathbb {N} }$ 自然數
• ${\displaystyle \mathbb {Z} }$ 整數
• ${\displaystyle \mathbb {Q} }$ 有理數
• ${\displaystyle \mathbb {R} }$ 實數
• ${\displaystyle \mathbb {C} }$ 複數
• ${\displaystyle \mathbb {H} }$ 四元數

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

## 算術

${\displaystyle e_{0}}$ ${\displaystyle e_{1}}$ ${\displaystyle e_{2}}$ ${\displaystyle e_{3}}$ ${\displaystyle e_{4}}$ ${\displaystyle e_{5}}$ ${\displaystyle e_{6}}$ ${\displaystyle e_{7}}$ ${\displaystyle e_{8}}$ ${\displaystyle e_{9}}$ ${\displaystyle e_{10}}$ ${\displaystyle e_{11}}$ ${\displaystyle e_{12}}$ ${\displaystyle e_{13}}$ ${\displaystyle e_{14}}$ ${\displaystyle e_{15}}$

${\displaystyle x=x_{0}e_{0}+x_{1}e_{1}+x_{2}e_{2}+\cdots +x_{14}e_{14}+x_{15}e_{15}.}$

${\displaystyle e_{i}e_{j}}$  ${\displaystyle e_{j}}$
${\displaystyle e_{0}}$  ${\displaystyle e_{1}}$  ${\displaystyle e_{2}}$  ${\displaystyle e_{3}}$  ${\displaystyle e_{4}}$  ${\displaystyle e_{5}}$  ${\displaystyle e_{6}}$  ${\displaystyle e_{7}}$  ${\displaystyle e_{8}}$  ${\displaystyle e_{9}}$  ${\displaystyle e_{10}}$  ${\displaystyle e_{11}}$  ${\displaystyle e_{12}}$  ${\displaystyle e_{13}}$  ${\displaystyle e_{14}}$  ${\displaystyle e_{15}}$
${\displaystyle e_{i}}$  ${\displaystyle e_{0}}$  ${\displaystyle e_{0}}$  ${\displaystyle e_{1}}$  ${\displaystyle e_{2}}$  ${\displaystyle e_{3}}$  ${\displaystyle e_{4}}$  ${\displaystyle e_{5}}$  ${\displaystyle e_{6}}$  ${\displaystyle e_{7}}$  ${\displaystyle e_{8}}$  ${\displaystyle e_{9}}$  ${\displaystyle e_{10}}$  ${\displaystyle e_{11}}$  ${\displaystyle e_{12}}$  ${\displaystyle e_{13}}$  ${\displaystyle e_{14}}$  ${\displaystyle e_{15}}$
${\displaystyle e_{1}}$  ${\displaystyle e_{1}}$  ${\displaystyle -e_{0}}$  ${\displaystyle e_{3}}$  ${\displaystyle -e_{2}}$  ${\displaystyle e_{5}}$  ${\displaystyle -e_{4}}$  ${\displaystyle -e_{7}}$  ${\displaystyle e_{6}}$  ${\displaystyle e_{9}}$  ${\displaystyle -e_{8}}$  ${\displaystyle -e_{11}}$  ${\displaystyle e_{10}}$  ${\displaystyle -e_{13}}$  ${\displaystyle e_{12}}$  ${\displaystyle e_{15}}$  ${\displaystyle -e_{14}}$
${\displaystyle e_{2}}$  ${\displaystyle e_{2}}$  ${\displaystyle -e_{3}}$  ${\displaystyle -e_{0}}$  ${\displaystyle e_{1}}$  ${\displaystyle e_{6}}$  ${\displaystyle e_{7}}$  ${\displaystyle -e_{4}}$  ${\displaystyle -e_{5}}$  ${\displaystyle e_{10}}$  ${\displaystyle e_{11}}$  ${\displaystyle -e_{8}}$  ${\displaystyle -e_{9}}$  ${\displaystyle -e_{14}}$  ${\displaystyle -e_{15}}$  ${\displaystyle e_{12}}$  ${\displaystyle e_{13}}$
${\displaystyle e_{3}}$  ${\displaystyle e_{3}}$  ${\displaystyle e_{2}}$  ${\displaystyle -e_{1}}$  ${\displaystyle -e_{0}}$  ${\displaystyle e_{7}}$  ${\displaystyle -e_{6}}$  ${\displaystyle e_{5}}$  ${\displaystyle -e_{4}}$  ${\displaystyle e_{11}}$  ${\displaystyle -e_{10}}$  ${\displaystyle e_{9}}$  ${\displaystyle -e_{8}}$  ${\displaystyle -e_{15}}$  ${\displaystyle e_{14}}$  ${\displaystyle -e_{13}}$  ${\displaystyle e_{12}}$
${\displaystyle e_{4}}$  ${\displaystyle e_{4}}$  ${\displaystyle -e_{5}}$  ${\displaystyle -e_{6}}$  ${\displaystyle -e_{7}}$  ${\displaystyle -e_{0}}$  ${\displaystyle e_{1}}$  ${\displaystyle e_{2}}$  ${\displaystyle e_{3}}$  ${\displaystyle e_{12}}$  ${\displaystyle e_{13}}$  ${\displaystyle e_{14}}$  ${\displaystyle e_{15}}$  ${\displaystyle -e_{8}}$  ${\displaystyle -e_{9}}$  ${\displaystyle -e_{10}}$  ${\displaystyle -e_{11}}$
${\displaystyle e_{5}}$  ${\displaystyle e_{5}}$  ${\displaystyle e_{4}}$  ${\displaystyle -e_{7}}$  ${\displaystyle e_{6}}$  ${\displaystyle -e_{1}}$  ${\displaystyle -e_{0}}$  ${\displaystyle -e_{3}}$  ${\displaystyle e_{2}}$  ${\displaystyle e_{13}}$  ${\displaystyle -e_{12}}$  ${\displaystyle e_{15}}$  ${\displaystyle -e_{14}}$  ${\displaystyle e_{9}}$  ${\displaystyle -e_{8}}$  ${\displaystyle e_{11}}$  ${\displaystyle -e_{10}}$
${\displaystyle e_{6}}$  ${\displaystyle e_{6}}$  ${\displaystyle e_{7}}$  ${\displaystyle e_{4}}$  ${\displaystyle -e_{5}}$  ${\displaystyle -e_{2}}$  ${\displaystyle e_{3}}$  ${\displaystyle -e_{0}}$  ${\displaystyle -e_{1}}$  ${\displaystyle e_{14}}$  ${\displaystyle -e_{15}}$  ${\displaystyle -e_{12}}$  ${\displaystyle e_{13}}$  ${\displaystyle e_{10}}$  ${\displaystyle -e_{11}}$  ${\displaystyle -e_{8}}$  ${\displaystyle e_{9}}$
${\displaystyle e_{7}}$  ${\displaystyle e_{7}}$  ${\displaystyle -e_{6}}$  ${\displaystyle e_{5}}$  ${\displaystyle e_{4}}$  ${\displaystyle -e_{3}}$  ${\displaystyle -e_{2}}$  ${\displaystyle e_{1}}$  ${\displaystyle -e_{0}}$  ${\displaystyle e_{15}}$  ${\displaystyle e_{14}}$  ${\displaystyle -e_{13}}$  ${\displaystyle -e_{12}}$  ${\displaystyle e_{11}}$  ${\displaystyle e_{10}}$  ${\displaystyle -e_{9}}$  ${\displaystyle -e_{8}}$
${\displaystyle e_{8}}$  ${\displaystyle e_{8}}$  ${\displaystyle -e_{9}}$  ${\displaystyle -e_{10}}$  ${\displaystyle -e_{11}}$  ${\displaystyle -e_{12}}$  ${\displaystyle -e_{13}}$  ${\displaystyle -e_{14}}$  ${\displaystyle -e_{15}}$  ${\displaystyle -e_{0}}$  ${\displaystyle e_{1}}$  ${\displaystyle e_{2}}$  ${\displaystyle e_{3}}$  ${\displaystyle e_{4}}$  ${\displaystyle e_{5}}$  ${\displaystyle e_{6}}$  ${\displaystyle e_{7}}$
${\displaystyle e_{9}}$  ${\displaystyle e_{9}}$  ${\displaystyle e_{8}}$  ${\displaystyle -e_{11}}$  ${\displaystyle e_{10}}$  ${\displaystyle -e_{13}}$  ${\displaystyle e_{12}}$  ${\displaystyle e_{15}}$  ${\displaystyle -e_{14}}$  ${\displaystyle -e_{1}}$  ${\displaystyle -e_{0}}$  ${\displaystyle -e_{3}}$  ${\displaystyle e_{2}}$  ${\displaystyle -e_{5}}$  ${\displaystyle e_{4}}$  ${\displaystyle e_{7}}$  ${\displaystyle -e_{6}}$
${\displaystyle e_{10}}$  ${\displaystyle e_{10}}$  ${\displaystyle e_{11}}$  ${\displaystyle e_{8}}$  ${\displaystyle -e_{9}}$  ${\displaystyle -e_{14}}$  ${\displaystyle -e_{15}}$  ${\displaystyle e_{12}}$  ${\displaystyle e_{13}}$  ${\displaystyle -e_{2}}$  ${\displaystyle e_{3}}$  ${\displaystyle -e_{0}}$  ${\displaystyle -e_{1}}$  ${\displaystyle -e_{6}}$  ${\displaystyle -e_{7}}$  ${\displaystyle e_{4}}$  ${\displaystyle e_{5}}$
${\displaystyle e_{11}}$  ${\displaystyle e_{11}}$  ${\displaystyle -e_{10}}$  ${\displaystyle e_{9}}$  ${\displaystyle e_{8}}$  ${\displaystyle -e_{15}}$  ${\displaystyle e_{14}}$  ${\displaystyle -e_{13}}$  ${\displaystyle e_{12}}$  ${\displaystyle -e_{3}}$  ${\displaystyle -e_{2}}$  ${\displaystyle e_{1}}$  ${\displaystyle -e_{0}}$  ${\displaystyle -e_{7}}$  ${\displaystyle e_{6}}$  ${\displaystyle -e_{5}}$  ${\displaystyle e_{4}}$
${\displaystyle e_{12}}$  ${\displaystyle e_{12}}$  ${\displaystyle e_{13}}$  ${\displaystyle e_{14}}$  ${\displaystyle e_{15}}$  ${\displaystyle e_{8}}$  ${\displaystyle -e_{9}}$  ${\displaystyle -e_{10}}$  ${\displaystyle -e_{11}}$  ${\displaystyle -e_{4}}$  ${\displaystyle e_{5}}$  ${\displaystyle e_{6}}$  ${\displaystyle e_{7}}$  ${\displaystyle -e_{0}}$  ${\displaystyle -e_{1}}$  ${\displaystyle -e_{2}}$  ${\displaystyle -e_{3}}$
${\displaystyle e_{13}}$  ${\displaystyle e_{13}}$  ${\displaystyle -e_{12}}$  ${\displaystyle e_{15}}$  ${\displaystyle -e_{14}}$  ${\displaystyle e_{9}}$  ${\displaystyle e_{8}}$  ${\displaystyle e_{11}}$  ${\displaystyle -e_{10}}$  ${\displaystyle -e_{5}}$  ${\displaystyle -e_{4}}$  ${\displaystyle e_{7}}$  ${\displaystyle -e_{6}}$  ${\displaystyle e_{1}}$  ${\displaystyle -e_{0}}$  ${\displaystyle e_{3}}$  ${\displaystyle -e_{2}}$
${\displaystyle e_{14}}$  ${\displaystyle e_{14}}$  ${\displaystyle -e_{15}}$  ${\displaystyle -e_{12}}$  ${\displaystyle e_{13}}$  ${\displaystyle e_{10}}$  ${\displaystyle -e_{11}}$  ${\displaystyle e_{8}}$  ${\displaystyle e_{9}}$  ${\displaystyle -e_{6}}$  ${\displaystyle -e_{7}}$  ${\displaystyle -e_{4}}$  ${\displaystyle e_{5}}$  ${\displaystyle e_{2}}$  ${\displaystyle -e_{3}}$  ${\displaystyle -e_{0}}$  ${\displaystyle e_{1}}$
${\displaystyle e_{15}}$  ${\displaystyle e_{15}}$  ${\displaystyle e_{14}}$  ${\displaystyle -e_{13}}$  ${\displaystyle -e_{12}}$  ${\displaystyle e_{11}}$  ${\displaystyle e_{10}}$  ${\displaystyle -e_{9}}$  ${\displaystyle e_{8}}$  ${\displaystyle -e_{7}}$  ${\displaystyle e_{6}}$  ${\displaystyle -e_{5}}$  ${\displaystyle -e_{4}}$  ${\displaystyle e_{3}}$  ${\displaystyle e_{2}}$  ${\displaystyle -e_{1}}$  ${\displaystyle -e_{0}}$

### 十六元數特性

${\displaystyle e_{i}e_{i}=-e_{0}\,\,{\text{for}}\,\,i\neq 0}$ ，且
${\displaystyle e_{i}e_{j}=-e_{j}e_{i}\,\,{\text{for}}\,\,i\neq j\,\,{\text{with}}\,\,i,j\neq 0}$

#### 反結合

${\displaystyle (ij)(kl)=-((ij)k)l=(i(jk))l=-i((jk)l)=i(j(kl))=-(ij)(kl),}$

### 四元子代數

{ {1, 2, 3}, {1, 4, 5}, {1, 7, 6}, {1, 8, 9}, {1, 11, 10}, {1, 13, 12}, {1, 14, 15},
{2, 4, 6}, {2, 5, 7}, {2, 8, 10}, {2, 9, 11}, {2, 14, 12}, {2, 15, 13}, {3, 4, 7},
{3, 6, 5}, {3, 8, 11}, {3, 10, 9}, {3, 13, 14}, {3, 15, 12}, {4, 8, 12}, {4, 9, 13},
{4, 10, 14}, {4, 11, 15}, {5, 8, 13}, {5, 10, 15}, {5, 12, 9}, {5, 14, 11}, {6, 8, 14},
{6, 11, 13}, {6, 12, 10}, {6, 15, 9}, {7, 8, 15}, {7, 9, 14}, {7, 12, 11}, {7, 13, 10} }

84組由十六元數單位組成的零因子數組${\displaystyle \{e_{a},e_{b},e_{c},e_{d}\}}$ 列舉如下，其中 ${\displaystyle (e_{a}+e_{b})\circ (e_{c}+e_{d})=0}$

${\displaystyle {\begin{array}{c}{\begin{array}{ccc}1\leq a\leq 6,&c>a,&9\leq b\leq 15\\9\leq d\leq 15&&-9\geq d\geq -15\end{array}}\\{\begin{array}{ll}\{e_{1},e_{10},e_{5},e_{14}\}&\{e_{1},e_{10},e_{4},-e_{15}\}\\\{e_{1},e_{10},e_{7},e_{12}\}&\{e_{1},e_{10},e_{6},-e_{13}\}\\\{e_{1},e_{11},e_{4},e_{14}\}&\{e_{1},e_{11},e_{6},-e_{12}\}\\\{e_{1},e_{11},e_{5},e_{15}\}&\{e_{1},e_{11},e_{7},-e_{13}\}\\\{e_{1},e_{12},e_{2},e_{15}\}&\{e_{1},e_{12},e_{3},-e_{14}\}\\\{e_{1},e_{12},e_{6},e_{11}\}&\{e_{1},e_{12},e_{7},-e_{10}\}\\\{e_{1},e_{13},e_{6},e_{10}\}&\{e_{1},e_{13},e_{7},-e_{14}\}\\\{e_{1},e_{13},e_{7},e_{11}\}&\{e_{1},e_{13},e_{3},-e_{15}\}\\\{e_{1},e_{14},e_{2},e_{13}\}&\{e_{1},e_{14},e_{4},-e_{11}\}\\\{e_{1},e_{14},e_{3},e_{12}\}&\{e_{1},e_{14},e_{5},-e_{10}\}\\\{e_{1},e_{15},e_{3},e_{13}\}&\{e_{1},e_{15},e_{2},-e_{12}\}\\\{e_{1},e_{15},e_{4},e_{10}\}&\{e_{1},e_{15},e_{5},-e_{11}\}\\\{e_{2},e_{9},e_{4},e_{15}\}&\{e_{2},e_{9},e_{5},-e_{14}\}\\\{e_{2},e_{9},e_{6},e_{13}\}&\{e_{2},e_{9},e_{7},-e_{12}\}\\\{e_{2},e_{11},e_{5},e_{12}\}&\{e_{2},e_{11},e_{4},-e_{13}\}\\\{e_{2},e_{11},e_{6},e_{15}\}&\{e_{2},e_{11},e_{7},-e_{14}\}\\\{e_{2},e_{12},e_{3},e_{13}\}&\{e_{2},e_{12},e_{5},-e_{11}\}\\\{e_{2},e_{12},e_{7},e_{9}\}&\{e_{2},e_{13},e_{3},-e_{12}\}\\\{e_{2},e_{13},e_{4},e_{11}\}&\{e_{2},e_{13},e_{6},-e_{9}\}\\\{e_{2},e_{14},e_{5},e_{9}\}&\{e_{2},e_{14},e_{3},-e_{15}\}\\\{e_{2},e_{14},e_{3},e_{14}\}&\{e_{2},e_{15},e_{4},-e_{9}\}\\\{e_{2},e_{15},e_{3},e_{14}\}&\{e_{2},e_{15},e_{6},-e_{11}\}\\\{e_{3},e_{9},e_{6},e_{12}\}&\{e_{3},e_{9},e_{4},-e_{14}\}\\\{e_{3},e_{9},e_{7},e_{13}\}&\{e_{3},e_{9},e_{5},-e_{15}\}\\\{e_{3},e_{10},e_{4},e_{13}\}&\{e_{3},e_{10},e_{5},-e_{12}\}\\\{e_{3},e_{10},e_{7},e_{14}\}&\{e_{3},e_{10},e_{6},-e_{15}\}\\\{e_{3},e_{12},e_{5},e_{10}\}&\{e_{3},e_{12},e_{6},-e_{9}\}\\\{e_{3},e_{14},e_{4},e_{9}\}&\{e_{3},e_{13},e_{4},-e_{10}\}\\\{e_{3},e_{15},e_{5},e_{9}\}&\{e_{3},e_{13},e_{7},-e_{9}\}\\\{e_{3},e_{15},e_{6},e_{10}\}&\{e_{3},e_{14},e_{7},-e_{10}\}\\\{e_{4},e_{9},e_{7},e_{10}\}&\{e_{4},e_{9},e_{6},-e_{11}\}\\\{e_{4},e_{10},e_{5},e_{11}\}&\{e_{4},e_{10},e_{7},-e_{9}\}\\\{e_{4},e_{11},e_{6},e_{9}\}&\{e_{4},e_{11},e_{5},-e_{10}\}\\\{e_{4},e_{13},e_{6},e_{15}\}&\{e_{4},e_{13},e_{7},-e_{14}\}\\\{e_{4},e_{14},e_{7},e_{13}\}&\{e_{4},e_{14},e_{5},-e_{15}\}\\\{e_{4},e_{15},e_{5},e_{14}\}&\{e_{4},e_{15},e_{6},-e_{13}\}\\\{e_{5},e_{10},e_{6},e_{9}\}&\{e_{5},e_{9},e_{6},-e_{10}\}\\\{e_{5},e_{11},e_{7},e_{9}\}&\{e_{5},e_{9},e_{7},-e_{11}\}\\\{e_{5},e_{12},e_{7},e_{14}\}&\{e_{5},e_{12},e_{6},-e_{15}\}\\\{e_{5},e_{15},e_{6},e_{12}\}&\{e_{5},e_{14},e_{7},-e_{12}\}\\\{e_{6},e_{11},e_{7},e_{10}\}&\{e_{6},e_{10},e_{7},-e_{11}\}\\\{e_{6},e_{13},e_{7},e_{12}\}&\{e_{6},e_{10},e_{7},-e_{13}\}\end{array}}\end{array}}}$

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