Sedenion

the Sedenionen (symbol <math> \ mathbb S< /math>) are 16dimensionale hypercomplex numbers.

They result from the application of the doubling procedure from the Oktonionen.

The multiplication of the Sedenionen is neither commutative nor associative and isalso not alternatively. It is only power associative. Furthermore possesses the Sedenionen zero-divisors.

Each Sedenion is a real linear combination of the units 1, e 1, e 2, e 3, e 4, e 5, e 6, e 7, e 8, e 9, e 10, e 11, e 12, e 13, e 14 and e 15.

The multiplication board of the units is:

×< /td> 1< /td>
e 1
< /td>
e 2
< /td>
e 3
< /td>
e 4
< /td>
e 5
< /td>
e 6
< /td>
e 7
< /td>
e 8
< /td>
e 9
< /td>
e 10
< /td>
e 11
< /td>
e 12
< /td>
e 13
< /td>
e 14
< /td>
e 15
< /td>

</tr>

</td>< /tr>
1
< /td>
1
< /td>
e 1
< /td>
e 2
< /td>
e 3
< /td>
e 4
< /td>
e 5
< /td>
e 6
< /td>
e 7
< /td>
e 8
< /td>
e 9
< /td>
e 10
< /td>
e 11
< /td>
e 12
< /td>
e 13
< /td>
e 14
< /td>
e 15
< /td>

</tr>

e 1
< /td>
e 1
< /td>
-1
< /td>
e 3
< /td>
- e 2
< /td>
e 5
< /td>
- e 4
< /td>
- e 7
< /td>
e 6
< /td>
e 9
< /td>
- e 8
< /td>
- e 11
< /td>
e 10
< /td>
- e 13
< /td>
e 12
< /td>
e 15
< /td>
- e 14
< /td>

</tr>

e 2
< /td>
e 2
< /td>
- e 3
< /td>
-1
< /td>
e 1
< /td>
e 6
< /td>
e 7
< /td>
- e 4
< /td>
- e 5
< /td>
e 10
< /td>
e 11
< /td>
- e 8
< /td>
- e 9
< /td>
- e 14
< /td>
- e 15
< /td>
e 12
< /td>
e 13
< /td>

</tr>

e 3
< /td>
e 3
< /td>
e 2
< /td>
- e 1
< /td>
-1
< /td>
e 7
< /td>
- e 6
< /td>
e 5
< /td>
- e 4
< /td>
e 11
< /td>
- e 10
< /td>
e 9
< /td>
- e 8
< /td>
- e 15
< /td>
e 14
< /td>
- e 13
< /td>
e 12
< /td>

</tr>

e 4
< /td>
e 4
< /td>
- e 5
< /td>
- e 6
< /td>
- e 7
< /td>
-1
< /td>
e 1
< /td>
e 2
< /td>
e 3
< /td>
e 12
< /td>
e 13
< /td>
e 14
< /td>
e 15
< /td>
- e 8
< /td>
- e 9
< /td>
- e 10
< /td>
- e 11
< /td>

</tr>

e 5
< /td>
e 5
< /td>
e 4
< /td>
- e 7
< /td>
e 6
< /td>
- e 1
< /td>
-1
< /td>
- e 3
< /td>
e 2
< /td>
e 13
< /td>
- e 12
< /td>
e 15
< /td>
- e 14
< /td>
e 9
< /td>
- e 8
< /td>
e 11
< /td>
- e 10
< /td>

</tr>

e 6
< /td>
e 6
< /td>
e 7
< /td>
e 4
< /td>
- e 5
< /td>
- e 2
< /td>
e 3
< /td>
-1
< /td>
- e 1
< /td>
e 14
< /td>
- e 15
< /td>
- e 12
< /td>
e 13
< /td>
e 10
< /td>
- e 11
< /td>
- e 8
< /td>
e 9
< /td>

</tr>

e 7
< /td>
e 7
< /td>
- e 6
< /td>
e 5
< /td>
e 4
< /td>
- e 3
< /td>
- e 2
< /td>
e 1
< /td>
-1
< /td>
e 15
< /td>
e 14
< /td>
- e 13
< /td>
- e 12
< /td>
e 11
< /td>
e 10
< /td>
- e 9
< /td>
- e 8
< /td>

</tr>

e 8
< /td>
e 8
< /td>
- e 9
< /td>
- e 10
< /td>
- e 11
< /td>
- e 12
< /td>
- e 13
< /td>
- e 14
< /td>
- e 15
< /td>
-1
< /td>
e 1
< /td>
e 2
< /td>
e 3
< /td>
e 4
< /td>
e 5
< /td>
e 6
< /td>
e 7
< /td>

</tr>

e 9
< /td>
e 9
< /td>
e 8
< /td>
- e 11
< /td>
e 10
< /td>
- e 13
< /td>
e 12
< /td>
e 15
< /td>
- e 14
< /td>
- e 1
< /td>
-1
< /td>
- e 3
< /td>
e 2
< /td>
- e 5
< /td>
e 4
< /td>
e 7
< /td>
- e 6
< /td>

</tr>

e 10
< /td>
e 10
< /td>
e 11
< /td>
e 8
< /td>
- e 9
< /td>
- e 14
< /td>
- e 15
< /td>
e 12
< /td>
e 13
< /td>
- e 2
< /td>
e 3
< /td>
-1
< /td>
- e 1
< /td>
- e 6
< /td>
- e 7
< /td>
e 4
< /td>
e 5
< /td>

</tr>

e 11
< /td>
e 11
< /td>
- e 10
< /td>
e 9
< /td>
e 8
< /td>
- e 15
< /td>
e 14
< /td>
- e 13
< /td>
e 12
< /td>
- e 3
< /td>
- e 2
< /td>
e 1
< /td>
-1
< /td>
- e 7
< /td>
e 6
< /td>
- e 5
< /td>
e 4
< /td>

</tr>

e 12
< /td>
e 12
< /td>
e 13
< /td>
e 14
< /td>
e 15
< /td>
e 8
< /td>
- e 9
< /td>
- e 10
< /td>
- e 11
< /td>
- e 4
< /td>
e 5
< /td>
e 6
< /td>
e 7
< /td>
-1
< /td>
- e 1
< /td>
- e 2
< /td>
- e 3
< /td>

</tr>

e 13
< /td>
e 13
< /td>
- e 12
< /td>
e 15
< /td>
- e 14
< /td>
e 9
< /td>
e 8
< /td>
e 11
< /td>
- e 10
< /td>
- e 5
< /td>
- e 4
< /td>
e 7
< /td>
- e 6
< /td>
e 1
< /td>
-1
< /td>
e 3
< /td>
- e 2
< /td>

</tr>

e 14
< /td>
e 14
< /td>
- e 15
< /td>
- e 12
< /td>
e 13
< /td>
e 10
< /td>
- e 11
< /td>
e 8
< /td>
e 9
< /td>
- e 6
< /td>
- e 7
< /td>
- e 4
< /td>
e 5
< /td>
e 2
< /td>
- e 3
< /td>
-1
< /td>
e 1
< /td>

</tr>

e 15
< /td>
e 15
< /td>
e 14
< /td>
- e 13
< /td>
- e 12
< /td>
e 11
< /td>
e 10
< /td>
- e 9
< /td>
e 8
< /td>
- e 7
< /td>
e 6
< /td>
- e 5
< /td>
- e 4
< /td>
e 3
< /td>
e 2
< /td>
- e 1
< /td>
-1
< /td>

</tr>

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