The
octonions are a non-
associative extension of the
quaternions.
They were discovered by
John T. Graves[?] in
1843, and independently by
Arthur Cayley[?], who published the first paper on them in
1845.
They are sometimes referred to as
Cayley numbers or the
Cayley algebra.
The octonions form an 8-dimensional algebra over the real numbers, and can therefore be thought of as octets of real numbers.
Every octonion is a real linear combination of the unit octonions 1, e1, e2, e3, e4, e5, e6 and e7,
the multiplication table for which looks as follows.
| ·
| 1
| e1
| e2
| e3
| e4
| e5
| e6
| e7
|
| 1
| 1
| e1
| e2
| e3
| e4
| e5
| e6
| e7
|
| e1
| e1
| -1
| e4
| e7
| -e2
| e6
| -e5
| -e3
|
| e2
| e2
| -e4
| -1
| e5
| e1
| -e3
| e7
| -e6
|
| e3
| e3
| -e7
| -e5
| -1
| e6
| e2
| -e4
| e1
|
| e4
| e4
| e2
| -e1
| -e6
| -1
| e7
| e3
| -e5
|
| e5
| e5
| -e6
| e3
| -e2
| -e7
| -1
| e1
| e4
|
| e6
| e6
| e5
| -e7
| e4
| -e3
| -e1
| -1
| e2
|
| e7
| e7
| e3
| e6
| -e1
| e5
| -e4
| -e2
| -1
|
See also Hypercomplex numbers.
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