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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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