Octonion

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In mathematics, the octonions are a non-associative extension of the quaternions.

History

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.

Algebraic operations

The octonions form an 8-dimensional (non-associative) division algebra over the real numbers, and can therefore be thought of as octets (or 8-tuples) of real numbers. Every octonion is a real linear combination of the unit octonions 1, e₁, e₂, e₃, e₄, e₅, e₆ and e₇.

Addition of octonions is accomplished by adding corresponding coefficients, as with the complex numbers and quaternions. By linearity, multiplication of octonions is completely determined by the multiplication table for the unit octonions; this table is given below.

�	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

The Fano plane can be used as a mnemonic for remembering the products of unit octonions. See Fano plane mnemonic.

Properties

The octonions are the only alternative but not associative finite-dimensional division algebra over the reals. The only finite-dimensional associative division algebras over the reals are the real numbers, the complex numbers, and the quaternions.

The group of automorphisms (symmetries) of the octonions is called G2.