Octonion

In mathematics, the octonion number system extends the complex numbers into eight dimensions. It is represented using the symbol . The 16-dimensional sedenions come after the octonions.

History
The octonions were first described by Irish mathematician John T. Graves in 1843, who originally called them "octaves". They were independently described by Arthur Cayley in 1845.

Introduction
The octonions take on the following form, with 8 total elements.  is a real number, and the others are imaginary units belonging to 7 different dimensions.

Multiplication

The Fano plane is a diagram that shows how another octonion element is obtained when two octonion elements are multiplied with each other.

The two examples below illustrate how a positive product is obtained when moving along with directions of the arrows in the Fano plane.

The two examples below illustrate how a negative product is obtained when moving against the directions of the arrows in the Fano plane.

Both quaternions and octonions are non-commutative, but octonions are also non-associative. However, quaternions are associative. The example below shows how the results of multiplying , ,  change when they are grouped differently (in order words, when the order of operations differ).

References

Other websites

Category:Algebra
Category:Numbers