Octonion

The (real) octaves or octonions also Cayleyzahlen, are an extension of quaternions and have the quantity symbol. They result from the application of the doubling method of quaternions and form an alternative body. Thus, they provide a range of coordinates is an example of a real, that is nichtdesarguessche Moufangebene in synthetic geometry.

History

The octonions were described in a letter to William Rowan Hamilton for the first time in 1843 by John Thomas Graves. Regardless, they were of Arthur Cayley published in 1845 ( as the first ).

Multiplication table

The octonions are an 8 -dimensional algebra over the real numbers. A multiplication is possible - with the base - is given as follows:

That calculates the product of the units

Properties

The octonions are a division algebra with unit element.

They do not form a skew field (and thus no body ), because they violate the

It is, however, for all the octaves A and B:

This property is called alternativeness. The octonions form an alternative body.

From alternativeness the relationship follows

This relationship is also called Flexibility Act.

The octonions also satisfy the Moufang identities

And

Application of the doubling method to the octaves provides the Sedenionen.

Representations

Each octave can be represented ...

The field of real numbers can be considered as a sub- structure:

Corresponds

The field of complex numbers can be considered as a sub- structure:

Corresponds

The skew field of quaternions can be considered as a structure of:

Corresponds

For the octaves addition and multiplication are defined so that they are backwards compatible, which means ...

Possible applications

Octaves could be used to describe an eight-dimensional supersymmetry. This also potential applications would result in the context of string theory and M-theory, since both are based on supersymmetry.