There are 7 hypercomplex octonion units: e

e

e

This is a property they share with the complex algebra.

Like the quaternion algebra the hypercomplex units anticommute:

e

for distinct a and b from 1 to 7. But unlike either the complexes or

quaternions, the octonions do not associate.

I'll look at the D(+) and C(+) multiplication tables here.

They are dual to each other, each sharing the same elegant properties.

In particular, in both cases, if one has

e

(a,b,c integers from 1 to 7), then that immediately implies that

e

and

e

and

e

(k an integer), where in all cases the indices are taken from 1 to 7, modulo 7.

Note, because of this modulo 7 property, e

so those first two properties hold in general for any power of 2 in the subscript,

but nothing new is gained after 2

It is worth noting that these

properties make it easy to generalize particular products and properties to the whole algebra.

Page 0. | Page 3. | Contents | Octoshop III page.