Yesterday I wrote about the fact that quaternions, unlike complex numbers, can form conjugates via a series of multiplications and additions. This post will show that you can do something similar with octonions.

If *x* is an octonion

*x* = *r*_{0} *e*_{0} + *r*_{1} *e*_{1} + … + *r*_{7} *e*_{7}

where all the *r*‘s are real numbers. The conjugate flips the signs of all the components except the real component:

*x** = *r*_{0} *e*_{0} – *r*_{1} *e*_{1} – … – *r*_{7} *e*_{7}

The conjugate theorem is

*x** = – (*x* + (*e*_{1} *x*) *e*_{1} + … (*e*_{7} *x*) *e*_{7}) / 6

which is analogous to the theorem

*q** = – (*q* + *iqi* + *jqj* + *kqk*) /2

for quaternions.

The internal parentheses are necessary because multiplication in octonions is not associative:

*xyz*

is ambiguous because it could mean

(*xy*)*z*

or

*x*(*yz*)

and the two are not necessarily equal.

The proof is also analogous to the proof given in the earlier post for quaternions. First work out what the effect is of multiplying on the left and right by one of the imaginary units, then add everything up. You’ll find that the real component is multiplied by -6 and the rest of the components are multiplied by 6.

While octonionic multiplication is not generally associative, i.e.

(a*b)*c != a*(b*c)

it is alternative, so

(a*b)*a == a*(b*a)

and the internal parentheses are unnecessary so you can just write

x* = – (x + e1 x e1 + … e7 x e7) / 6