Octonion

In 1845 Cayley made a further generalisation when he created the octonions O. (In fact they were also independently invented by J.J. Graves slightly earlier.) This time O is 8-dimensional over R. A typical octonion is a 2x2 matrix

 --  --
| a  r |
| s  b |
 --  --

 --  --  --  --       --                         --
| a  r || c  t |  =  | ac - r.v       at + dr +sxv |
| s  b || v  d |     | sc + bv + rxt  bd - s.t     |
 --  --  --  --       --                         --

In O most of the axioms for a ring still hold (the identity element is the 2x2 identity matrix) and each nonzero element has an inverse but associativity does not hold. The main interest in the octonions is that they can be used to construct g₂, an important non-classical Lie algebra. (Note that the same construction will work with any base field replacing R.)

Also known as Cayley numbers, after their 19th century inventor Arthur Cayley.

Octonions make use of seven unique roots of -1, labelled i₀ to i₆. Addition and subtraction of them is very straight forward, just as for complex numbers and quaternions. However, multiplication is more complex. In short, each of the following triplets behaves like the i, j and k of quaternions. Follow that link if you're unsure.

(i₀, i₁, i₃)
(i₁, i₂, i₄)
(i₂, i₃, i₅)
(i₃, i₄, i₆)
(i₄, i₅, i₀)
(i₅, i₆, i₁)
(i₆, i₀, i₂)

Multiplication of two general octonions is a pretty arduous business by hand, but it is theoretically possible. You can read about it over on multiplying octonions.

Octonions have the highest number of units with which division can be everywhere defined, except by zero. Therefore progressing to 16-ions and beyond is somewhat futile.

However, associativity does not hold for octonions. Therefore (ab)c != a(bc). This takes some getting used, as with complex numbers and even quaternions, we automatically expect it to work.

One of the few practical uses for octonions is for describing rotation and translation in seven and eight dimensional space, but I wouldn't claim to be the authority on that.