# Definition:Octonion

## Definition

The set of **octonions**, usually denoted $\Bbb O$, can be defined by using the Cayley-Dickson construction from the quaternions $\Bbb H$ as follows:

From Quaternions form Algebra, $\Bbb H$ forms a nicely normed $*$-algebra.

Let $a, b \in \Bbb H$.

Then $\tuple {a, b} \in \Bbb O$, where:

- $\tuple {a, b} \tuple {c, d} = \tuple {a c - d \overline b, \overline a d + c b}$
- $\overline {\tuple {a, b} } = \tuple {\overline a, -b}$

where:

- $\overline a$ is the conjugate on $a$

and

- $\overline {\tuple {a, b} }$ is the conjugation operation on $\Bbb O$.

### Octonion Addition

Let $x = \tuple {a, b}$ and $y = \tuple {c, d}$ be octonions, where $a, b, c, d \in \H$ are quaternions.

The **sum** of $x$ and $y$ is defined as:

- $x + y = \tuple {a, b} + \tuple {c, d} = \tuple {a + c, b + d}$

## Also known as

The **octonions** are sometimes referred to as the **Cayley numbers**, for Arthur Cayley.

Some sources report them as the **Graves-Cayley Numbers**, for John Thomas Graves, who actually discovered them.

## Also see

- Results about
**octonions**can be found**here**.

## Historical Note

The **octonions** were discovered by John Thomas Graves in December $1843$, in the wake of the work he did with William Rowan Hamilton on the quaternions.

However, he never made it into print, and it was Arthur Cayley who published his own work on **octonions** in $1845$, hence claiming the credit.

Graves' precedent was discovered in $1847$ on the evidence of a letter he had written to Hamilton on the occasion of his initial discovery of them.

## Sources

- 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {B}.26$: Extensions of the Complex Number System. Algebras, Quaternions, and Lagrange's Four Squares Theorem - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**octonion**