Editing Octonion (section)

===Conjugate, norm, and inverse===
The ''conjugate'' of an octonion

:<math> x = x_0\ e_0 + x_1\ e_1 + x_2\ e_2 + x_3\ e_3 + x_4\ e_4 + x_5\ e_5 + x_6\ e_6 + x_7\ e_7 </math>

is given by

:<math> x^* = x_0\ e_0 - x_1\ e_1 - x_2\ e_2 - x_3\ e_3 - x_4\ e_4 - x_5\ e_5 - x_6\ e_6 - x_7\ e_7 ~.</math>
Conjugation is an [[involution (mathematics)|involution]] of <math>\ \mathbb{O}\ </math> and satisfies {{math|(''xy'')* {{=}} ''y''*''x''*}} (note the change in order).

The ''real part'' of {{mvar|x}} is given by

:<math>\frac{x + x^*}{2} = x_0\ e_0</math>

and the ''imaginary part'' (sometimes called the ''pure part'') by

:<math> \frac{x - x^*}{2} = x_1\ e_1 + x_2\ e_2 + x_3\ e_3 + x_4\ e_4 + x_5\ e_5 + x_6\ e_6 + x_7\ e_7 ~.</math>

The set of all purely imaginary octonions [[linear span|spans]] a 7&nbsp;[[dimension (vector space)|dimensional]] [[linear subspace|subspace]] of <math>\ \mathbb{O}\ ,</math> denoted <math>\ \operatorname\mathcal{I_m}\bigl[\ \mathbb{O}\ \bigr] ~.</math>

Conjugation of octonions satisfies the equation

:<math> -6 x^* = x + (e_1x)e_1 + (e_2x)e_2 + (e_3x)e_3 + (e_4x)e_4 + (e_5x)e_5 + (e_6x)e_6 + (e_7x)e_7 ~.</math>

The product of an octonion with its conjugate, {{nobr| {{math|''x''*''x'' {{=}} ''xx''*}} ,}} is always a nonnegative real number:

:<math>x^*x = x_0^2 + x_1^2 + x_2^2 + x_3^2 + x_4^2 + x_5^2 + x_6^2 + x_7^2 ~.</math>

Using this, the norm of an octonion is defined as

:<math>\|x\| = \sqrt{x^*x} ~.</math>

This norm agrees with the standard 8&nbsp;dimensional [[Euclidean norm]] on {{math|ℝ<sup>8</sup>}}.

The existence of a norm on <math>\ \mathbb{O}\ </math> implies the existence of [[inverse element|inverses]] for every nonzero element of <math>\ \mathbb{O} ~.</math> The inverse of{{nobr| {{math| ''x'' ≠ 0}} ,}} which is the unique octonion {{math|''x''<sup>−1</sup>}} satisfying {{nobr|{{math| ''x x''<sup>−1</sup> {{=}} ''x''<sup>−1</sup>''x'' {{=}} 1}} ,}} is given by

:<math>x^{-1} = \frac {x^*}{\|x\|^2} ~.</math>