Chapter 7.2, Problem 41E

Exercise $38-42$ are stated using the notation in the last paragraph of Example $3$.

Prove or disprove $\overline{q_1{q}_2}=\overline{q_1}\overline{q_2}$ for all $q_1,q_2\in H$.

Example $3$: In this example we outline the development of the quaternions as the set

$H=\{(x,y,z,w)|x,y,z,w\in ℝ\}$,

For each $q=x+yi+zj+wk$ in $H$, we imitate conjugates in $ℂ$ and write

$\bar{q}=x-yi-zj-wk$,

It is left as exercise to verify that

$\bar{q}q=q\bar{q}=x^2+y^2+z^2+w^2$

If $q\ne 0$, then $\bar{q}q\ne 0$, and

$q^{-1}=\left(\frac{1}{\bar{q}q}\right)\bar{q}$.

Thus $H$ has all the field properties except commutative multiplication.