I'm pretty new to quantum mechanics (I would like to understand it broadly as an hobbyist). I'm trying to reading Principles of Quantum Mechanics by Dirac. I've found difficult to understand a particular piece on p. 28 in the II chapter. It says that:
the conjugate complex of the product of two linear operators equals the product of the conjugate complexes of the factors in reverse order: $$\overline{\beta} \overline{\alpha} = \overline{\alpha \beta}\tag{$*$} $$
As simple examples of this result, it should be noted that, if $\xi$ and $\eta$ are real, in general $\xi \eta$ is not real. This is an important difference from classical mechanics. However, $\xi \eta + \eta \xi$ is real, and so is $i(\xi \eta - \eta \xi)$. Only when $\xi$ and $\eta$ commute is $\xi \eta$ itself also real.
I didn't understand why $\xi \eta$ is not real in general and why $\xi \eta + \eta \xi$ and $i(\xi \eta - \eta \xi)$ are, even if $\xi$ and $\eta$ are real.