Finding conjugate of a complex number

Question

I am stuck with a really silly question :

What is the conjugate of $a\bar c-\bar ac$ ?

I calculated it as $\bar ac-a\bar c$ but according to my lecture notes, its conjugate is $a\bar c-\bar ac$ itself, i don't understand how's this happening ?

Can someone please shed some light on this.Thanks.

Answer 1

Your notes are wrong, your calculations are correct. Take $a = 1, c = i$. Then

$$ a\overline c - \overline a c = 1\overline i - \overline 1 i = -i - i = -2i $$

which is not its own conjugate.

On the other hand

$$ \overline a c - a \overline c = 1i - 1\overline i = i + i = 2i $$

gives you the correct result. (of course this doesn't demonstrate the rule in general, but examples are good for verifying your calculations)

Answer 2

Generally $\overline{\overline{z}} = z$, $\overline{zw} = \overline{z}\, \overline{w}$ and $\overline{z+w} = \overline{z} + \overline{w}$.

Hence $\overline{a\overline{c}-\overline{a} c} = \overline{a\overline{c}} - \overline{\overline{a} c } = \overline{a} \overline{\overline{c}} - \overline{\overline{a}}\overline{c} = \overline{a} c - a \overline{c}$.