TITLE: Proving that a complex number is nonreal.
QUESTION [5 upvotes]: Let $m$ be a nonzero complex number such that and $z=-1+im$ and $w=-1-im$.
Prove that the number $$\frac{m-w}{z-w}$$ is nonreal.
I've tried all sorts of approaches to this question but there seems to be something I'm missing.
Any help would be appreciated, thank you in advance.
EDIT: My bad, I made a mistake in the text.

REPLY [5 votes]: This statement seems to be wrong.
$${m-\omega\over z-\omega}={-i\over 2m}+{-(1+i)i\over 2}={-i\over 2m}+{1-i\over 2}.$$
Now, write $m$ as $m=x+iy\;$ for some real numbers $x$ and $y$ with $x\neq 0\neq y$. Then,
$${-i\over 2m}+{1-i\over 2}={1-i\over 2} - {i\over 2(x + i y)}.$$
Out of this you get that,
$$\text{Re}\left({m-\omega\over z-\omega}\right)={1\over 2}-{y\over 2(x^2+y^2)},$$
which is equal to zero only for certain values of $x$ and $y$. Therefore is not true that ${m-\omega\over z-\omega}$ is imaginary.