Proving that a complex number is nonreal.

Question

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.

Answer 1

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.

Answer 2

$$\forall m \in \mathbb{R}^*, Re\left(\frac{m-w}{z-w}\right)=\frac{1}{2}\ne 0$$ either you made a typo or the claim is wrong