Why take the complex conjugate

Question

If I have the equation:

$$\frac{250+0.915j}{350+0.915j}$$

Why do I have to take the complex conjugate to rid of the complex part on the bottom?

Why can't I just times by:

$$ \frac{250+0.915j}{350+0.915j} \frac{-j}{-j}$$

Answer 1

Because then you get $$\frac {-250j+0.915}{-350j+0.915}$$ and you still don't have a real number in the denominator.

The conjugate method depends on the fact that $(a+jb)(a-jb)=a^2+b^2$ is a non-zero (positive) real number unless $a=b=0$.