[Math] Complex numbers – separate real/imaginary parts

Question

$$K(\omega) = \frac{1}{1 + j\omega RC}$$

Uhm…How do I separate the real part from the imaginary part here? :U And how can I find the argument? I mean, if the document I got this formula from is right, the argument should be negative…

Answer 1

If you have $\frac{z_1}{z_2}$ a ratio of complex numbers, multiplying $$ \frac{z_1}{z_2}\cdot\frac{\overline{z_2}}{\overline{z_2}} = \frac{z_1\overline{z_2}}{z_2\overline{z_2}} $$ will give you a real denominator ($\overline{z_2}$ is the complex conjugate of $z_2$). Once you do that, you need only write $z_1\overline{z_2} = x + jy$, $x,y\in\Bbb R$, as then $$ \frac{z_1}{z_2} = \frac{x + jy}{z_2\overline{z_2}} = \frac{x}{z_2\overline{z_2}} + j\frac{y}{z_2\overline{z_2}}. $$