We can use the concept of complex conjugate to give a strategy for dividing two complex numbers, $z_1 = x_1 + i y_1$ and $z_2 = x_2 + i y_2$. The trick is to multiply by the number 1, in a special form that simplifies the denominator to be a real number and turns division into multiplication.

Consider: \begin{eqnarray} \frac{z_1}{z_2}&=&\frac{z_1}{z_2} \frac{z_2^*}{z_2^*}\\ &=&\frac{z_1 z_2^*}{\vert z \vert ^2} \end{eqnarray}

You should try this by writing out the real an imaginary components separately. \begin{eqnarray} \frac{z_1}{z_2} &=&\frac{x_1 + i y_1}{x_2 +i y_2}\\ &=&\left(\frac{x_1 + i y_1}{x_2 +i y_2}\right) \left(\frac{x_2 - i y_2}{x_2 - i y_2}\right)\\ &=&\frac{(x_1 y_1 - x_2 y_2) + i (x_1 y_2 + x_2 y_1)}{x_2^2 +y_2^2}\\ &=&\left(\frac{x_1 y_1 - x_2 y_2}{x_2^2 + y_2^2}\right) + i\left(\frac{x_1 y_2 + x_2 y_1}{x_2^2 + y_2^2}\right) \end{eqnarray}