Operations on two complex numbers

Let $z = x + j y = r e^{j \theta}$, $z_1 = x_1 + j y_1 = r_1 e^{j \theta_1}$ and $z_2 = x_2 + j y_2 = r_2 e^{j \theta_2}$

(1)
\begin{eqnarray} z_1 \pm z_2 & = & (x_1 \pm x_2) + j (y_1 \pm y_2) \\ z_1 z_2 & = & (x_1 x_2 - y_1 y_2) + j (x_1 y_2 + x_2 y_1) = r_1 r_2 e^{j (\theta_1+\theta_2)} \\ \nonumber z z^* &=& x^2 + y^2 = r^2 \\ \nonumber |z| &=& \sqrt{zz^*} = r\\ \nonumber \frac{z_1}{z_2} &=& \frac{(x_1+jy_1)}{(x_2+jy_2)} = \frac{(x_1+jy_1)(x_2-jy_2)}{x_2^2+y_2^2} = \frac{r_1}{r_2} e^{j (\theta_1-\theta_2)} \end{eqnarray}

Based on the above operations, the following facts about complex number can be verified.

(2)
\begin{eqnarray} \nonumber (z_1+z_2)^* & = & z_1^* + z_2^*\\ \nonumber (z_1 \ z_2)^* & = & z_1^* \ z_2^* \\ \nonumber \left(\frac{z_1}{z_2}\right)^* & = & \frac{z_1^*}{z_2^*} \\ \nonumber |z_1-z_2| &=& \sqrt{(x_1-x_2)^2+(y_1-y_2)^2}\\ \nonumber |z_1z_2| &=& |z_1||z_2| = r_1 r_2 \\ \nonumber \left|\frac{z_1}{z_2}\right| &=& \frac{|z_1|}{|z_2|} = \frac{r_1}{r_2} \end{eqnarray}

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