Elementary Relationships

From the above definitions, one can quickly verify

$$\begin{eqnarray*} z+\overline{z} &=& 2 \, \mbox{re}\left\{z\right\} \\ z-\overline{z} &=& 2\, j\,\, \mbox{im}\left\{z\right\} \\ z\overline{z} &=& \left\vert z\right\vert^2. \end{eqnarray*}$$

Let's verify the third relationship which states that a complex number multiplied by its conjugate is equal to its magnitude squared:

$$z \overline{z} \isdef (x+jy)(x-jy) = x^2-(jy)^2 = x^2 + y^2 \isdef \vert z\vert^2 \protect$$ (2.4)