(related to Proposition: Calculating with Complex Conjugates)
The calculation rules follow immediately from the definition of complex conjugates. As an illustrative example, we verify the rule \((3)\). For two complex numbers \[\begin{array}{rcl} z_1 & := & \alpha_1 + \beta_1 i\\ z_2 & := & \alpha_2 + \beta_2 i\\ \end{array}\] we have \[\begin{array}{rcl} (z_1\cdot z_2)^*&=&((\alpha_1 + \beta_1 i)\cdot(\alpha_2 + \beta_2 i))^*\\ &=&((\alpha_1\alpha_2 - \beta_1\beta_2)+(\alpha_1\beta_2+\beta_1\alpha_2) i)^*\\ &=&(\alpha_1\alpha_2 - \beta_1\beta_2)-(\alpha_1\beta_2+\beta_1\alpha_2) i\\ &=&(\alpha_1 - \beta_1 i)\cdot(\alpha_2 - \beta_2 i)\\ &=&(\alpha_1 + \beta_1 i)^*\cdot (\alpha_2 + \beta_2 i)^*\\ &=&z_1^*\cdot z_2^*. \end{array}\]
