Prove que $\overline{Z_1 + Z_2} = \overline{Z_1} +\overline{ Z_2}$, onde $Z_1$ e $Z_2 \in \mathbb{C}$.

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ITA IIIT 27/02/2022 22:25
$• \ \text{Forma geométrica:}$ $Z_1 = (a,b) \ \ , \ \ Z_2 = (x,y)$ \begin{matrix} \overline{Z_1} = (a,-b) &,& \overline{Z_2} = (x,-y) \\ \\ \overline{Z_1} + \overline{Z_1} &=& (a+x,-b-y) \end{matrix} Por outro lado, \begin{matrix} Z_1 + Z_2 &=& (a+x,b+y) \\ \\ \overline{Z_1 + Z_2} &=& (a+x,-b-y) \end{matrix} \begin{matrix} \overline{Z_1 + Z_2} &=& \overline{Z_1} + \overline{Z_1} \end{matrix} $• \ \text{Forma algébrica:}$ $Z_1 = a + bi\ \ , \ \ Z_2 = x + yi$ \begin{matrix} \overline{Z_1} = a - bi &,& \overline{Z_2} = x - yi\\ \\ \overline{Z_1} + \overline{Z_1} &=& (a+x) -(b+y)i \end{matrix} Por outro lado, \begin{matrix} Z_1 + Z_2 &=& (a+x) + (b+y)i \\ \\ \overline{Z_1 + Z_2} &=& (a+x) - (b+y)i \end{matrix} \begin{matrix} \overline{Z_1 + Z_2} &=& \overline{Z_1} + \overline{Z_1} \end{matrix}
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