Sum of Complex Conjugates

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Theorem

Let $z_1, z_2 \in \C$ be complex numbers.

Let $\overline z$ denote the complex conjugate of the complex number $z$.


Then:

$\overline {z_1 + z_2} = \overline {z_1} + \overline {z_2}$


Proof

Let $z_1 = x_1 + i y_1, z_2 = x_2 + i y_2$.

Then:

\(\displaystyle \overline {z_1 + z_2}\) \(=\) \(\displaystyle \overline {\paren {x_1 + x_2} + i \paren {y_1 + y_2} }\)
\(\displaystyle \) \(=\) \(\displaystyle \paren {x_1 + x_2} - i \paren {y_1 + y_2}\) Definition of Complex Conjugate
\(\displaystyle \) \(=\) \(\displaystyle \paren {x_1 - i y_1} + \paren {x_2 - i y_2}\) Definition of Complex Addition
\(\displaystyle \) \(=\) \(\displaystyle \overline {z_1} + \overline {z_2}\) Definition of Complex Conjugate

$\blacksquare$


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