Sum of Complex Number with Conjugate

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Theorem

Let $z \in \C$ be a complex number.

Let $\overline z$ be the complex conjugate of $z$.

Let $\map \Re z$ be the real part of $z$.


Then:

$z + \overline z = 2 \, \map \Re z$


Proof

Let $z = x + i y$.

Then:

\(\ds z + \overline z\) \(=\) \(\ds \paren {x + i y} + \paren {x - i y}\) Definition of Complex Conjugate
\(\ds \) \(=\) \(\ds 2 x\)
\(\ds \) \(=\) \(\ds 2 \, \map \Re z\) Definition of Real Part

$\blacksquare$


Also defined as

This result is also reported as:

$\map \Re z = \dfrac {z + \overline z} 2$


Sources

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