Sum of Complex Number with Conjugate

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:

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

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

$\blacksquare$

This result is also reported as:

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

1957: E.G. Phillips: Functions of a Complex Variable (8th ed.) ... (previous) ... (next): Chapter $\text I$: Functions of a Complex Variable: $\S 2$. Conjugate Complex Numbers
1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Supplementary Problems: Fundamental Operations with Complex Numbers: $58 \ \text{(a)}$
1990: H.A. Priestley: Introduction to Complex Analysis (revised ed.) ... (previous) ... (next): $1$ The complex plane: Complex numbers $\S 1.3$ Complex conjugation: $(2)$
2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): conjugate (of a complex number)
2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): conjugate (of a complex number)