Sum of Complex Conjugates

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:

\(\ds \overline {z_1 + z_2}\)

\(=\)

\(\ds \overline {\paren {x_1 + x_2} + i \paren {y_1 + y_2} }\)

\(\ds \)

\(=\)

\(\ds \paren {x_1 + x_2} - i \paren {y_1 + y_2}\)

Definition of Complex Conjugate

\(\ds \)

\(=\)

\(\ds \paren {x_1 - i y_1} + \paren {x_2 - i y_2}\)

Definition of Complex Addition

\(\ds \)

\(=\)

\(\ds \overline {z_1} + \overline {z_2}\)

Definition of Complex Conjugate

$\blacksquare$

Sources

• 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
• 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 1.2$. The Algebraic Theory: $(1.6)$
• 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Solved Problems: Fundamental Operations with Complex Numbers: $4 \ \text{(a)}$
• 1990: H.A. Priestley: Introduction to Complex Analysis (revised ed.) ... (previous) ... (next): $1$ The complex plane: Complex numbers $\S 1.3$ Complex conjugation: $(4)$
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): conjugate (of a complex number)