Complex Conjugation is Involution

Theorem

Let $z = x + i y$ be a complex number.

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

Then the operation of complex conjugation is an involution:

$\overline {\paren {\overline z} } = z$

Proof

\(\ds \overline {\paren {\overline z} }\)

\(=\)

\(\ds \overline {\paren {\overline {x + i y} } }\)

Definition of $z$

\(\ds \)

\(=\)

\(\ds \overline {x - i y}\)

Definition of Complex Conjugate

\(\ds \)

\(=\)

\(\ds x + i y\)

Definition of Complex Conjugate

\(\ds \)

\(=\)

\(\ds z\)

Definition of $z$

$\blacksquare$

Sources

• 1990: H.A. Priestley: Introduction to Complex Analysis (revised ed.) ... (previous) ... (next): $1$ The complex plane: Complex numbers $\S 1.3$ Complex conjugation: $(1)$
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): conjugate (of a complex number)