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)
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