Difference of Complex Number with Conjugate

Theorem

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

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

Let $\map \Im z$ be the imaginary part of $z$.


Then

$z - \overline z = 2 i \, \map \Im z$


Proof

Let $z = x + i y$.

Then:

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

$\blacksquare$


Also defined as

This result is also reported as:

$\map \Im z = \dfrac {z - \overline z} {2 i}$


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
  • 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Supplementary Problems: Fundamental Operations with Complex Numbers: $58 \ \text{(b)}$
  • 1990: H.A. Priestley: Introduction to Complex Analysis (revised ed.) ... (previous) ... (next): $1$ The complex plane: Complex numbers $\S 1.3$ Complex conjugation: $(2)$
  • 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): conjugate (of a complex number)
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