Complex conjugate

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The complex conjugate of a complex number $z = a + bi$ is the complex number $\overline{z} = a - bi$.

Geometrically, if $z$ is a point in the complex plane, $\overline z$ is the reflection of $z$ across the real axis.

Properties

Conjugation is its own functional inverse and commutes with the usual operations on complex numbers:

  • $\overline{(\overline z)} = z$.
  • $\overline{(w \cdot z)} = \overline{w} \cdot \overline{z}$. ($\overline{(\frac{w}{z})}$ is the same as $\overline{(w \cdot \frac{1}{z})}$)
  • $\overline{(w + z)} = \overline{w} + \overline{z}$. ($\overline{(w - z)}$ is the same as $\overline{(w + (-z))}$)

It also interacts in simple ways with other operations on $\mathbb{C}$:

  • $|\overline{z}| = |z|$.
  • $\overline{z}\cdot z = |z|^2$.
  • If $z = r\cdot e^{it}$ for $r, t \in \mathbb{R}$, $\overline z = r\cdot e^{-it}$. That is, $\overline z$ is the complex number of same absolute value but opposite argument of $z$.
  • $z + \overline z = 2 \mathrm{Re}(z)$ where $\mathrm{Re}(z)$ is the real part of $z$.
  • $z - \overline{z} = 2i \mathrm{Im}(z)$ where $\mathrm{Im}(z)$ is the imaginary part of $z$.
  • If a complex number $z$ is a root of a polynomial with real coefficients, then so is $\overline z$.

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