Complex conjugate: Difference between revisions

Latest revision as of 17:40, 28 September 2024

The conjugate of a complex number $z = a + bi$ is $a - bi$ , denoted by $\overline{z}$ . Geometrically, $\overline z$ is the reflection of $z$ across the real axis if both points were plotted in the complex plane.For all polynomials with real coefficients, if a complex number $z$ is a root of the polynomial its conjugate $\overline{z}$ will be a root as well.

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 $\overline z$ .