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Complex conjugate

From AoPSWiki

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.

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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