Definition:Complex Conjugate Coordinates

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Definition

Let $P$ be a point in the complex plane.


$P$ may be located using complex conjugate coordinates $\tuple {z, \overline z}$ based on:

\(\displaystyle x\) \(=\) \(\displaystyle \dfrac {z + \overline z} 2\) $\quad$ Sum of Complex Number with Conjugate $\quad$
\(\displaystyle y\) \(=\) \(\displaystyle \dfrac {z - \overline z} {2 i}\) $\quad$ Difference of Complex Number with Conjugate $\quad$


where $P = \tuple {x, y}$ is expressed in Cartesian coordinates.


Examples

Example: $2 x + y = 5$

The equation of the straight line in the plane:

$2 x + y = 5$

can be expressed in complex conjugate coordinates as:

$\paren {2 i + 1} z + \paren {2 i - 1} \overline z = 10 i$


Example: $x^2 + y^2 = 36$

The equation of the circle:

$x^2 + y^2 = 36$

can be expressed in complex conjugate coordinates as:

$z \overline z = 36$


Sources