Definition:Complex Conjugate Coordinates

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: $2 x + y = 5$

$2 x + y = 5$

can be expressed in complex conjugate coordinates as:

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

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

$x^2 + y^2 = 36$

can be expressed in complex conjugate coordinates as:

$z \overline z = 36$