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:

\(\ds x\)

\(=\)

\(\ds \dfrac {z + \overline z} 2\)

Sum of Complex Number with Conjugate

\(\ds y\)

\(=\)

\(\ds \dfrac {z - \overline z} {2 i}\)

Difference of Complex Number with Conjugate

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

• 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Complex Conjugate Coordinates