Distance between complex numbers

$|z_1-z_2|$ , the ordinary plane distance between the points $z_1$ and $z_2$ .

The distance between two complex numbers $z_1$ and $z_2$ is the modulus of their difference, $|z_1-z_2|$ , which by the Pythagorean theorem equals the ordinary Euclidean distance between the points $(x_1,y_1)$ and $(x_2,y_2)$ they represent. Every geometric shape built from a distance condition in the real plane -- circles, disks, perpendicular bisectors -- carries over to the complex plane unchanged, just rewritten with $|\,\cdot\,|$ in place of a square-root distance formula.

Examples

$|(3+4i)-0|=5$ , the distance from $3+4i$ to the origin.
The equation $|z-2|=|z+2|$ says $z$ is equidistant from $2$ and $-2$ -- the imaginary axis.