Transformations of the complex plane

A transformation of the complex plane is a rule that takes each complex number $z$ and assigns it to another complex number $w$. Common transformations include translations, rotations, dilations (scaling), and reflections. These can be described using algebraic formulas involving $z$ and sometimes its conjugate $\overline{z}$.

Important properties

• Translations: $w = z + a$ shifts every point by the complex number $a$.

• Rotations: $w = e^{i\theta}z$ rotates every point around the origin by angle $\theta$.

• Dilations: $w = kz$ (with $k > 0$ real) scales distances from the origin by $k$.

• Reflections: $w = \overline{z}$ reflects points across the real axis.

• Combinations: Transformations can be combined (e.g., rotate then translate).