Exercise 427Algebra › Complex numbersHigh schoolAdded 3 Nov 2026

Geometric interpretation of operations

#complex plane#geometric interpretation#translation

Statement

Interpret geometrically the operations $z\mapsto z+z_0$ , $z\mapsto \bar z$ , $z\mapsto \lambda z$ (for $\lambda>0$ real), and $z\mapsto e^{i\theta}z$ .

Hint

Think of complex numbers as points in the plane.

Show solutionHide solution

Solution

$z\mapsto z+z_0$ . Translation by the vector of affix $z_0$ .

$z\mapsto\bar z$ . Reflection across the real axis.

$z\mapsto\lambda z$ (Î»>0). Homothety (scaling) centered at the origin with ratio $\lambda$ .

$z\mapsto e^{i\theta}z$ . Rotation centered at the origin with angle $\theta$ .

Conclusion. Translation, reflection, scaling, and rotation are all realized as simple complex operations.

Geometric interpretation of operations

Statement

Solution

More problems in Complex numbers