Geometric Meaning of Complex Number Multiplication: Practice and Visualization

This example demonstrates the geometric behavior of complex multiplication on the complex plane: represent complex numbers as vectors (directed segments) and visualize how the product’s direction and magnitude are determined by the factors. The key concepts are complex numbers in the complex plane, modulus (absolute value), argument (phase/angle), and the polar form rule: for z1 and z2, the product z1·z2 has modulus |z1||z2| and argument arg(z1)+arg(z2), corresponding to a rotation plus a scaling. In GeoGebra, the construction typically involves creating complex-number objects (or equivalent point/vector representations), plotting them on the complex plane, and linking algebraic multiplication with geometric measurements such as vector length and angle. The main construction focus is to display the factors z1 and z2, compute z1z2, and show—via angle and magnitude cues—how the product emerges from combining the two inputs. Since no GeoGebra command “construction summary” was provided, the description is inferred from the standard workflow implied by the topic: complex arithmetic, complex-plane rendering (points/vectors), and dynamic visualization of modulus/argument for interpretation. Suitable for late high school through early university, especially for teaching complex multiplication with geometric insight and for practice-based learning.