Geometry of the complex plane

The complex plane is a way to represent complex numbers as points on a flat surface. Each complex number $z = a + bi$ corresponds to the point $(a, b)$, where $a$ is the real part and $b$ is the imaginary part. The geometry of the complex plane lets us visualize operations like addition, subtraction, and multiplication as movements or transformations of points.

Important properties

• Addition of complex numbers corresponds to vector addition in the plane.

• The modulus $|z|$ of a complex number $z = a + bi$ is its distance from the origin: $|z| = \sqrt{a^2 + b^2}$.

• The argument $\arg(z)$ is the angle the line from the origin to $(a, b)$ makes with the positive real axis.

• Multiplying by a real number scales the distance from the origin.

• Multiplying by $i$ rotates a point by 90 degrees counterclockwise.