Complex Multiplication Geometry

Geometric Interpretation of Complex Multiplication

Multiplication of complex numbers has an elegant geometric interpretation involving rotation and scaling.

Multiplication as Rotation and Scaling

Let \(z_1 = r_1e^{i\theta_1}\) and \(z_2 = r_2e^{i\theta_2}\). Their product is:

This shows:

• The moduli multiply: \(|z_1 z_2| = |z_1| \cdot |z_2|\)
• The arguments add: \(\arg(z_1 z_2) = \arg(z_1) + \arg(z_2)\)

Geometric Meaning

Multiplication by a complex number \(re^{i\theta}\) performs:

1. Scaling by factor \(r\) (stretch or shrink)
2. Rotation by angle \(\theta\) counterclockwise

For example, multiplying by \(i\) rotates by \(\frac{\pi}{2}\):

• \(i \cdot (a + bi) = -b + ai\)

Special Cases

• Multiplying by \(e^{i\theta}\) rotates by \(\theta\) without scaling (\(r = 1\))
• Multiplying by \(-1\) rotates by \(\pi\) (180°)
• Multiplying by \(i\) rotates by \(\frac{\pi}{2}\) (90°)

Complex multiplication unifies scaling and rotation into a single operation in the complex plane.