Complex Number Arithmetic

Beyond basic addition and multiplication, advanced arithmetic involves interpreting operations geometrically as transformations of the plane.

Geometric Transformations

1. Translation

Adding a complex number $w=u+iv$ to $z$ translates the point $z$ by the vector $⟨u,v⟩$.

2. Rotation

Multiplying $z$ by a complex number with modulus 1 rotates the point. Multiplying by $e^{i\theta }=\cos \theta +i\sin \theta$ rotates $z$ counterclockwise by angle $\theta$ about the origin.

3. Spiral Similarity

Multiplying by a general complex number $w=r{e}^{i\theta }$ performs two operations:

1. Dilation (Scaling): Scales distance from origin by factor $r=|w|$.
2. Rotation: Rotates by angle $\theta =\arg (w)$.

This is called a spiral similarity centered at the origin.

Rotating Around a General Point

To rotate point $z$ by angle $\theta$ around a center $c$:

1. Translate center to origin: $z-c$.
2. Rotate: $(z-c)e^{i\theta }$.
3. Translate back: $(z-c)e^{i\theta }+c$.

Practice Problems