What happens when you multiply two complex numbers?
Multiplying two arrows has a beautiful rule: the lengths multiply and the angles add. An arrow of length 2 at 30° times an arrow of length 3 at 60° is an arrow of length 6 at 90°.
Some multiplications are pure turns. Multiplying by i (length 1, angle 90°) turns any arrow by a quarter turn. Multiplying by −1 (length 1, angle 180°) turns it round. And multiplying by any arrow of length 1 at angle φ turns an arrow by φ without changing its length.
You multiply an arrow by i four times in a row. Where does it point? Four quarter turns make a full turn: i × i × i × i = 1, and the arrow is back where it started.
This is how optical components act on light. A thin glass plate delays the light a little: it shifts its phase by some angle φ, which turns its arrow by φ. A mirror flips the wave: it multiplies its arrow by −1. A beam splitter shortens the arrow (only part of the light goes each way) and turns the reflected part.
Every component on the lab's optical table is a multiplication of arrows, and every path through the table is a chain of them. You now have both operations that quantum physics needs: multiplying arrows along each path, and adding the arrows of different paths.
In polar form: (r1 e^iθ1) × (r2 e^iθ2) = r1 r2 e^i(θ1 + θ2).
In parts: (a + bi)(c + di) = (ac − bd) + (ad + bc)i. Check with i × i: (0 + 1i)(0 + 1i) = (0 − 1) + 0i = −1.
A phase plate multiplies by e^iφ, a 50:50 beam splitter by 1/√2 (transmitted) or i/√2 (reflected) in the lab's convention, a mirror by −1.
Multiplying arrows multiplies lengths and adds angles; optical components act on light by multiplying its arrow.