Multiplication: Rotation and Scaling

Multiplying z by w = re^(iα) rotates z by angle α anticlockwise and scales its modulus by r. Multiplying by i (where r=1, α=π/2) is a pure 90° anticlockwise rotation. This geometric view of multiplication is the key to understanding the Coni method.

Algebraically, multiplying two complex numbers uses the distributive law and $i^2 = -1$. Geometrically, the same operation is a rotation and a scaling. The Euler form makes this visible.

The Polar Form of Multiplication

Let $z_1 = r_1 e^{i\theta_1}$ and $z_2 = r_2 e^{i\theta_2}$. Then:

$$z_1 z_2 = r_1 e^{i\theta_1} \cdot r_2 e^{i\theta_2} = r_1 r_2\, e^{i(\theta_1 + \theta_2)}$$

The product has:

• Modulus $r_1 r_2$ — the moduli multiply
• Argument $\theta_1 + \theta_2$ — the arguments add

Geometric Interpretation

Fix $z_1$ and think of multiplication by $z_1$ as a transformation acting on $z_2$:

$$z_2 \xrightarrow{\times z_1} r_1 r_2\, e^{i(\theta_1 + \theta_2)}$$

This transformation does two things simultaneously:

1. Rotates $z_2$ anticlockwise by $\theta_1 = \arg(z_1)$
2. Scales $z_2$ by $r_1 = |z_1|$

When $|z_1| = 1$ (i.e., $z_1$ lies on the unit circle), the transformation is a pure rotation — no change in distance from the origin.

Special Cases

Multiplication by $i$:

$i = e^{i\pi/2}$, so $|i| = 1$ and $\arg(i) = \pi/2$.

$$i \cdot z = e^{i\pi/2} \cdot re^{i\theta} = re^{i(\theta + \pi/2)}$$

Multiplying by $i$ rotates any complex number by $90°$ anticlockwise. This is why $i \cdot (a + ib) = -b + ia$ — the point $(a, b)$ moves to $(-b, a)$, a quarter turn.

Multiplication by $-1$:

$-1 = e^{i\pi}$ rotates by $180°$. A point $z$ maps to the point diametrically opposite through the origin.

Multiplication by $e^{i\alpha}$:

Pure rotation by $\alpha$, any angle. This is the seed of the Coni method — to rotate a segment by $\alpha$, multiply the displacement vector by $e^{i\alpha}$.

Why This Matters

Division is the inverse operation: dividing by $z_1$ rotates by $-\theta_1$ and scales by $1/r_1$. The ratio $(z_3 - z_1)/(z_2 - z_1)$ therefore encodes the angle from segment $z_1 z_2$ to segment $z_1 z_3$ and the ratio of their lengths — which is exactly the Coni method.