Toolkit 120

Complex Plane Transformations

z_1=r_1e^{i\theta_1},\qquad z_2=r_2e^{i\theta_2}

z_1z_2=r_1r_2e^{i(\theta_1+\theta_2)},\qquad \frac{z_1}{z_2}=\frac{r_1}{r_2}e^{i(\theta_1-\theta_2)}

Multiplication by $1+i$ stretches the length of $z$ by $\sqrt2$ and rotates it counterclockwise by $\frac{\pi}{4}$ .

1+i=\sqrt2e^{i\pi/4}

Division by $1+i$ shortens the length of $z$ by a factor of $\frac{1}{\sqrt2}$ and rotates it clockwise by $\frac{\pi}{4}$ .

\frac{1}{1+i}=\frac{1}{\sqrt2}e^{-i\pi/4}