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Complex Plane Transformations

Lesson · Intermediate

Algebra: Complex Numbers

z1=r1eiθ1,z2=r2eiθ2 z_1=r_1e^{i\theta_1}, \qquad z_2=r_2e^{i\theta_2}
z1z2=r1r2ei(θ1+θ2),z1z2=r1r2ei(θ1−θ2) 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 √2 and rotates it counterclockwise by π/4.

1+i=2eiπ/41+i=\sqrt2e^{i\pi/4}

Division by 1 + i shortens the length of z by a factor of 1/√2 and rotates it clockwise by π/4.

11+i=12e−iπ/4\frac{1}{1+i}=\frac{1}{\sqrt2}e^{-i\pi/4}
Multiplication and division by 1 plus i on the complex plane

In other words,

z↦wz⟹scale by ∣w∣ and rotate counterclockwise by arg⁡(w) z\mapsto wz \qquad\Longrightarrow\qquad \text{scale by }|w|\text{ and rotate counterclockwise by }\arg(w)
z↦zw⟹scale by 1∣w∣ and rotate clockwise by arg⁡(w) z\mapsto\frac{z}{w} \qquad\Longrightarrow\qquad \text{scale by }\frac1{|w|}\text{ and rotate clockwise by }\arg(w)
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