high School⏱ 4 min

Complex Multiplication

Rotation and scaling in the plane

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z₁ (r=1.5, 30°)z₂ (r=2.0, 45°)z₁·z₂ (r=3.00, 75°)

z₁

Modulus r₁: 1.5
Argument θ₁: 30°

z₂

Modulus r₂: 2.0
Argument θ₂: 45°
z1⋅z2:∣z∣=1.5⋅2.0=3.00,arg⁡=30°+45°=75°z_1 \cdot z_2: \quad |z| = 1.5 \cdot 2.0 = 3.00, \quad \arg = 30°+45°=75°

Multiplying complex numbers = ROTATE by the sum of angles + SCALE by the product of moduli. Change the angles: moduli multiply, arguments add.