high School⏱ 4 min

Complex Division

Geometry of complex division

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z₁z₂z₁/z₂

z₁ = 3+2i

z₂ = 1+2i

\ racz1z2=\ racz1⋅\arz2∣z2∣2\ rac{z_1}{z_2} = \ rac{z_1 \cdot \ar{z_2}}{|z_2|^2}
=(3+2i)(1−2i)5=1.400+−0.800i= \frac{(3+2i)(1-2i)}{5} = 1.400+-0.800i

Result modulus: 1.612

Result angle: -29.7°

Division = subtract angles and divide moduli. Trick: multiply both numerator and denominator by the conjugate of z₂ – the denominator becomes real.