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Multiplying uses i² = −1, giving (ac − bd) + (ad + bc)i. In polar terms the moduli multiply and the arguments add, so multiplication is a rotation combined with a scaling.
Complex multiplication
(a + bi)(c + di) = (ac − bd) + (ad + bc)i
Polar form
|z₁z₂| = |z₁||z₂| and arg(z₁z₂) = arg z₁ + arg z₂
(3 − 8) + (12 + 2)i = −5 + 14i, of modulus √221 ≈ 14.8660687.
Because eⁱᵃ · eⁱᵇ = eⁱ⁽ᵃ⁺ᵇ⁾, which is Euler's formula turning multiplication into rotation.