Theory of Intelligence
11TH•Mathematics•Complex Numbers•Exercise 1.5

Theory

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Theory

If z=r(cos⁡θ+isin⁡θ)z=r(\cos\theta+i\sin\theta) then Arg⁡(z)=θ\operatorname{Arg}(z)=\theta (up to adding multiples of 2π2\pi).

  • Product:
Arg⁡(z1z2)=Arg⁡(z1)+Arg⁡(z2) (mod 2π)\operatorname{Arg}(z_1z_2)=\operatorname{Arg}(z_1)+\operatorname{Arg}(z_2)\ (\text{mod }2\pi)
  • Quotient:
Arg⁡(z1z2)=Arg⁡(z1)−Arg⁡(z2) (mod 2π)\operatorname{Arg}\left(\frac{z_1}{z_2}\right)=\operatorname{Arg}(z_1)-\operatorname{Arg}(z_2)\ (\text{mod }2\pi)

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