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Toolkit 97

Basic Complex Numbers

Complex number on the complex plane
Rectangular Form: z=a+bi\text{Rectangular Form: }\quad z=a+bi
Polar Form: z=reiθ\text{Polar Form: }\quad z=re^{i\theta}
r=∣z∣=a2+b2r=|z|=\sqrt{a^2+b^2}
θ=arg⁡z=tan⁡−1(ba)\theta=\arg z=\tan^{-1}\left(\frac{b}{a}\right)
a=rcos⁡θ,b=rsin⁡θa=r\cos\theta,\qquad b=r\sin\theta
(a+bi)+(c+di)=(a+c)+(b+d)i(a+bi)+(c+di)=(a+c)+(b+d)i
(a+bi)2=a2−b2+2abi(a+bi)^2=a^2-b^2+2abi
(a+bi)(c+di)=(ac−bd)+(ad+bc)i(a+bi)(c+di)=(ac-bd)+(ad+bc)i
z1z2=r1r2ei(θ1+θ2)z_1z_2=r_1r_2e^{i(\theta_1+\theta_2)}
z1z2=r1r2ei(θ1−θ2)\frac{z_1}{z_2}=\frac{r_1}{r_2}e^{i(\theta_1-\theta_2)}
(reiθ)n=rneinθ(re^{i\theta})^n=r^ne^{in\theta}
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