Complex Numbers

Basic idea

Numbers of the form $z=a+bi$ with $i^2=-1$ . They unify rotation and scaling via polar form $z=re^{i\theta}$ and make the reals algebraically closed.

$i^2=-1$ , $i^3=-i$ , $i^4=1$
Sum/product: $(a+bi)+(c+di)=(a+c)+(b+d)i$ ; $(a+bi)(c+di)=(ac-bd)+(ad+bc)i$
Conjugate: $\bar z = a - bi$ ; $z\bar z=|z|^2$
Modulus: $|z|=\sqrt{a^2+b^2}$
Argument: $\arg z = \operatorname{atan2}(b,a)$
Polar form: $z = r(\cos\theta+i\sin\theta)=re^{i\theta}$
Euler’s formula: $e^{i\theta}=\cos\theta+i\sin\theta$ ; Euler’s identity: $e^{i\pi}+1=0$
de Moivre: $(\cos\theta+i\sin\theta)^n=\cos(n\theta)+i\sin(n\theta)$
Multiplication in polar: $r_1 e^{i\theta_1}\cdot r_2 e^{i\theta_2}=r_1 r_2 e^{i(\theta_1+\theta_2)}$
$n$ -th roots of unity: $z_k=e^{2\pi i k/n}$ for $k=0,\dots,n-1$