Complex numbers

$z=a+bi$

• Addition: $(a+bi)+(c+di)=(a+c)+(b+d)i$

• Subtraction: $(a+bi)-(c+di)=(a-c)+(b-d)i$

• Multiplication: (FOIL) $(a+bi)(c+di)=(ac-bd)+(ad+bc)i$

  • $(a+ib{)}^2=a^2-b^2+i(2ab)$

• Dividing: To simplify the quotient $\frac{a+bi}{c+di}$ multiply the numerator and the denominator by the complex conjugate of the denominator: $\frac{a+bi}{c+di}=\frac{(a+bi)(c-di)}{(c+di)(c-di)}=\frac{(ac+bd)+(bc-ad)i}{c^2+d^2}$

• Inverse $\frac{1}{z}=z^{-1}=\frac{\bar{z}}{|z{|}^2}$

• Absolute value $|z|=\sqrt{a^2+b^2}$

  • $|w||z|=|wz|$
  • $|w+z|\le |w|+|z|$
  • $|z^{-1}|=|z{|}^{-1}$ where ($z\ne 0$)

• Complex conjugate $\overline{z}=a-bi$

  • $\overline{z+w}=\overline{z}+\overline{w}$
  • $\overline{z-w}=\overline{z}-\overline{w}$
  • $\overline{zw}=\overline{z}\cdot \overline{w}$
  • $\overline{\left(\frac{z}{w}\right)}=\frac{\overline{z}}{\overline{w}}$ if $w\ne 0$
  • $0\le z\overline{z}={\left|z\right|}^2\in \mathbb{R}$
  • $|z|=\left|\overline{z}\right|$
  • $z+\overline{z}=2a$
  • $z-\overline{z}=2ib$
  • $z=\overline{z}⟺z\in \mathbb{R}$

• Polar representation $z=r(\cos \theta +i\sin \theta )$

  • $r=\sqrt{a^2+b^2}$
  • $\tan \theta =\frac{b}{a}$ or $\theta =\arctan \left(\frac{b}{a}\right)$
  • $a=r\cos \theta$
  • $b=r\sin \theta$

• De Moivre’s formula - $(\cos x+i\sin x{)}^n=\cos nx+i\sin nx$