Complex numbers

Representation

Cartesian Form

$z=a+bi$

• $a$ is the real part of $z$
• $b$ is the imaginary part of $z$

Polar Form

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

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

Exponential Form

$z=r{e}^{i\theta }$

Operations

• 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: $\frac{a+bi}{c+di}=\frac{(a+bi)(c-di)}{(c+di)(c-di)}=\frac{(ac+bd)+(bc-ad)i}{c^2+d^2}$

• (Power of integer) $z^n=r^n\left(\cos n\theta +i\sin n\theta \right)$

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

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

• ${\mathbb{S}}^1=\{z\in \mathbb{C}\mid |z|=1\}=\{e^{i\theta }\mid \theta \in [0,2\pi )\}$ is the unit circle in $\mathbb{C}$

  • Multiplication by a complex number $e^{i\theta }$ is a rotation by $\theta$ radians about the origin

• An $n$th root of unity, where $n\in \mathbb{N}$, is a complex number $z$ such that $z^n=1$

  • The $n$th roots of unity are:
    • $1,\omega ,{\omega }^2,\dots ,{\omega }^{n-1}$ where $\omega =e^{i\frac{2\pi }{n}}$, or equivalently
    • $\{\cos \left(\frac{2\pi k}{n}\right)+i\sin \left(\frac{2\pi k}{n}\right)\mid k\in \{0,1,\dots ,n-1\}\}$
  • An $n$th root of unity $z$ is primitive if $\forall k\in \{1,\dots ,n-1\},z^k\ne 1$
  • If $z$ is a primitive $n$th root of unity, then $\{z^k\mid k\in \{0,1,\dots ,n-1\}\}$ are the $n$th roots of unity
  • $z^k$ is a primitive $n$th root of unity if and only if $\gcd (k,n)=1$