Complex Number

Description:

• $i^2=-1$ or $\sqrt{i}=-1$
  • Imaginary Number
• Plotted in cartesian form $x+yi$ in Argand diagram
• Calculations of 2d vector can be applied
• $s=t⟹Re(s)=Re(t)$ and $Im(z)=Im(s)$
• Complex root of polynomials are always in conjugate pairs

Cartesian form: $a+bi=r\cos \theta +ir\sin \theta$

Modulus form: $r(\cos \theta +i\sin \theta )$

Exponential form: $r{e}^{i\theta }$

Modulus:

• $r=\mid z\mid =a^2+b^2$
• Argument $\theta$: angle of z to positive real axis $⟹-\pi \le \theta \le \pi$
• $\theta ={\tan }^{-1}(\frac{b}{a})$
• Draw diagram and check which quadrant is the number at based on sign of a and b
• $\theta =\theta \pm \pi \mid \mid \pi -\theta$; use common sense to decide
• give sign for a and b depends on the argument when converting to cartesian form

Complex Conjugate

Complex Root of Unity