Complex Numbers. $ \fcolorbox{white}{yellow}{$i = \sqrt{-1}, \m i^2 = -1$}$.
Sections 1 - 4:

• $ e^{i \theta } = \cos\theta + i \sin\theta $
• 5 ways of representing a complex number.
  $\fcolorbox{white}{yellow}{$ z =(x,y)$}$, $\fcolorbox{white}{yellow}{$ z =x+ i y$}$, Polar coordinate: $\fcolorbox{white}{yellow}{$ (r,\theta) $}$, $\fcolorbox{white}{yellow}{$ z = r(\cos\theta+i \sin\theta) $}$, $\fcolorbox{white}{yellow}{$ z = r e^{i \theta} $}$. $$ r = x^2+y^2, \m \tan \theta = \frac{y}{x} .$$
• Conjugate: changing the sign of imaginary part. $$ \fcolorbox{white}{yellow}{$\bar z = x- i y = r(\cos\theta-i\sin\theta) = r e^{-i\theta}. $} $$
• $$ \fcolorbox{white}{yellow}{$|z|=\sqrt{x^2+y^2} = r$}, \m \fcolorbox{white}{yellow}{Re $ z = x$}, \m \fcolorbox{white}{yellow}{Im $z = y$}, \m \fcolorbox{white}{yellow}{angle of $z = \theta$} $$

5. Complex Algebra:
   1. Elementary operations, solving complex algebraic equations,
   2. $ \fcolorbox{white}{yellow}{$|z| = \sqrt{z \bar z}$}$,
   3. Motion of a particle in a complex plane, $ \fcolorbox{white}{yellow}{computing magnitude: $ v = \bigg| \frac{dz}{dt}\bigg|, \m a = \bigg|\frac{d^2z}{dt^2}\bigg|$}$.