Key Points

• 1. Watch the video:

• 2. Read this presentation

Key Points

• 1. Watch the video:

• 2. Read this presentation

Key Points

• 1. Watch the video:

• 2. Read this presentation

Key Points

• 1. For \( z = x + iy \):
  modulus \( r = \sqrt{x^2 + y^2} \),
  argument \( \tan \theta = \frac{y}{x} \).

• 2. Polar form: \( z = r(\cos\theta + i\sin\theta) \).

• 3. Modulus–argument rules:
  \( |zw| = |z||w| \),
  \( \arg(zw) = \arg z + \arg w \);
  \( \left|\frac{z}{w}\right| = \frac{|z|}{|w|} \),
  \( \arg\!\left(\frac{z}{w}\right) = \arg z - \arg w \).

• 4. Euler form:
  \( z = re^{i\theta} \),
  \( e^{i\theta} = \cos\theta + i\sin\theta \),
  \( z^* = re^{-i\theta} \).

• 5. A useful short-cut when working with complex conjugate roots is
  
  \[ (x - z)(x - z^*) = x^2 - 2\operatorname{Re}(z)\,x + |z|^2. \]

• 6. De Moivre’s theorem:
  \( (r(\cos\theta + i\sin\theta))^n = r^n(\cos n\theta + i\sin n\theta) \), \( n \in \mathbb{Z} \).

• 7. You should be able to use De Moivre’s theorem to find roots of complex numbers.
  More generally, the solutions of \( z^n = w \) form a regular \( n \)-gon with vertices on a circle of radius \( |z| \) centred at the origin.

• 8. De Moivre identities:
  if \( z = \cos\theta + i\sin\theta \), then
  \( z^n + z^{-n} = 2\cos(n\theta) \),
  \( z^n - z^{-n} = 2i\sin(n\theta) \).