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Polar Form of Complex Numbers

Main definition and formulas:

• Complex numbers can be written in rectangular form z = x + yi, representing the rectangular coordinates of the point.
• They can be also be written in polar form z = re^iθ , representing the polar coordinates of the point.
• r represents the distance from the origin and θ represents the angle it makes with the positive x-axis.
• Conversions:
  

  r = √   x2 + y2   θ = arctan y x , if x > 0 (Quadrants I and IV)     π + arctan y x , if x < 0 (Quadrants II and III) x = r cosθ y = r sinθ

• Usually, r ≥ 0, but not necessarily.
• Usually, 0 ≤ θ < 2π , but not necessarily.
• Euler's formula:
  

  eiθ = cosθ + i sinθ

• If z = r₁e^{iθ ₁} and w = r₂e^{iθ ₂} are two complex numbers given in polar form, then we can easily multiply them:
  

  zw = r1eiθ 1r2eiθ 2 = r1r2 ei(θ 1+θ 2)