Notes on Complex Numbers - Polar Form

Complex Numbers in Polar Form

• Imaginary Numbers

  • Form: $a + bi$ (where $a$ is real, $bi$ is imaginary)
  • Example: For $2 - 7i$, identify real part: $2$, imaginary part: $-7i$.

• Graphing Complex Numbers

  • Plotting example:
  • $4 + 3i$
  • $-5 - 2i$

• Finding Distance

  • Distance formula involves calculating the magnitude of complex numbers.
  • Example: Distance of $3 - 4i$ through structured steps.
  • Steps:
  1. Identify values
  2. Plug into the distance formula
  3. Solve

• Polar Form

  • Important for complex number representation.
  • Use $r$ and $heta$ to transform into polar representation.

• Formulas:

  • Polar Form: $r( ext{cos}( heta) + i ext{sin}( heta))$

• Example Conversions:

  • Polar form of $-4 + 4i$:
  1. Solve for $r$
  2. Use $an( heta) = \frac{y}{x}$ to find $heta$
  3. Write in polar form
  • Convert $12( ext{cos}(\frac{ ext{π}}{6}) + i ext{sin}(\frac{ ext{π}}{6}))$
  1. Calculate using unit circle or triangles
  2. Simplify to $x + yi$ and round if necessary
  • For $r = 13$ and $an( heta) = \frac{5}{12}$:
  1. Find $heta$ using inverse tangent
  2. Simplify into rectangular form.