AP Precalculus: Complex Plane Formulas & Properties

1. Introduction & Plotting

• Complex plane: horizontal axis = real (\( \mathrm{Re} \)), vertical axis = imaginary (\( \mathrm{Im} \))
• Plot \( z = a + bi \) as point \( (a, b) \)

2. Addition & Subtraction in the Complex Plane

• \( (a+bi) + (c+di) = (a+c) + (b+d)i \)
• \( (a+bi) - (c+di) = (a-c) + (b-d)i \)
• Vector addition: combine real and imaginary parts separately

3. Complex Conjugate (Graphically)

• Conjugate of \( z = a+bi \) is \( \overline{z} = a-bi \)
• Plotted as mirror image across real axis

4. Absolute Value in the Complex Plane

• \( |z| = \sqrt{a^2 + b^2} \)
• Represents distance from origin \( (0,0) \) to \( (a,b) \)

5. Midpoint Formula

• Midpoint between \( z_1 = a+bi \) and \( z_2 = c+di \) is:
  \( z_m = \frac{z_1 + z_2}{2} = \left( \frac{a+c}{2} \right) + \left( \frac{b+d}{2} \right)i \)

6. Distance Formula

• Distance between \( z_1 = a+bi \) and \( z_2 = c+di \):
  \( d = |z_1 - z_2| = \sqrt{(a-c)^2 + (b-d)^2} \)