AP Precalculus: Complex Numbers Formulas & Properties
1. Definition & Standard Form
- Complex number: \( z = a + bi \), \( a, b \in \mathbb{R} \), \( i = \sqrt{-1} \)
- Real part: \( a \), Imaginary part: \( b \)
2. Addition & Subtraction
- \( (a+bi) \pm (c+di) = (a\pm c) + (b\pm d)i \)
3. Complex Conjugates
- Conjugate: \( \overline{z} = a - bi \)
- \( z\overline{z} = a^2 + b^2 \)
4. Multiplication
- \( (a+bi)(c+di) = (ac-bd) + (ad+bc)i \)
5. Division
- \( \frac{a+bi}{c+di} = \frac{(a+bi)(c-di)}{(c+di)(c-di)} = \frac{ac+bd}{c^2+d^2} + \frac{bc-ad}{c^2+d^2}i \)
6. Absolute Value (Modulus)
- \( |z| = \sqrt{a^2 + b^2} \)
7. Powers of \( i \)
- \( i^1 = i \)
- \( i^2 = -1 \)
- \( i^3 = -i \)
- \( i^4 = 1 \)
- Powers repeat every 4: \( i^n = i^{n \mod 4} \)