Lesson 38: Complex Numbers (Introduction, Operations)
Complex numbers extend the real number system to include solutions to equations like x² + 1 = 0. A complex number has the form a + bi, where a and b are real numbers and i = √−1.
Addition and Subtraction
To add or subtract complex numbers, combine like terms (real with real, imaginary with imaginary):
- (3 + 2i) + (5 − 4i) = (3+5) + (2−4)i = 8 − 2i
- (7 − 3i) − (2 + 5i) = (7−2) + (−3−5)i = 5 − 8i
Multiplication
Use distributive property (FOIL) and remember that i² = −1:
- (2 + 3i)(1 − 4i) = 2*1 + 2*(−4i) + 3i*1 + 3i*(−4i) = 2 − 8i + 3i −12i² = 2 − 5i + 12 = 14 − 5i
Division
To divide complex numbers, multiply numerator and denominator by the conjugate of the denominator:
- (3 + 2i) / (1 − i) → multiply top and bottom by (1 + i)
- [(3 + 2i)(1 + i)] / [(1 − i)(1 + i)] = (3 + 3i + 2i + 2i²) / (1 − i²) = (3 + 5i − 2) / (1 + 1) = (1 + 5i)/2 = 0.5 + 2.5i
Practice Problems
- Add: (4 + 3i) + (−2 + 5i)
- Subtract: (7 − i) − (3 + 4i)
- Multiply: (1 + 2i)(2 − i)
- Divide: (3 + i) / (1 − 2i)
- Simplify: iⴠ+ 2i³ − i² + 5i