Learn it
The imaginary unit is i, where i² = −1. A complex number is a + bi, with a real part a and an imaginary part b.
- Add/subtract: combine real parts and imaginary parts.
- Multiply: use FOIL and replace i² with −1.
- √(−16) = 4i.
Complex numbers let us solve every quadratic: x² + 9 = 0 gives x = ±3i.
Worked examples
Example 1
Simplify (3 + 2i)(1 − 4i).
- FOIL: 3 − 12i + 2i − 8i²
- i² = −1, so −8i² = 8
- 3 + 8 − 10i
Answer: 11 − 10i
Example 2
Solve x² + 4x + 13 = 0.
- D = 16 − 52 = −36
- x = (−4 ± 6i)/2
Answer: x = −2 ± 3i
Practice
Try these on paper first, then check your work.
- (5 − i) + (2 + 7i)
- √(−49)
- i³
- (2 + i)(2 − i)
Show the answer key
- 7 + 6i
- 7i
- −i
- 5